The modified version of the Schrödinger equation that describes deep water waves is known as the Nonlinear Schrödinger Equation (NLS). In fluid dynamics, this equation captures the evolution of complex wave envelopes in deep water and is particularly useful in modeling phenomena like rogue waves, solitons, and wave packet modulation. The deep-water version of the NLS is given by:

where:

The NLS effectively captures how wave packets in deep water undergo both dispersion (due to (\alpha)) and nonlinearity (due to (\beta)), creating a balance that can lead to stable structures like solitons or unstable ones like rogue waves. This version of the equation is essential in understanding how energy distributes within wave packets in deep water.

The rate of energy loss via radiation from a body can be calculated using the Stefan-Boltzmann law, which is applied to thermal radiation emitted by an object. Here's the formula:

P=σ⋅A⋅ε⋅T4

where:

• ( P ) is the power radiated (energy per unit time, in watts).
• ( \sigma ) is the Stefan-Boltzmann constant, approximately ( 5.67 \times 10^{-8} , \text{W} , \text{m}^{-2} , \text{K}^{-4} ).
• ( A ) is the surface area of the radiating body in square meters.
• ( \varepsilon ) is the emissivity of the surface (a value between 0 and 1, where 1 represents a perfect blackbody).
• ( T ) is the absolute temperature of the body in Kelvin.

Steps:

1. Determine Surface Area (A): Calculate or measure the surface area of the object.
2. Determine Temperature (T): Measure the temperature of the object in Kelvin.
3. Determine Emissivity (ε): Use the material’s emissivity; typical values can be found in tables based on material type.
4. Calculate Power Radiated (P): Substitute values into the Stefan-Boltzmann formula.

This gives the rate at which energy is radiated. If you want the total energy radiated over a certain time ( t ), multiply ( P ) by ( t ):

[ E = P \cdot t ]

For complex cases (e.g., non-uniform temperature), you might integrate over the surface with varying ( T ) values.

You're right; Apollo 13 was indeed rotated at intervals to prevent one side from overheating while the other side froze. This maneuver, known as a "Passive Thermal Control" (PTC) or "barbecue roll," was crucial for evenly distributing solar heating around the spacecraft, avoiding excessive temperature gradients that could damage the craft.

This rolling motion would have periodically changed the areas exposed to the Sun, slightly adjusting the effective radiating and absorbing surfaces over time, which helped to manage the temperature distribution without active heating systems. It’s a great example of how crucial thermal management is in space, especially in emergencies.

So How Could They Remain in Shade?

 For Apollo 13 to remain in the shade as much as possible while balancing temperature, they would rely on the "barbecue roll" to rotate the craft at intervals, presenting each part to sunlight briefly. This maneuver minimizes prolonged exposure of any single area to direct sunlight while ensuring all sides receive some heating, keeping extreme temperatures in check.

The challenge of maintaining shade involves:

1. Slow and Controlled Rotation: By rotating every 30 degrees at intervals, the spacecraft's orientation toward the Sun changes frequently. This balances out the heating, reducing hotspots while still preventing prolonged exposure to space's intense cold.

2. Optimal Sun Angle Management: By controlling the angle of rotation, Apollo 13's crew ensured the craft stayed in a position where direct sunlight was only glancing off portions of the surface. This setup let them balance temperature while preventing excessive heating that could occur if one side faced the Sun continuously.

3. Minimal Surface Area in Direct Sunlight: Since they maintained intervals in the craft's rotation, no side stayed in full sunlight for too long. This method allowed indirect, periodic warming without full exposure, effectively keeping much of the craft in shade relative to intense solar heating.

This delicate balance between sun exposure and shade was crucial to prevent freezing in shaded areas and overheating in sunlit areas, all without active temperature control systems.

Let's recalculate just for the LEM

Let's focus on the Lunar Module (LEM) of Apollo 13 alone. We’ll estimate its energy loss through radiation and its solar energy absorption, taking into account the rotating maneuvers.

Revised Parameters for the LEM

1. Surface Area (A): The LEM’s surface area was smaller than the combined modules. We can approximate it at around ( 20 , \text{m}^2 ).

2. Temperature (T): The outer surface temperature during the Apollo 13 crisis could still be around ( 250 , \text{K} ).

3. Emissivity (ε): Similar to before, we’ll assume ( \varepsilon = 0.5 ) for its metallic surface.

4. Absorptive Side (A_sunlit): Since we’ll consider only half of the LEM facing the Sun, we’ll use ( 10 , \text{m}^2 ) for the sunlit side.

5. Absorptivity (α): We’ll keep ( \alpha = 0.3 ).

Let’s recalculate the radiated and absorbed power for the LEM under these conditions.

For the Lunar Module (LEM) of Apollo 13:

• Power radiated from the shaded side: approximately ( 1107 , \text{W} ).
• Power absorbed from the Sun on the sunlit side: approximately ( 4083 , \text{W} ).

This results in a net energy gain of about ( 2976 , \text{W} ). Despite this net gain, the rotating maneuver (barbecue roll) was still essential for managing the internal temperatures, as it distributed this absorbed energy across the LEM more evenly, preventing one side from overheating and the other from freezing.

One side was a 3Kw heater and they were cold?

 Yes, the paradox of a "3 kW heater" on one side, yet experiencing intense cold, underscores the unique thermal challenges in space. Despite the LEM absorbing significant energy from sunlight, this energy didn't translate effectively into warmth for the crew inside due to several factors:

1. Vacuum of Space: Space lacks an atmosphere to conduct or convect heat, so the only way for heat to reach the shaded and interior areas was through conduction within the spacecraft materials. Unfortunately, the LEM’s outer insulation was designed to prevent heat transfer, both in and out, which reduced the transfer of warmth to the interior.

2. Thermal Isolation: The materials and design of the spacecraft prioritized insulating the interior from both extreme heat and cold. This was effective under normal mission conditions, where internal systems provided heating. However, with Apollo 13’s power systems mostly shut down, the insulation worked against the crew by trapping the cold inside.

3. Uneven Heating with Sunlight Intervals: The "barbecue roll" allowed the craft to avoid overheating by distributing sunlight exposure across surfaces. While it prevented damage, it also limited how much solar energy could contribute to internal warmth, as no surface was in the Sun long enough to conductively transfer significant heat inward.

4. Insufficient Internal Power: Normally, electrical systems and equipment in the Command Module generated heat, balancing out any external cold. With those systems offline, there was minimal internal heat generation, making the crew reliant on residual warmth and passive solar exposure.

In essence, while the sunlit side absorbed solar energy, the spacecraft's design for thermal isolation and the lack of active heating systems resulted in much of that energy remaining outside, leaving the crew to endure the freezing cold.

Maurice Ravel was not only a brilliant composer but also a masterful arranger, known for his intricate and highly detailed orchestrations. His ability to capture color, texture, and emotion through orchestration has earned him a reputation as one of the great orchestrators in classical music.

One of the most famous examples of Ravel's arranging prowess is his orchestration of Modest Mussorgsky's "Pictures at an Exhibition". Originally composed for solo piano, Ravel transformed it into a vibrant and richly colored orchestral suite in 1922. His version is now arguably the most widely performed and recorded rendition of the piece. In this arrangement, Ravel showcases his ability to use the orchestra to convey a wide range of moods and atmospheres, using unique combinations of instruments to highlight specific moments. For example, the use of brass and percussion in "The Great Gate of Kiev" creates a grand and triumphant sound, while the delicate string and woodwind work in "The Old Castle" evokes a haunting, mysterious mood.

Ravel was also highly skilled at orchestrating his own piano works. His "Alborada del gracioso" (originally for piano) and the "Mother Goose Suite" are perfect examples of how he expanded the textures and colors in his compositions when arranging them for orchestra, often using subtle but ingenious touches to bring out new details.

In addition to orchestrating his own works and the works of others, Ravel often emphasized clarity in his arrangements, ensuring that each instrument’s voice could be heard distinctly, which gave his orchestrations a crystalline quality.

Ravel's meticulous approach to orchestration has made his arrangements some of the finest in the classical canon. His ability to bring out emotional nuance and sonic beauty through the orchestra is central to his legacy.

A TV satellite in geostationary orbit typically generates the power for transmission, rather than simply reflecting signals. Here's how that works:

1. Satellite Transmission and Power Generation:

• TV satellites actively receive signals from Earth (from ground-based stations), process them, and then transmit those signals back to Earth (to individual TV receivers, such as satellite dishes). This process requires power to amplify and retransmit the signals over long distances.
• The power needed for this comes from solar panels on the satellite, which generate electricity from solar energy. The satellite’s onboard systems, including its transmitters and communication payloads, use this electricity to operate.

2. Active vs. Passive Satellites:

• Active Satellites: Most modern TV communication satellites are active satellites. This means they not only relay signals but also amplify the signals they receive from Earth before sending them back down to receivers. The onboard electronics require power to function, which is why the satellite needs to generate its own electricity via solar panels.
• Passive Satellites: These types of satellites, which were more common in the early days of space exploration, would simply reflect signals without amplifying or processing them. An example is Echo 1, an early communication satellite launched in 1960, which was essentially a large metallic balloon that reflected radio signals. However, this type of satellite is not commonly used today for TV or communications.

3. Solar Panels as Power Source:

• Solar panels are mounted on most modern communication satellites, including TV satellites. These panels capture sunlight and convert it into electricity, which powers the satellite’s internal systems, including the transponders that amplify and transmit signals.
• The electricity generated by the solar panels is stored in batteries to ensure continuous operation during periods when the satellite passes through the Earth’s shadow and is not exposed to sunlight.

4. How TV Signals Are Processed:

• Uplink: A ground station sends a signal (uplink) to the satellite.
• Amplification and Processing: The satellite’s transponder receives this signal, amplifies it, and changes its frequency to avoid interference with the uplink signal.
• Downlink: The amplified signal is then transmitted back to Earth (downlink) to reach individual receivers, like TV satellite dishes.

Summary:

TV satellites generate their own power using solar panels to operate their systems, including the transmitters that broadcast signals back to Earth. They do not merely reflect signals; instead, they actively receive, process, amplify, and transmit signals, requiring a constant power source to function.

Solar Panels Give it the Output Power So it can Run in the Shade Too?

Yes, you're correct in the general concept! Satellites, including TV satellites, rely on solar panels to generate power, and they need to produce enough power to operate continuously, even when they are in the Earth's shadow (referred to as the eclipse period). Here’s how this works in more detail:

1. Solar Panels and Continuous Power:

• In geostationary orbit, a satellite spends the majority of its time in direct sunlight, but for part of the year (typically around the equinoxes), it will pass through the Earth’s shadow for about 45 to 72 minutes each day. During this time, the satellite's solar panels can’t generate power because they are not exposed to sunlight.
• To ensure continuous operation, the satellite’s solar panels must generate more power than is immediately needed while in sunlight. This extra power is used to charge the onboard batteries during the sunlight phase, which then provide power when the satellite is in the shadow.

2. Battery Backup for Eclipse Periods:

• Satellites are equipped with rechargeable batteries (typically lithium-ion or nickel-hydrogen batteries) that store energy when the solar panels are exposed to sunlight. These batteries are designed to provide enough power to run the satellite during periods of darkness.
• The solar panels are usually designed to generate about twice the required operational power to ensure that:
  • The satellite can operate normally while in sunlight.
  • There is enough excess power to charge the batteries for operation during the shadow periods.

3. Solar Panel Output and Design Considerations:

• The solar panels on satellites are sized and engineered to account for the eclipse periods. This means they are intentionally built to generate excess power when in sunlight, ensuring there’s a surplus available to store in the batteries.
• For example, if a satellite needs 5 kW of power to operate, the solar panels might be designed to generate 10 kW during sunlight. This way, it can use half the generated power for immediate operation and the other half to charge the batteries.

4. Eclipse Periods:

• A geostationary satellite generally encounters eclipse periods around the spring and fall equinoxes, when the Earth directly blocks sunlight to the satellite for a short time each day. These eclipse periods last for up to 72 minutes per orbit.
• For most of the satellite’s orbit, it’s outside the Earth's shadow and exposed to continuous sunlight, but the battery capacity must be sufficient to keep the satellite running during those daily eclipse phases.

5. Power Balance and Efficiency:

• The satellite’s power management system ensures that battery charging and power consumption are balanced. During the sunlight phase, the excess power generated by the solar panels is diverted to the batteries, ensuring that they are fully charged by the time the satellite enters the eclipse.
• The system also has to account for the degradation of solar panels over time (due to radiation exposure in space), meaning that the initial power generation capacity may be higher to account for gradual efficiency loss.

Summary:

Yes, solar panels on satellites are designed to provide twice the required power during sunlight to ensure continuous operation during the eclipse periods when the satellite is in the Earth's shadow. The extra power is stored in onboard batteries that provide energy when the satellite can't rely on solar panels. This design ensures that the satellite can operate 24/7, even when sunlight isn’t available for part of the orbit.

What Frequencies Do They Transmit At?

TV satellites typically transmit signals in specific frequency bands within the microwave portion of the electromagnetic spectrum. These frequency bands are chosen to balance signal quality, power efficiency, and atmospheric interference. The two most common frequency bands used for TV satellite transmissions are the C-band and the Ku-band. In some cases, the Ka-band is also used for newer services, including broadband satellite communication.

Here’s a breakdown of the main frequency bands used by TV satellites:

1. C-Band:

• Uplink Frequency (Earth to satellite): 5.925 to 6.425 GHz
• Downlink Frequency (Satellite to Earth): 3.7 to 4.2 GHz
• Bandwidth: Typically 500 MHz
• Usage:
  • C-band is widely used for satellite TV broadcasts, especially in regions with heavy rain or atmospheric interference because these frequencies are less affected by rain fade (signal loss due to rain).
  • C-band is also more commonly used for large satellite dishes (often 2–3 meters in diameter) and was the primary frequency band for early satellite TV services.

2. Ku-Band:

• Uplink Frequency (Earth to satellite): 14.0 to 14.5 GHz
• Downlink Frequency (Satellite to Earth): 10.7 to 12.75 GHz
• Bandwidth: Typically 1,000 MHz (varies by region and service)
• Usage:
  • Ku-band is more commonly used for direct-to-home (DTH) satellite TV services, including popular services like DirecTV, Dish Network, and Sky.
  • Ku-band supports smaller dishes (60–90 cm) because the shorter wavelength allows for more focused signals, making it suitable for consumer-grade satellite dishes.
  • However, Ku-band frequencies are more susceptible to rain fade, meaning signal quality can degrade during heavy rain or storms.

3. Ka-Band:

• Uplink Frequency (Earth to satellite): 27.5 to 31.0 GHz
• Downlink Frequency (Satellite to Earth): 17.7 to 21.2 GHz
• Bandwidth: Typically 3,500 MHz
• Usage:
  • The Ka-band is used in some modern satellite TV systems and for high-speed broadband services via satellite. It's becoming increasingly popular for internet connectivity and video broadcasting, as it allows for higher data rates.
  • Ka-band offers more bandwidth and higher frequency, which enables greater data transmission capacity but is more susceptible to atmospheric interference than lower frequency bands.
  • Some newer TV and satellite internet services are shifting to Ka-band due to the increased capacity and the potential for higher-quality video transmission, including HD and 4K broadcasting.

4. L-Band (Less Common for TV):

• Uplink Frequency (Earth to satellite): 1.6 to 1.7 GHz
• Downlink Frequency (Satellite to Earth): 1.5 to 1.6 GHz
• Usage:
  • The L-band is more commonly used for mobile satellite services (like GPS and satellite phones) rather than direct-to-home TV. However, in some cases, it can be used for satellite TV transmission in narrowband services (though this is relatively rare).

5. S-Band (Less Common for TV):

• Uplink Frequency (Earth to satellite): 2.025 to 2.12 GHz
• Downlink Frequency (Satellite to Earth): 2.2 to 2.3 GHz
• Usage:
  • S-band is not typically used for TV broadcasting but is more commonly found in mobile satellite communications or some specialized satellite radio and broadcasting systems (such as SiriusXM).

Summary of Common Bands for TV Satellites:

1. C-Band (3.7 to 4.2 GHz downlink, 5.925 to 6.425 GHz uplink) - Used for robust satellite TV services, large dishes, less affected by weather.
2. Ku-Band (10.7 to 12.75 GHz downlink, 14.0 to 14.5 GHz uplink) - Widely used for direct-to-home TV services, smaller dishes, susceptible to rain fade.
3. Ka-Band (17.7 to 21.2 GHz downlink, 27.5 to 31.0 GHz uplink) - Used for high-speed broadband and newer satellite TV services, higher data rates, more atmospheric interference.

Why Different Frequencies Are Used:

• C-band provides better resilience to weather conditions but requires larger satellite dishes, while Ku-band allows for smaller consumer dishes, which is more practical for home users but can experience rain fade during bad weather. Ka-band offers the potential for higher bandwidth and data rates, making it ideal for modern high-definition broadcasts and satellite internet services, though it's more prone to weather-related signal degradation.

Each of these frequency bands is carefully selected based on the needs of the service (e.g., coverage area, bandwidth, weather resilience, and signal quality).

How Could Solar Panels Generate all that Power?

Solar panels on TV satellites and other communications satellites generate the required power through efficient design, advanced solar cell technology, and by optimizing the use of space on the satellite for power generation. Let’s explore how solar panels can generate enough power to support the satellite’s communications, transmission, and other operations, even in the harsh environment of space:

1. Solar Energy Availability in Space:

• In geostationary orbit, a satellite is exposed to direct sunlight for most of the year, except during brief eclipse periods when the Earth blocks the sunlight (typically twice a year, around the equinoxes).
• In space, without atmospheric interference, the amount of solar energy received is about 1,360 watts per square meter (the solar constant), much higher than what is available on Earth due to the lack of atmospheric absorption and scattering.
• This constant exposure to intense sunlight makes solar power a reliable energy source for satellites.

2. Efficiency of Solar Cells:

• Satellites use high-efficiency solar cells that are far more advanced than typical commercial-grade solar panels used on Earth. The solar cells on satellites are usually made of gallium arsenide (GaAs) or multi-junction solar cells.
• Multi-junction solar cells are designed to capture a wider spectrum of sunlight. These cells are made up of multiple layers of materials, each of which is tuned to capture different wavelengths of solar energy, making them much more efficient than standard single-junction silicon cells.
• These high-efficiency cells can convert around 30% to 35% of the available solar energy into electricity, compared to about 15% to 20% for typical Earth-based silicon solar panels.

3. Size of the Solar Array:

• To meet the satellite’s power demands, the solar panels are often large arrays that deploy once the satellite is in orbit. These panels are folded up during launch and then unfolded in space to maximize their surface area exposed to the Sun.
• The total surface area of the solar panels on a typical geostationary satellite can be quite large, often several tens of square meters (20-40 m² or more), allowing the satellite to capture a significant amount of solar energy.
• The size and surface area of the solar panels are calculated based on the power requirements of the satellite and the efficiency of the solar cells.

4. Power Requirements of the Satellite:

• TV satellites typically require several kilowatts (kW) of power to operate. Older satellites might require 2 to 5 kW, while newer, more advanced satellites with more transponders and features might need 10 to 15 kW or more.
• The power generated by the solar panels must be enough to power all of the satellite’s systems, including:
  • Communication transponders for uplink and downlink.
  • Signal processing and transmission equipment.
  • Attitude control systems (to maintain correct orientation).
  • Thermal control systems to regulate the satellite’s temperature.
  • Battery charging for operation during eclipse periods (when the satellite is in Earth’s shadow).

5. Energy Storage for Eclipse Periods:

• During eclipse periods, when the satellite passes through Earth’s shadow and its solar panels are not exposed to the Sun, the satellite relies on batteries to maintain continuous operation.
• Satellites use high-capacity rechargeable batteries, typically lithium-ion or nickel-hydrogen batteries. These batteries are charged during the periods when the satellite is in sunlight and can store enough energy to power the satellite through the eclipse periods, which last for 45 to 72 minutes per orbit, twice a year.
• The solar array is designed to generate excess power during sunlight periods, not only to meet the satellite’s operational needs but also to charge the batteries.

6. Power Management Systems:

• The satellite’s power management system ensures that the energy generated by the solar panels is used efficiently. It regulates the distribution of power between the satellite’s systems and manages the charging and discharging of the batteries.
• Redundancy is also built into the system to ensure continuous operation even if part of the solar array fails or becomes less efficient over time.

7. Durability and Longevity:

• Satellite solar panels are designed to operate in the harsh environment of space for 15 to 20 years or more. Over time, exposure to radiation in space can degrade the efficiency of the solar cells, so satellites are typically built with extra capacity in their solar arrays to compensate for this gradual decline.
• Solar cells and panels are engineered to withstand thermal cycling (extreme temperature changes between sunlight and shadow), cosmic radiation, and micrometeoroid impacts.

Example of Power Generation on a Satellite:

A modern geostationary satellite might be equipped with a solar array that can generate 10 to 15 kW of electrical power, which is typically enough to power its communication systems, onboard computers, attitude control, and other subsystems. If the satellite requires 10 kW of power to operate, the solar panels may generate twice that amount during sunlight (20 kW) to allow for battery charging and to account for efficiency losses over time.

Summary of Key Points:

• High-efficiency solar cells (like multi-junction GaAs cells) convert about 30% to 35% of solar energy into electricity.
• Large solar arrays (often 20-40 m² in size) are deployed to maximize the satellite’s exposure to sunlight.
• Excess power generated during sunlight periods is stored in rechargeable batteries for use during eclipse periods.
• Satellites require several kilowatts of power (5-15 kW), which is efficiently managed by the satellite’s power management system.

Through advanced solar technology, large solar arrays, and efficient power management, TV satellites are able to generate and store enough power to support their communication functions and remain operational for decades in orbit.

A rebreather is a type of breathing apparatus that allows the user to reuse some or all of the exhaled breath by removing carbon dioxide (CO₂) and replenishing oxygen (O₂). This makes it different from traditional open-circuit breathing systems, where the exhaled air is expelled into the environment. Here's how it works step by step:

1. Inhalation: The user breathes in from the breathing loop, which contains a mix of gases, usually oxygen or an oxygen-nitrogen mix.

2. Exhalation: The exhaled air contains unused oxygen, nitrogen, and carbon dioxide. Instead of being expelled, it is directed into the rebreather’s circuit.

3. CO₂ Scrubbing: The exhaled gas passes through a chemical scrubber, usually containing a material like soda lime. The scrubber absorbs the carbon dioxide, preventing it from building up to toxic levels in the breathing loop.

4. Oxygen Replenishment: As the user consumes oxygen during respiration, the oxygen levels in the breathing loop drop. The rebreather has a mechanism (manual or automatic) to add oxygen to maintain the proper levels. Some advanced rebreathers can add oxygen based on the depth of the dive to keep the oxygen partial pressure safe.

5. Recycled Air: After scrubbing the CO₂ and replenishing the oxygen, the cleaned and oxygen-enriched gas is fed back into the breathing loop, ready for the user to inhale again.

Advantages of Rebreathers:

• Efficiency: They allow longer dive times since the oxygen is not wasted with each breath.
• Stealth: They are quieter and produce fewer bubbles than open-circuit systems, making them useful for military, technical, or wildlife diving.
• Compactness: Because gas is reused, the cylinders carried can be smaller, making the setup more lightweight.

Types of Rebreathers:

• Oxygen rebreathers: Used in shallow diving, they add pure oxygen.
• Semi-closed circuit rebreathers (SCR): These add a gas mix based on the user's consumption but vent some of the gas to maintain the balance.
• Closed-circuit rebreathers (CCR): These completely recycle exhaled gases and manage oxygen levels precisely, making them suitable for deep and technical diving.

Safety is crucial, as improper oxygen or CO₂ management can be dangerous in rebreathers.

Do astronauts use a CCR?

Yes, astronauts use a form of closed-circuit rebreather (CCR) technology in their life support systems, although these systems are more advanced and specifically designed for space environments. The Extravehicular Mobility Unit (EMU), which is the spacesuit worn by astronauts during spacewalks (extravehicular activities or EVAs), uses a closed-loop life support system to maintain breathable air. Here's how it relates to a CCR and the specifics for astronauts:

How the Life Support System Works in Space:

1. Closed-Circuit Rebreather Functionality: Similar to a CCR, the spacesuit’s life support system recycles exhaled air. The astronaut breathes oxygen from the suit's tanks, and when exhaling, the system collects the exhaled gas, removes carbon dioxide (CO₂), and replenishes oxygen as needed.

2. CO₂ Scrubbing: Like in diving rebreathers, CO₂ must be removed from the breathing loop. In spacesuits, this is done using lithium hydroxide (LiOH) canisters, which absorb CO₂. The scrubbed air, now free of CO₂, is recirculated back into the breathing loop.

3. Oxygen Replenishment: The suit continuously monitors oxygen levels. As the astronaut consumes oxygen during respiration, the system releases more from onboard tanks to keep oxygen levels optimal for breathing.

4. Cooling and Humidity Control: The EMU also has to manage other environmental factors that aren’t an issue underwater. The suit’s life support system removes excess moisture (humidity) from the exhaled breath and body perspiration. It also has a cooling system, usually liquid-cooled garments, to prevent overheating in the vacuum of space.

5. Pressure Control: Unlike underwater CCRs, which balance gas pressures to avoid issues related to depth, the spacesuit maintains a constant internal pressure, since astronauts operate in a near-vacuum environment where maintaining suit integrity is critical to prevent decompression and protect the astronaut’s body.

Why a CCR-Type System is Used in Space:

• Oxygen Conservation: In space, resources are limited. A closed-circuit system minimizes the need for large oxygen reserves by recycling and reusing the air the astronaut breathes.
• CO₂ Removal: Just like in diving, the accumulation of CO₂ in a closed environment can quickly become toxic. Effective scrubbing is essential for astronaut safety.
• Suit Mobility: Space suits need to be compact and efficient to allow for maximum mobility during spacewalks. A closed-loop system means less oxygen needs to be stored, reducing the suit's bulk and weight.

While the basic principles of gas recycling and CO₂ scrubbing are shared between a CCR used by divers and an astronaut’s life support system, the engineering of space suits is far more complex, addressing the unique challenges of operating in a zero-gravity vacuum.

What were the specifics of the system used by the Apollo crews?

The Apollo astronauts used a sophisticated life support system both in their spacecraft and within their spacesuits during extravehicular activities (EVAs). Here's a breakdown of the specifics of the system used by the Apollo crews, especially in relation to the Portable Life Support System (PLSS) used during lunar surface operations:

Portable Life Support System (PLSS) for EVAs

The Apollo astronauts used a closed-loop system similar to modern closed-circuit rebreathers (CCR), but adapted for the unique challenges of space exploration. Here's a detailed look at how it worked:

1. Oxygen Supply:

• Primary and Backup Tanks: The PLSS carried two oxygen tanks—a primary tank that provided a continuous flow of oxygen to the astronaut's suit and a secondary tank as a backup in case of emergencies.
• Constant Flow: The system delivered a constant flow of oxygen, which the astronaut breathed in. Oxygen was provided at a rate of about 0.84 to 1.2 kilograms per hour.

2. CO₂ Scrubbing:

• Lithium Hydroxide (LiOH) Canisters: The exhaled air, rich in carbon dioxide (CO₂), was passed through filters containing lithium hydroxide (LiOH) to chemically absorb the CO₂. This ensured that dangerous levels of CO₂ did not build up inside the suit.
• Scrubbing Efficiency: The LiOH canisters were highly efficient, capable of scrubbing CO₂ for up to 7 hours during a lunar EVA, with spare canisters available for replacement.

3. Cooling System:

• Liquid-Cooled Garment (LCG): To prevent overheating, astronauts wore a liquid-cooled garment under their spacesuits. The LCG was a network of tubes that circulated water cooled by a sublimator.
• Sublimator Cooling System: The sublimator used the vacuum of space to boil away water into vapor (sublimation), removing excess heat from the circulating water. This heat exchange helped maintain a stable body temperature during physically demanding EVAs on the lunar surface.

4. Humidity and Moisture Control:

• The life support system also removed moisture from the exhaled air. Humidity was a concern in the closed environment of the spacesuit, as too much moisture could lead to fogging of the helmet visor and affect the astronaut’s comfort. Excess moisture was condensed out of the air and managed by the suit’s system.

5. Suit Pressure and Structural Integrity:

• The spacesuit maintained a pressure of around 3.7 psi (25.5 kPa), which was a partial pressure of oxygen, providing enough to breathe without risking decompression sickness. The suit had to maintain this pressure while allowing enough flexibility for the astronaut to move on the Moon's surface.

6. Waste Management:

• The Apollo astronauts had to deal with waste products while wearing the suit for long periods. Urine was collected in a special device called the Urine Collection Transfer Assembly (UCTA), which channeled liquid waste away from the body into a storage unit in the suit.

7. Communication and Power:

• The PLSS also housed the communication equipment, with the system relaying voice communications back to the Lunar Module (LM) and the mission control on Earth. It was powered by batteries that provided enough energy for the life support and communication systems during lunar walks.

Environmental Control System in the Command Module (CM):

When inside the Command Module (CM), the life support system was designed to handle the unique conditions of space, maintaining a controlled cabin environment.

1. Oxygen Supply:

• The Command Module had oxygen tanks that supplied the astronauts with breathable air. It operated on a closed-loop system where oxygen levels were maintained by onboard controls.

2. CO₂ Scrubbing:

• Lithium Hydroxide (LiOH) Canisters: As in the PLSS, the Command Module used LiOH canisters to scrub CO₂ from the air. After the Apollo 13 incident, where an oxygen tank explosion compromised the Command Module, the crew had to adapt the LiOH canisters from the Lunar Module to scrub CO₂ from the air while they awaited rescue.

3. Temperature and Humidity Control:

• The CM had an environmental control system that maintained comfortable temperature and humidity levels. Excess heat generated inside the spacecraft by both the astronauts and onboard equipment was managed by a combination of radiators and heat exchangers.

Lunar Module (LM) Life Support:

The Lunar Module (LM), which took astronauts to the Moon's surface, also had its own life support system.

• Oxygen Supply and CO₂ Scrubbing: The LM had its own oxygen supply and CO₂ scrubbing system similar to the Command Module, relying on LiOH canisters to remove CO₂.
• Water and Power: The LM carried water for drinking and batteries to power its systems, but was designed for short-duration missions, meaning life support systems only needed to last for a few days.

Emergency Systems:

The Apollo spacesuits and modules had built-in redundancy. In case of failure, astronauts had emergency oxygen supplies and backup CO₂ scrubbers to ensure survival.

Key Takeaways:

• Efficient CO₂ Removal: Lithium hydroxide canisters were the primary method for CO₂ scrubbing both in the PLSS during EVAs and in the spacecraft.
• Closed-Loop Oxygen System: The astronauts used a highly efficient closed-loop system that recycled air, replenished oxygen, and removed CO₂.
• Cooling System in Vacuum: The PLSS used a sublimator system to remove excess heat, leveraging the vacuum of space to dissipate heat through water sublimation.

This system allowed Apollo astronauts to survive in the vacuum of space and the extreme conditions on the lunar surface for extended periods during their moonwalks.

How did sublimation work through the outer spacesuit from the internal tubing?

The sublimation cooling system in the Apollo astronauts’ Portable Life Support System (PLSS) was a crucial mechanism for controlling body temperature during extravehicular activities (EVAs) on the Moon. The system relied on the principle of sublimation, which occurs when a substance transitions directly from a solid to a gas, bypassing the liquid phase. Here's a breakdown of how this process worked through the layers of the spacesuit:

Sublimator in the Cooling System

1. Liquid-Cooled Garment (LCG):

   • The astronaut wore a Liquid-Cooled Garment (LCG) under their spacesuit, which consisted of a network of narrow plastic tubes woven into a fabric that circulated water over the astronaut's skin. This garment absorbed excess body heat during physical activity or exposure to the Sun on the lunar surface.
   • The water was cooled via a sublimation process, which was carried out in the PLSS.

2. Heat Transfer to Water:

   • The heated water, having absorbed heat from the astronaut’s body, was pumped through the tubes and into a heat exchanger in the PLSS.

3. Sublimator Unit:

   • The heart of the cooling system was the sublimator, a device located within the PLSS. This sublimator was filled with a porous metal plate or porous material, which held ice formed from water introduced into the system.
   • The ice was kept in a special compartment within the sublimator where it could be exposed to the vacuum of space.

4. Sublimation Process:

   • Once in the vacuum of space, the water ice in the sublimator was exposed to an environment with extremely low pressure. In this vacuum, ice does not melt into liquid water; instead, it sublimates directly into water vapor.
   • This phase change from solid ice to water vapor required heat energy. The heat was drawn from the circulating water that had absorbed the astronaut's body heat. As the ice sublimated, the heat was removed from the water, cooling it down.

5. Venting Water Vapor:

   • The resulting water vapor, formed from sublimation, was vented directly into space. The continuous sublimation process provided a steady removal of heat as long as ice and water were supplied to the system.

6. Cooling the Circulating Water:

   • As the water vapor was vented out, the remaining water in the heat exchanger was cooled, and this cooler water was then circulated back through the tubes in the LCG. The astronaut's body heat was once again absorbed by the circulating water, and the cycle repeated.

Key Points of the Sublimation Process:

• Ice as a Heat Sink: The sublimator acted as a heat sink by using the astronaut's body heat to transform solid ice into vapor. The cooling effect was direct, and no mechanical pumps or refrigeration units were necessary to expel heat, making it efficient in the vacuum of space.

• Porous Plate Design: The porous structure in the sublimator ensured that the water formed ice evenly across the plate's surface, maximizing the exposure of ice to the vacuum and making the sublimation process highly efficient.

• Vacuum of Space: The lack of atmospheric pressure in space enabled sublimation, as water ice directly transitions to vapor in such conditions, which wouldn't be possible in Earth’s atmosphere.

• Continuous Operation: As long as the PLSS had a supply of water for the sublimator and the environment remained a vacuum, the system could continue cooling effectively. The system worked by maintaining a balance between body heat removal and the sublimation of ice.

Spacesuit Layer Interaction:

While the sublimator was located in the PLSS backpack and not directly integrated into the outer layers of the spacesuit, the cooling was transferred to the astronaut through the water circulating within the tubes of the Liquid-Cooled Garment (LCG). The cooling water was isolated from the external environment by the multiple layers of the spacesuit, which provided thermal insulation and protection. These outer layers did not interfere with the sublimation process happening in the PLSS backpack because the vented water vapor exited through designated ports in the PLSS.

In summary, the sublimation cooling system removed heat by using the vacuum of space to sublimate ice, cooling water that was then circulated through the astronaut’s Liquid-Cooled Garment. The process was efficient and vital for keeping astronauts comfortable during EVAs in the harsh lunar environment.

Ice Sublimated from the Backpack/PLSS?

Yes, that's correct! The ice that sublimated during the cooling process was located inside the Portable Life Support System (PLSS), which was the backpack worn by the Apollo astronauts during extravehicular activities (EVAs) on the Moon. Here's a clearer explanation of how the sublimation process worked within the PLSS:

Sublimation Process in the PLSS:

1. Ice Formation in the PLSS Sublimator:

   • The PLSS had a dedicated sublimator unit, which was essentially a heat exchanger that used ice to cool the astronaut’s body via a Liquid-Cooled Garment (LCG).
   • Water from the astronaut’s suit cooling system was fed into the sublimator, where a portion of it was frozen into ice on a porous plate.

2. Exposing Ice to the Vacuum:

   • This ice, located within the PLSS sublimator, was then exposed to the vacuum of space via a venting mechanism. In the vacuum, the ice could not melt into water due to the extremely low pressure. Instead, it sublimated directly into water vapor.
   • Sublimation is the phase transition where a solid (in this case, ice) changes directly into a gas (water vapor) without passing through the liquid phase. This phase change absorbed heat from the system.

3. Cooling the Circulating Water:

   • The sublimation of ice absorbed the heat from the water circulating in the system. This cooled the water, which was then pumped back through the tubes of the Liquid-Cooled Garment worn by the astronaut to regulate body temperature.

4. Venting Water Vapor into Space:

   • The water vapor generated by the sublimation process was vented out directly into space through a small exhaust port in the PLSS. This allowed the system to continually remove heat from the astronaut’s body as long as ice was available.

Why Ice Was in the PLSS:

The reason the ice was located in the PLSS (and not directly in the suit) is that the PLSS housed all of the astronaut’s life support equipment, including the oxygen supply, CO₂ scrubbing system, communication systems, and cooling systems. The sublimator was an efficient way to remove excess heat in the vacuum of space by harnessing the properties of sublimation.

Since the Moon has no atmosphere, the vacuum environment around the astronaut allowed the ice to sublimate directly, creating an effective method of cooling without requiring complex mechanical refrigeration systems. The heat from the astronaut’s body was carried by the water in the LCG to the sublimator, where it was released as water vapor into space, thus cooling the astronaut.

In summary:

• The sublimator unit in the astronaut’s PLSS froze water into ice, which was exposed to the vacuum of space.
• The ice in the PLSS sublimated (turned directly into vapor) when exposed to the vacuum, absorbing heat from the circulating water.
• The water vapor was vented out into space, while the cooled water was recirculated through the astronaut’s cooling garment to regulate body temperature.

This process allowed astronauts to stay cool while performing physically demanding tasks on the Moon, where extreme temperature fluctuations occur.

How Could They Still Have Ice After Four Days in Direct Solar Radiation?

Great question! It seems surprising that the Apollo astronauts could still have ice in the Portable Life Support System (PLSS) after days in the intense solar radiation on the Moon. However, several factors allowed the PLSS sublimator to maintain ice and perform effective cooling, despite these extreme conditions. Here's how it worked:

1. Water, Not Pre-Frozen Ice:

• The astronauts did not start with ice in the PLSS; instead, they carried liquid water in a tank inside the PLSS.
• The sublimator created ice on-demand by freezing a small portion of this water in a porous plate inside the sublimator, which was constantly exposed to the vacuum of space.
• Since the water was not exposed directly to the Sun and was stored in an insulated tank, it remained in liquid form. Ice was formed only when needed during the cooling process.

2. Vacuum of Space and Sublimator Design:

• The vacuum of space around the astronauts played a crucial role. In the vacuum, water can sublimate (turn directly from ice into vapor) without first melting. The sublimator harnessed this principle.
• The sublimator had a porous metal plate where water from the tank was slowly fed. As the water entered the sublimator, the extreme cold of space caused the water to freeze immediately on the plate.
• This ice was then exposed to the vacuum of space, causing it to sublimate (turn directly into water vapor). This phase change absorbed heat from the water circulating in the astronaut’s Liquid-Cooled Garment (LCG), cooling it.

3. Thermal Control of the PLSS:

• The PLSS and the spacesuit had multiple layers of thermal insulation that protected the astronaut and the equipment from extreme temperatures on the lunar surface.
• The outer layer of the spacesuit was designed to reflect solar radiation using a highly reflective surface made from aluminized Mylar and other insulating materials. This helped prevent excessive heating of both the astronaut and the equipment, including the water reservoir inside the PLSS.

4. The Moon’s Day-Night Cycle:

• Although the Moon experiences extreme temperature changes (ranging from about +127°C (260°F) in direct sunlight to -173°C (-280°F) in shadow), each Apollo mission was carefully timed to take place during the lunar day, when the Sun was lower in the sky.
• The astronauts were exposed to less intense solar radiation because they were conducting operations during the early lunar morning or late afternoon, avoiding the hottest part of the day.

5. Efficiency of the Sublimation Process:

• The sublimator process was extremely efficient in cooling the circulating water by sublimating small amounts of ice, only when heat needed to be removed.
• The water supply in the PLSS was adequate for the duration of the lunar EVAs (which lasted about 7 hours per EVA). The cooling system was designed to maintain enough ice through sublimation to keep the astronaut cool without the need for large amounts of pre-stored ice.

6. Duration of the Moonwalks:

• The Apollo astronauts were not continuously exposed to the Sun for four straight days while on the lunar surface. Each lunar EVA (Extravehicular Activity) lasted around 4 to 7 hours, after which they would return to the Lunar Module (LM), which had its own life support and thermal management systems.
• Between EVAs, the astronauts were inside the Lunar Module, where they had access to a separate thermal control system and could replenish their resources.

Conclusion:

• The astronauts carried liquid water in their PLSS, which was insulated and protected from direct solar radiation by the suit’s reflective and insulating layers.
• The sublimator inside the PLSS created ice on-demand from this water and sublimated it in the vacuum of space, cooling the astronauts through the heat exchange process.
• The thermal control measures of the PLSS and careful timing of the missions ensured that ice could be continuously generated, even after days of exposure to sunlight.

In essence, the design of the PLSS and the sublimator allowed for the creation and sublimation of ice only when needed, rather than relying on pre-stored ice, making the cooling system highly efficient and well-suited for the extreme environment of the lunar surface.

I Meant Four Days in Translunar Space

Ah, I see! You're asking how the Apollo astronauts managed to have liquid water for the sublimator system after spending several days in translunar space, where they were exposed to direct solar radiation without the Moon's shadow. This is a great point, and the answer lies in the thermal management systems of both the Command Module (CM) and the Portable Life Support System (PLSS). Let's break down how they dealt with the extreme temperatures in deep space:

1. Thermal Management in Translunar Space

In space, especially during the journey from Earth to the Moon (translunar space), there is no atmosphere to regulate temperature. The spacecraft and life support systems were exposed to extreme conditions—solar radiation on one side and the cold of space on the other. To maintain a suitable environment for both the astronauts and their equipment, including the water for the PLSS, the Apollo spacecraft used several strategies:

Active Thermal Control in the Command Module (CM):

• During the multi-day journey to the Moon, the astronauts were inside the Command Module (CM), where the water for the PLSS was stored.
• The Command Module had a heat exchanger system that regulated the temperature of the spacecraft's interior and prevented any single component from overheating or freezing.
• The CM used a combination of radiators and a liquid cooling system to dump excess heat from the spacecraft into space. This system kept the spacecraft’s interior within a comfortable temperature range (around 18–24°C or 65–75°F), ensuring that the water in the PLSS remained in liquid form during transit.

The "Barbecue Roll" Maneuver:

• One of the most well-known techniques used during the Apollo missions was the Passive Thermal Control (PTC) maneuver, commonly referred to as the "barbecue roll."
• This maneuver involved rotating the entire spacecraft along its long axis slowly and continuously (about 1 revolution per hour). This slow roll ensured that the spacecraft's surface was evenly heated by the Sun, rather than one side baking in the sunlight and the other side freezing in deep space.
• By rotating the spacecraft, the Apollo engineers prevented excessive heating or cooling on any part of the spacecraft, maintaining stable internal temperatures for the astronauts, the equipment, and the life support systems.

2. Water Storage and Insulation in the PLSS

• The water tank for the PLSS was stored in the Command Module during the journey to the Moon and was transferred to the PLSS only when the astronauts prepared for their extravehicular activities (EVAs) on the lunar surface.
• During transit, the water tanks were insulated to protect the contents from external temperature fluctuations. Even though space has extreme temperatures, the insulation and thermal control measures in the spacecraft prevented the water from freezing or boiling.

3. Solar Radiation Protection:

• The Command Module’s outer shell was equipped with thermal blankets and insulation layers that reflected solar radiation and minimized heat absorption from the Sun.
• These materials protected the spacecraft from excessive heating due to direct sunlight, allowing the internal systems (including the water supply) to remain within safe temperature ranges.

4. Water Cooling in the PLSS After Lunar Orbit Insertion:

• Once the astronauts were in lunar orbit and preparing for their lunar EVA, the water for the PLSS sublimator was transferred to the PLSS backpack.
• By that point, they were closer to the Moon, which presented different thermal conditions compared to the translunar phase. However, the design of the PLSS ensured that once the water was inside the system, it remained stable, only sublimating small amounts of ice as needed during the EVA.

Key Points:

• Thermal control systems inside the Command Module, such as heat exchangers, radiators, and the rotating "barbecue roll" maneuver, prevented extreme temperatures from affecting the astronauts and their equipment during the trip to the Moon.
• Water storage tanks inside the Command Module were well insulated and maintained at safe temperatures for the duration of the mission.
• Upon reaching the Moon, the water was transferred to the PLSS backpack, where it was used to generate ice and cool the astronauts via the sublimation process during EVAs.

Conclusion:

The thermal management systems aboard the Apollo spacecraft were specifically designed to handle the extreme environment of space. By carefully regulating the spacecraft's internal temperature through a combination of insulation, active cooling, and passive maneuvers like the "barbecue roll," the Apollo missions ensured that essential resources, such as the liquid water needed for the PLSS sublimator, remained in a usable state throughout the multi-day journey to the Moon.

Space isn't Cold, It's Just a Vacuum

You're absolutely right! Space itself isn’t "cold" in the way we typically think of it; it’s a vacuum, which means it doesn't conduct heat because there are no molecules to transfer thermal energy. Instead, the way objects in space gain or lose heat is primarily through radiation, not through conduction or convection as on Earth.

Here’s a breakdown of what this means and how it affects temperature management in space, especially during Apollo missions:

1. How Heat Works in a Vacuum:

• In a vacuum like space, objects can only exchange heat through radiation, meaning they emit infrared energy to lose heat or absorb it if exposed to sunlight.
• Radiative heat transfer is the only way for an object to cool down or heat up. If an object is in the shade, it cools down by radiating its heat into space. If it's in direct sunlight, it absorbs the Sun's energy and heats up.

2. Solar Radiation in Space:

• Without an atmosphere to filter or block sunlight, solar radiation is extremely intense. Objects in direct sunlight can get very hot, while objects in the shade or facing away from the Sun can cool to very low temperatures because they radiate heat away into the vacuum of space.
• For instance, objects in direct sunlight in space can reach temperatures over 120°C (248°F), while objects in shadow can plummet to temperatures as low as -150°C (-238°F) or even lower, depending on their environment and distance from a heat source.

3. Thermal Regulation in Spacecraft:

• To maintain livable temperatures in spacecraft like the Apollo Command Module (CM) or to keep equipment (like the PLSS water supply) stable, thermal regulation systems are critical.
• These systems manage both the intense solar heating and the cooling effects of radiating heat into space. They use techniques like:
  • Insulation: Thermal blankets and multi-layer insulation (MLI) reflect solar radiation and prevent excessive heat from being absorbed.
  • Radiators: Heat from inside the spacecraft is actively pumped to radiators, which emit it as infrared radiation into space.
  • Thermal control coatings: Special coatings on the spacecraft’s surface reflect solar radiation or emit heat efficiently to maintain a stable temperature.

4. Thermal Control for Astronauts and Equipment:

• In translunar space, as you mentioned, the Apollo astronauts were exposed to direct solar radiation for days. However, the Command Module’s heat shields and insulation layers protected the internal environment, keeping it within a comfortable temperature range. This was essential for the astronauts and the water in the PLSS.
• The "barbecue roll" maneuver helped spread the heat evenly around the spacecraft, preventing one side from overheating and the other from getting too cold due to excessive radiative cooling.

5. Why Water Didn't Freeze or Boil:

• Water in the PLSS or the Command Module didn’t freeze or boil in space because of the combination of thermal insulation and the spacecraft’s internal temperature control systems. Although space is a vacuum and water would behave unpredictably if exposed directly, the Apollo spacecraft maintained a controlled environment to prevent this from happening.
• Thermal management systems ensured that the interior of the spacecraft (including the water tanks) stayed in a temperature range that allowed water to remain in its liquid state, despite the intense solar radiation outside.

In Summary:

• Space isn’t "cold" in the traditional sense; it’s a vacuum, meaning there’s no matter to conduct heat.
• In space, temperature is managed by radiation, with objects heating up in sunlight and cooling down when radiating heat into the vacuum.
• The Apollo missions used thermal insulation, active heat management systems, and maneuvers like the "barbecue roll" to protect both the astronauts and the equipment from extreme temperature variations.
• The water in the PLSS remained in liquid form because it was stored in a carefully regulated environment, protected from both excessive heating and cooling.

So, while space itself doesn't "freeze" objects, managing heat in space is all about balancing radiation absorption (like from the Sun) and heat emission (through radiators or insulation).

But it's a Closed System Though, Radiation Out=Radiation In

You're touching on a key principle of thermal equilibrium, but the situation in space isn't quite as simple as radiation out always equaling radiation in, especially in the context of spacecraft and life support systems like those used on the Apollo missions. Let’s dive deeper into the nuances of thermal management in a closed system in space:

1. Closed System vs. Open System:

• Closed system: In the context of the Apollo spacecraft, the Command Module (CM) or Portable Life Support System (PLSS) was indeed a closed system in terms of mass (i.e., no matter or gases entering or leaving). However, energy transfer still occurred between the spacecraft or suit and the external environment, primarily through radiation.
• Radiation as the Only Transfer Mechanism: In space, since there's no air or matter for conduction or convection, the spacecraft can only gain or lose heat by radiating energy to or absorbing energy from the environment (like solar radiation).

2. Radiation In ≠ Radiation Out (Active Management):

• Solar Radiation Input: The spacecraft or astronaut's suit would naturally absorb solar radiation if exposed to direct sunlight. This absorption increases the internal energy (temperature) of the system.

• Radiation Output: The spacecraft or astronaut’s suit also radiates infrared energy out into space to balance the energy input. The problem is that without active thermal control, the amount of radiation out may not match the amount of radiation in.

  • If radiation in > radiation out, the system heats up.
  • If radiation out > radiation in, the system cools down.

3. Thermal Management in Space:

• In space, achieving thermal equilibrium (where radiation out = radiation in) is not automatic. Without intervention, one side of the spacecraft could become extremely hot from sunlight exposure while the other side, in shadow, could become extremely cold from radiating heat away into space.

• This is where active and passive thermal control systems come into play:

  • Reflective Insulation: Materials like multi-layer insulation (MLI) on spacecraft reflect much of the incoming solar radiation, reducing the amount of heat that is absorbed.
  • Radiators: Active radiators emit excess heat from the interior of the spacecraft into space. This helps prevent the spacecraft from overheating by increasing radiation out.
  • Barbecue Roll: As mentioned earlier, the slow roll of the Apollo spacecraft ensured that all sides were evenly heated and cooled, avoiding localized hot or cold spots. This maneuver distributed the solar input and radiative losses evenly, helping to balance radiation out and in.

4. Apollo Thermal Management:

• Inside the Command Module (CM) or Lunar Module (LM) during the journey to the Moon, thermal control systems maintained a stable internal environment. The crew compartment was kept at a comfortable temperature by regulating heat.
• This was not simply a matter of letting radiation out equal radiation in. The spacecraft actively controlled how much heat was retained or radiated using heat exchangers, thermal blankets, and radiators.

5. Why Apollo Wasn't a Perfectly Closed Radiative System:

• Controlled Heat Rejection: While the Apollo spacecraft and suits were "closed" in terms of keeping in air and other life support components, they were not a perfectly closed radiative system because heat rejection was carefully controlled. The goal was not to let natural radiative equilibrium occur passively, but to manage the balance between heat gain and heat loss.
• The sublimator in the PLSS used sublimation (converting ice into water vapor) as an additional method to remove heat, providing active cooling rather than relying solely on radiative cooling.
• Excess heat from equipment and the astronauts' bodies was transferred via cooling systems (water or glycol loops) to the radiators or sublimators, which then released that heat into space.

6. Radiation in Translunar Space:

• During the translunar phase, the spacecraft received heat primarily from the Sun. Without the Earth or Moon's atmosphere to filter out solar radiation, the spacecraft was exposed to significant energy.
• The Apollo Command Module's thermal systems had to reject this heat to maintain a livable environment. This involved reflective surfaces (to minimize heat absorption) and radiative panels (to increase heat loss) to ensure that radiation out was balanced with the incoming energy.
• If the spacecraft was left in one position, facing the Sun for too long, one side could overheat. The "barbecue roll" distributed this exposure more evenly, preventing localized temperature extremes.

7. Astronauts' PLSS (Portable Life Support System) during Lunar EVA:

• The PLSS wasn't just relying on radiative cooling to maintain temperature equilibrium. The system had an active cooling mechanism via the sublimator to ensure the astronauts didn’t overheat during EVAs.
• The sublimator vented water vapor into space, actively cooling the water that circulated through the Liquid-Cooled Garment (LCG) worn by the astronauts. This allowed the astronauts to control their temperature in a very dynamic environment, where both solar radiation and body heat could cause rapid overheating if not managed.

Conclusion:

While the Apollo spacecraft and suits were closed systems in terms of mass, they were not simple "radiative equilibrium" systems. They used active thermal control to balance heat gain and loss, with reflective surfaces, radiators, and even maneuvers like the "barbecue roll" to avoid overheating or excessive cooling.

In short, radiation out ≠ radiation in naturally, but the Apollo systems were carefully designed to manage this balance and maintain stable, safe temperatures for both astronauts and equipment.

In a Vacuum you can Perfectly Reflect Radiation Without Gaining Energy?

In a vacuum, perfect reflection of radiation without gaining energy into a system is not possible in practical terms. While highly reflective materials can significantly reduce the absorption of radiation, there is no material that can reflect 100% of all incoming radiation across all wavelengths. Let’s explore this idea in more detail:

1. Reflection vs. Absorption in a Vacuum:

• Reflection in the vacuum of space works similarly to reflection in any other environment. If an object is covered in a highly reflective material (like the thermal blankets used on spacecraft), it can reflect a large portion of the incoming radiation, particularly in the visible and infrared spectra.
• However, even highly reflective materials still absorb some fraction of the radiation. The reflected radiation helps reduce the net energy gain, but it is not a perfect shield. Some amount of energy is always absorbed, and this is where thermal management systems become important.

2. Perfect Reflectors Do Not Exist:

• Perfect reflection of radiation, meaning 100% reflection without any absorption, is not achievable with current materials. Even highly reflective materials, such as aluminized Mylar used in spacecraft, are not perfect reflectors.
• These materials can reflect most of the incident radiation (up to 95-99% depending on the wavelength and material), but they still absorb a small portion of it. This absorbed energy causes the object to heat up over time unless there’s a mechanism to remove the excess heat.

3. Energy Balance in Space:

• In space, the balance between the energy coming into a system (through solar radiation) and the energy leaving the system (through thermal radiation emitted by the object) determines the temperature of the object.
• Even with reflective surfaces, some amount of radiation will be absorbed, and that energy will cause the system to gain heat unless it is actively managed.
• The concept of thermal equilibrium comes into play: if an object absorbs more energy than it radiates, it heats up; if it radiates more energy than it absorbs, it cools down. Reflective materials help shift this balance by reducing the absorption of solar energy, but they don’t eliminate it.

4. Albedo and Emissivity:

• Reflective materials work by having a high albedo, meaning they reflect a large fraction of the incoming radiation. However, they still have some degree of emissivity, which measures how well an object emits thermal radiation.
• Even objects with a low emissivity (like polished metals) radiate energy, which means they can lose heat to space. In the vacuum of space, this radiation is the only mechanism through which an object can cool down.
• The albedo-emissivity ratio plays a significant role in determining how a material behaves thermally in space. The goal in spacecraft design is to balance these factors so that the system doesn't gain or lose too much heat.

5. Thermal Management on Spacecraft:

• Spacecraft use highly reflective surfaces to minimize the amount of solar radiation absorbed. Materials like gold, silver, or aluminized Mylar are commonly used because they reflect much of the Sun’s energy, particularly in the visible and infrared wavelengths.
• However, spacecraft also need to radiate heat away, which is why they have radiators that emit infrared radiation to prevent overheating.
• The balance between reflecting incoming solar energy and radiating away excess heat is essential to maintain a stable temperature inside the spacecraft.

6. Practical Example – Apollo Spacecraft:

• The Apollo Command Module and the Portable Life Support System (PLSS) used reflective materials to reduce solar heating. However, they were not immune to heat gain. That’s why the Apollo spacecraft employed active thermal control systems, such as radiators and heat exchangers, to maintain a balanced temperature.
• The barbecue roll maneuver was also used to expose the spacecraft to solar radiation evenly, ensuring that no side absorbed too much energy or radiated too much heat.

7. Active Heat Management is Necessary:

• Since no material is a perfect reflector, and some energy is always absorbed, active thermal management systems are critical in space.
• These systems ensure that any absorbed heat is removed, either by radiating it into space (through radiators) or, in the case of spacesuits, by sublimating water (in the PLSS) to carry heat away.

Conclusion:

In space, while highly reflective surfaces can significantly reduce the amount of solar radiation absorbed by a spacecraft or spacesuit, they cannot completely eliminate heat gain. Even in a vacuum, some radiation is absorbed, and this will lead to a net energy gain in the system unless active cooling methods (such as radiators or sublimators) are used.

In short, perfect reflection without energy absorption isn’t achievable in practice. Therefore, spacecraft rely on a combination of reflective materials and active thermal control systems to manage heat in the vacuum of space.

Why Were they So Cold in Apollo 13?

The extreme cold experienced by the crew of Apollo 13 was primarily due to the loss of electrical power, which crippled the spacecraft's ability to actively manage its thermal environment. Here’s a detailed breakdown of why the Apollo 13 astronauts faced such cold temperatures:

1. Loss of Power and Life Support:

• The Apollo 13 accident occurred when an oxygen tank exploded in the Service Module (SM), which contained the primary power and life support systems for the spacecraft. This explosion disabled the fuel cells that generated electricity by combining hydrogen and oxygen.
• With the fuel cells offline, the Command Module (CM) lost most of its power. The heaters and active thermal management systems were powered by these fuel cells, and without electricity, the systems couldn't function properly.

2. No Active Thermal Control:

• Normally, the Environmental Control System (ECS) in the Command Module and the Lunar Module (LM) would regulate the internal temperature of the spacecraft using radiators and active heat exchangers. These systems helped balance heat gain and heat loss, maintaining a livable temperature for the crew.
• After the explosion, these systems could not operate at full capacity due to limited power, so the spacecraft was unable to reject excess heat effectively or maintain an internal temperature.

3. Shutdown of Command Module:

• To conserve the limited battery power remaining in the Command Module, the crew was forced to shut down all non-essential systems, including those responsible for maintaining temperature.
• The astronauts moved into the Lunar Module (LM), which was designed to be a temporary habitat for use on the Moon, not for prolonged travel in space. The LM was not equipped to provide long-term thermal control for the extended period they had to remain in it.

4. The Moon’s Shadow and Cold of Space:

• The spacecraft had to navigate back to Earth after the accident, passing through the cold vacuum of space. The Lunar Module’s Passive Thermal Control (PTC) system, which relied on radiation to manage heat, was not sufficient to keep the crew warm under these conditions.
• Space is a vacuum, and while it doesn't have a temperature in the traditional sense, objects in space radiate their heat away into the vacuum. Without sufficient active heating, the Lunar Module and the Command Module became cold as they radiated more heat than they absorbed.

5. Lack of Solar Heating:

• Although the spacecraft was exposed to some sunlight, they were no longer using the barbecue roll maneuver, which was designed to distribute solar radiation evenly across the surface of the spacecraft. This led to uneven heating and allowed the spacecraft to lose more heat in shadowed areas.
• The astronauts didn’t receive enough solar radiation to offset the heat they were losing through radiation into space, especially after power was lost and active heating was no longer functioning.

6. Cold from Radiative Cooling:

• Without power to run the heaters or maintain environmental controls, the temperature inside the spacecraft dropped significantly. The spacecraft was designed to lose heat through radiative cooling into the vacuum of space, and without the ability to add heat back in, it continued cooling down.
• Temperatures inside the Lunar Module dropped to near-freezing levels, creating an extremely uncomfortable and challenging environment for the crew.

7. Water and Humidity Issues:

• As the temperature dropped, condensation began to form on the interior walls of the spacecraft. This was because the colder temperatures caused the water vapor in the air to condense, leading to high humidity levels.
• The cold, combined with the condensation, posed risks of electrical short circuits if power systems were reactivated. It also contributed to the discomfort of the crew, who were cold, wet, and trying to conserve energy.

8. Lack of Waste Heat:

• Normally, the systems onboard the spacecraft (especially the electrical systems) would generate waste heat that helped maintain a comfortable internal temperature. With most systems shut down, this source of heat was no longer available.
• The astronauts themselves were generating some body heat, but it was insufficient to counteract the heat loss from the spacecraft radiating heat into space.

Summary of Why They Were So Cold:

• The explosion of the oxygen tank caused a loss of electrical power, which disabled the heaters and thermal control systems that were necessary to maintain a livable temperature in the spacecraft.
• The crew was forced to shut down non-essential systems to conserve power, which meant the active thermal control system was also shut down.
• Without these systems, the spacecraft radiated heat into space and was unable to replace the lost heat, leading to extreme cold.
• The Lunar Module was not designed for long-term occupancy or thermal management in deep space, contributing to the cold environment.

In conclusion, the extreme cold during Apollo 13 was caused by the loss of power and the resulting inability to actively manage the spacecraft’s internal temperature. This left the astronauts at the mercy of radiative heat loss into the vacuum of space.

That Doesn't Really Seem to Fit with Our Discussion Up Until Now

You're right—it seems like the previous answer didn’t fully align with the concepts we were discussing regarding thermal regulation and radiation in space. Let's try to reconcile the details of the Apollo 13 situation with the broader principles of thermal management in space.

Revisiting Key Concepts:

In our earlier discussion, we talked about how radiation is the primary mechanism for heat transfer in space, where there’s no air for conduction or convection. Objects in space gain or lose heat primarily through the absorption and emission of radiation. A spacecraft, even in the vacuum of space, can manage its internal temperature using reflective materials, radiators, and active control systems to ensure that the heat balance is maintained. When those systems fail or are disrupted, temperature regulation becomes a serious challenge.

Why Apollo 13 Got So Cold:

Given that, let's take a closer look at Apollo 13 and why it became so cold, while keeping in mind the principles we’ve discussed.

1. Loss of Power → Loss of Active Thermal Control:

   • Active thermal control is crucial in space because you have to manage heat flows carefully. For Apollo 13, once the oxygen tank exploded, the crew had to shut down the Command Module (CM) to conserve battery power for re-entry. The active thermal systems, which normally would balance the heat generated internally and manage heat loss to space, were no longer operational.
   • Without these systems, the spacecraft was unable to control its internal temperature. It couldn’t regulate heat loss, so it began to radiate more heat into space than it could retain, causing the internal temperature to drop.

2. Heat Loss by Radiation:

   • Even though space itself is a vacuum and doesn’t "conduct" cold, objects in space naturally radiate heat away into the vacuum. Normally, the Apollo spacecraft would radiate excess heat through radiators, which were designed to manage this balance.
   • With the spacecraft in a low-power state and without active thermal regulation, it radiated heat out into space faster than it could replenish heat. Passive radiation in space can cause rapid cooling, especially when systems that generate heat (like electronics) are turned off or operating at low power.

3. The Role of the "Barbecue Roll":

   • The "barbecue roll" maneuver (slowly rotating the spacecraft) helped maintain thermal balance by evenly distributing the heat from the Sun around the spacecraft. During normal operations, this roll helped prevent one side from becoming too hot and the other from becoming too cold.
   • After the explosion, the maneuver was likely disrupted or abandoned, meaning that parts of the spacecraft were exposed to the coldness of space (radiative heat loss) for prolonged periods. This uneven exposure could lead to a faster rate of cooling in certain areas, compounding the overall drop in temperature.

4. No Heat Generation From Equipment:

   • Under normal conditions, spacecraft equipment, like computers and other electronics, generate heat as they operate. This heat contributes to maintaining the spacecraft’s internal temperature. However, once the Apollo 13 crew shut down non-essential systems to conserve power, there was very little waste heat being generated.
   • This lack of internal heat generation meant that the spacecraft had less heat to "lose" through radiation, which accelerated the cooling process.

5. Conservation Mode in the Lunar Module:

   • The astronauts moved into the Lunar Module (LM), which wasn’t designed for prolonged occupancy in deep space. The LM had a much smaller life support system, and it wasn’t equipped to handle the long-term thermal regulation needed for this emergency situation.
   • The Lunar Module’s systems were designed to work during lunar surface operations, where different thermal challenges exist (such as exposure to sunlight on the Moon or lunar surface cooling). In translunar space, with low power and no active thermal control, the LM wasn’t able to provide sufficient heat.

6. Limited Solar Heating:

   • Although the Sun was still present during parts of Apollo 13's journey back to Earth, the crew was not able to manage their orientation as carefully as they would under normal circumstances. They couldn’t optimize their exposure to the Sun for passive heating.
   • Without the barbecue roll or careful orientation, parts of the spacecraft were not receiving as much sunlight, meaning the crew couldn’t take advantage of solar heating.

So Why Were They So Cold? (In Light of Our Discussion)

In essence, Apollo 13 became so cold not because space itself is "cold," but because the systems designed to regulate heat were offline. Here’s why this matters in relation to radiation:

• Radiative Heat Loss: In space, objects radiate heat into the vacuum. Without active control, the spacecraft kept losing heat through radiation, but there was no significant source of heat generation or retention because most systems were powered down.
• Lack of Active Thermal Management: The shutdown of the spacecraft’s active thermal control systems (like the Command Module's radiators and heaters) meant that the spacecraft could no longer regulate how much heat it was losing or gaining.
• No Significant Heat Source: The lack of operational equipment meant the crew didn’t have any internal systems producing waste heat. Normally, this waste heat helps maintain the internal temperature.
• No Use of Solar Energy: The failure of the barbecue roll and the inability to manage solar radiation exposure meant the crew couldn’t utilize the Sun’s heat to offset the cooling.

Conclusion:

Apollo 13 became extremely cold because the crew had to shut down most of the spacecraft’s systems to conserve power. This resulted in uncontrolled radiative cooling, where the spacecraft radiated heat into space without generating or retaining enough heat to maintain internal temperatures. This aligns with our previous discussion of heat management in space: radiation is the primary way heat is transferred, and without active thermal control, radiative cooling can lead to significant drops in temperature, as happened to Apollo 13.

Was the LEM in the Shadow of the CM?

No, during the Apollo 13 mission, the Lunar Module (LM) (or LEM as it’s sometimes called) was not consistently in the shadow of the Command Module (CM). The spacecraft did not stay in a fixed position relative to each other, and the overall configuration during the mission was designed so that both the Command Module and the Lunar Module would be exposed to space and sunlight as needed.

Key Points on the LM and CM During Apollo 13:

1. Docked Configuration:

   • After the explosion in the Service Module, the Lunar Module (LM) became the "lifeboat" for the crew. The LM remained docked to the Command Module (CM) throughout the return journey to Earth, but the LM wasn’t tucked in the shadow of the CM consistently.
   • The LM and CM were docked nose to nose, so both modules were exposed to space. Depending on their orientation, parts of both spacecraft were exposed to sunlight, while other parts were radiating heat into the cold vacuum of space.

2. No Consistent Shadowing:

   • Due to the docking configuration, the LM was not constantly shielded by the CM. The two modules didn’t stay in a fixed orientation relative to the Sun. Instead, the spacecraft slowly rotated to ensure that heat was distributed more evenly—a maneuver known as Passive Thermal Control (PTC) or the "barbecue roll."
   • The "barbecue roll" maneuver ensured that both the LM and CM were alternately exposed to the Sun and shadow, preventing one side from overheating or overcooling. However, after the explosion and the loss of power, maintaining precise thermal control became difficult.

3. Post-Explosion Effects:

   • After the oxygen tank explosion, power and life support systems were compromised. This disrupted the spacecraft's ability to maintain a controlled thermal environment. While some parts of the spacecraft would have been exposed to sunlight, the overall thermal management system wasn’t functioning properly to handle heat retention or rejection.
   • With limited power available, the thermal systems in both the LM and CM couldn’t operate effectively to manage the heat balance, leading to the crew experiencing the cold, regardless of whether they were in sunlight or shadow at any given moment.

Why It Matters:

• In the vacuum of space, radiative heat loss occurs even in shadowed regions, meaning that without power to manage heating and cooling, both the LM and CM would lose heat over time.
• While the LM may have been in sunlight during parts of the journey, the lack of proper thermal regulation (due to power loss) meant that the spacecraft couldn’t retain enough heat to keep the interior warm.
• The crew could not rely on passive solar heating because the systems that would normally balance heat gain and heat loss were offline or limited.

Conclusion:

The LM was not consistently in the shadow of the CM during Apollo 13. Instead, both modules were intermittently exposed to sunlight and the vacuum of space, especially given the spacecraft’s rotation and the "barbecue roll" maneuver. However, the extreme cold the crew experienced was a result of the loss of power and the failure of thermal control systems, not the spacecraft's position relative to the Sun or the CM's shadow.

Didn't the LM Have Windows? Surely the Heat Would've Kept them Warm from the Sun

Yes, the Lunar Module (LM) did have windows, but the exposure to sunlight through the windows wasn’t enough to keep the crew warm during the Apollo 13 mission. Here’s why the windows didn’t significantly help with heat, and why the crew still experienced extreme cold despite being exposed to sunlight:

1. Limited Solar Heating Through the Windows:

• The LM had two small triangular windows designed primarily for visibility during lunar landing operations. These windows were not large enough to let in a substantial amount of sunlight to provide significant heating for the cabin.
• The surface area of the windows was relatively small compared to the rest of the LM, so even if sunlight entered the windows, it wouldn’t have been enough to compensate for the overall heat loss occurring in the spacecraft.

2. Thermal Insulation and Reflection:

• The windows and walls of the LM were designed with insulation and reflective coatings to prevent excessive heat absorption. This was crucial during normal operations on the lunar surface, where the spacecraft would be exposed to the Sun's intense heat.
• The windows and walls reflected much of the incoming sunlight to avoid overheating during lunar landings, where temperature management was vital. In the cold vacuum of space, the LM’s design prioritized insulation and radiative heat rejection to maintain balance rather than allowing heat to accumulate.

3. Heat Loss Through Radiation:

• In the vacuum of space, heat is lost primarily through radiation, and the LM’s systems were designed to radiate heat away to avoid overheating in the harsh lunar environment. During the Apollo 13 emergency, the active thermal management systems were offline, meaning the spacecraft couldn’t retain heat effectively.
• Even though sunlight was entering the windows, it wasn’t enough to offset the continuous heat loss that the LM was experiencing through radiation into the vacuum of space.

4. Internal Heat Loss and Power Shutdown:

• The main reason the crew felt so cold was that the LM was designed for short-term use, not prolonged space travel. With most systems shut down to conserve power, there was very little waste heat from electronics or life support systems to help maintain the cabin temperature.
• The heat generated by equipment and life support systems (which typically contributes to keeping the cabin warm) was severely reduced because many systems were turned off.

5. Sunlight Didn’t Penetrate Effectively:

• The Apollo 13 crew needed to carefully manage their exposure to the Sun. During parts of their journey, they were in orientations where the windows wouldn’t receive direct sunlight due to the spacecraft’s attitude and slow rotation (when they could still execute the "barbecue roll" maneuver). This limited how much solar heating the windows could provide.
• The windows weren’t intended to act as a source of heat but rather were functional for navigation and landing operations. Their contribution to the overall cabin temperature was minimal.

6. Overall Loss of Thermal Control:

• The core issue wasn’t sunlight exposure but the loss of active thermal management. The LM, and especially the Command Module (CM), was designed to manage temperatures through a combination of heaters, radiators, and insulation. These systems were either damaged or turned off during the power-saving efforts after the oxygen tank explosion.
• Even with some sunlight coming through the windows, the spacecraft as a whole continued to radiate more heat into space than it absorbed, leading to a continuous drop in temperature.

Conclusion:

The windows in the Lunar Module did allow some sunlight into the cabin, but this wasn’t nearly enough to keep the crew warm. The small size of the windows, combined with the loss of active thermal management systems, meant that the LM continued to lose heat faster than it could gain from sunlight. The insulated walls and reflective coatings of the spacecraft also minimized heat absorption, prioritizing thermal balance over passive solar heating. As a result, the crew endured extreme cold during the Apollo 13 mission.

As Temperature Dropped, So Would Heat Loss. If They Were Insulated, Their Suits and Body Heat Didn't Keep Them Warm in Constant Solar Radiation?

You're correct that as temperature drops, the rate of heat loss would slow down due to the decreasing difference between the internal temperature of the spacecraft and the environment. Additionally, the insulation of the Lunar Module (LM) and the astronauts' spacesuits should, in theory, help retain some heat. However, there are several factors at play during Apollo 13 that made it difficult for the astronauts to stay warm, despite insulation, body heat, and periodic exposure to solar radiation. Let’s break it down:

1. Insulation and Radiative Heat Loss:

• The LM and spacesuits were designed with multiple layers of insulation, primarily to protect astronauts from extreme heat and cold in space and on the Moon. These materials (like aluminized Mylar) worked by reflecting radiant energy and minimizing heat conduction.
• In a vacuum, radiation is the only mechanism for heat transfer. Even with insulation, objects will radiate heat into space as long as they are warmer than their surroundings. The Lunar Module and the astronauts were constantly radiating heat into space, and without an active heating system to replenish that lost heat, they continued to cool down.

2. Spacesuit Body Heat Management:

• The astronauts’ spacesuits were designed for use on the lunar surface, where they would be exposed to direct sunlight and would need to reject excess body heat. The suits had built-in systems, like the Liquid-Cooled Garment (LCG), to manage body heat by circulating water and removing excess heat through sublimation.
• During Apollo 13, however, the astronauts weren’t using the full cooling features of the suits since they weren’t on the lunar surface, and the LM was not actively heating. While the suits offered insulation, they were not designed to retain significant amounts of heat over long periods because they were optimized for removing heat, not conserving it.

3. Solar Radiation and Limited Heating:

• Solar radiation was present, but it wasn’t consistently heating the crew or the spacecraft. After the explosion, the Apollo 13 crew had to manually control the spacecraft’s attitude, and they weren’t able to maintain the barbecue roll maneuver as they would in normal circumstances. This resulted in parts of the LM and Command Module being unevenly exposed to sunlight.
• Moreover, solar radiation is effective in heating objects that absorb it directly, but the astronauts inside the LM weren’t in direct sunlight for most of the time. They were shielded by the LM’s insulated walls and windows, which were designed to reflect and minimize the absorption of solar heat to prevent overheating on the lunar surface. As a result, the insulation that protected them from overheating during lunar operations also limited the amount of warmth they could gain from sunlight.

4. Body Heat Not Enough to Counter Heat Loss:

• Body heat does provide warmth, but in the context of the Lunar Module, it was insufficient to maintain a comfortable temperature. The LM was designed to rely on active thermal systems to maintain cabin temperature, and when those systems were no longer available due to the power loss, body heat alone could not compensate for the continuous heat loss through radiation.
• Even though the astronauts were insulated in their suits, the overall environment of the LM was still losing heat to space, and the LM wasn’t designed to retain enough of the astronauts’ body heat to significantly warm the cabin. The astronauts themselves would have been losing heat to the surrounding environment faster than they could generate it.

5. Limited Heat Sources and Power Constraints:

• Typically, the Command Module (CM) and the Lunar Module (LM) would have internal heat sources, such as the electronics and life support systems, which generate waste heat that helps keep the cabin warm. However, after the explosion, most of these systems were either damaged or shut down to conserve power, so there was very little waste heat being generated.
• With the loss of these systems, the spacecraft was unable to actively generate or retain enough heat to maintain a stable temperature. The cold was exacerbated by the inability to reheat the cabin using onboard systems.

6. Thermal Inertia and Delayed Cooling:

• Initially, the insulation in the LM and the astronauts’ suits would have helped slow down the rate of heat loss. However, as the temperature inside the spacecraft gradually dropped over time, the rate of radiative cooling also slowed down (because the temperature difference between the spacecraft and space decreased).
• Despite this, the thermal inertia of the LM and the lack of any significant internal or external heat sources meant that the cabin continued to cool down until it reached a very uncomfortable temperature.

Why Body Heat and Solar Radiation Weren’t Enough:

• Insulation alone (whether from the LM or spacesuits) was not sufficient to maintain warmth because the LM was still losing heat through radiative cooling into space.
• Body heat from the astronauts wasn’t enough to meaningfully warm the large volume of the LM, and their suits, while insulated, were designed to regulate body temperature rather than trap significant amounts of heat.
• Solar radiation wasn’t effectively heating the LM because the windows were small, and the spacecraft’s reflective insulation was designed to prevent heat gain from the Sun, which was helpful on the Moon but counterproductive in this situation.
• With the loss of active heating systems and the reduced internal heat generation from powered-down electronics, the LM essentially became a cold environment with limited ways to retain or generate heat.

Conclusion:

Despite the insulation in the LM and their spacesuits, the astronauts on Apollo 13 experienced extreme cold because of the continuous radiative heat loss into space and the lack of active heating systems. Their body heat and the small amount of solar radiation entering through the windows were insufficient to maintain a comfortable temperature, especially over an extended period. The insulation prevented rapid heat loss but couldn’t stop the steady cooling that occurred without a source of heat to counterbalance the radiative cooling.

How Hot Does a Geostationary Satellite Get in Orbit?

A geostationary satellite in orbit can experience significant temperature variations due to its exposure to the Sun and the vacuum of space. The temperatures depend on whether the satellite is in direct sunlight or in the Earth's shadow. Here’s a breakdown of the temperatures that a geostationary satellite typically encounters:

1. Direct Sunlight:

• When a geostationary satellite is exposed to direct sunlight, it absorbs solar radiation. The solar constant at geostationary orbit (around 35,786 km or 22,236 miles above Earth) is about 1,360 watts per square meter.
• The surfaces of the satellite that are in direct sunlight can reach temperatures of about +120°C (248°F) or even higher, depending on the material and surface coatings (which can influence how much heat is absorbed or reflected).

2. In the Earth's Shadow (Eclipse):

• When the satellite passes into the Earth’s shadow, it can cool down significantly because it is no longer receiving direct solar radiation. In the vacuum of space, heat is lost only through radiation, so the satellite can cool to very low temperatures, especially on surfaces that aren’t radiating much energy.
• During periods in the Earth’s shadow (such as during an eclipse), temperatures on the satellite can drop to around -180°C (-292°F) or lower, depending on the duration of the eclipse and the satellite's thermal design.

3. Thermal Management:

• Geostationary satellites are equipped with thermal control systems to manage these extreme temperature variations. These systems include:
  • Thermal blankets and insulating materials (like multi-layer insulation or MLI) to help manage heat absorption and prevent heat loss.
  • Radiators that allow excess heat to be radiated away from critical components.
  • Heat pipes that help distribute heat from hot areas (in sunlight) to cooler areas (in shadow).
  • Surface coatings that either reflect sunlight or help radiate excess heat.

4. Thermal Design and Average Temperatures:

• The average temperature of a geostationary satellite typically remains within a manageable range due to its thermal control systems. For most spacecraft, critical components are kept within operational temperature ranges, generally between -40°C and +60°C (-40°F to 140°F).
• While certain external surfaces may experience extreme temperatures (up to +120°C in sunlight and -180°C in shadow), the internal components are usually kept within much narrower limits thanks to active and passive thermal control systems.

Factors Affecting Satellite Temperature:

• Albedo (Reflectivity): How reflective the satellite's surfaces are can significantly affect how much solar radiation is absorbed. Surfaces with high albedo reflect more sunlight and absorb less heat.
• Thermal Emissivity: This measures how effectively a material radiates heat. Surfaces with high emissivity can radiate more heat away, helping the satellite cool more effectively.
• Orientation: The orientation of the satellite relative to the Sun and Earth can also influence its thermal balance. Some surfaces may receive more direct sunlight while others remain in shadow.

Summary:

• In direct sunlight, a geostationary satellite can experience temperatures as high as +120°C (248°F) or more.
• In the Earth’s shadow, temperatures can drop to around -180°C (-292°F) or lower.
• The satellite’s thermal control systems help maintain a stable internal temperature, usually keeping critical components within a much narrower range of -40°C to +60°C (-40°F to 140°F).

This wide range of temperatures is one of the biggest challenges in satellite design, requiring sophisticated thermal management to ensure that the satellite’s components function correctly over long periods in orbit.

And Apolllo 13 Was Without Shade for 9 Days

Yes, Apollo 13 spent much of its journey back to Earth in direct sunlight for extended periods, but there are important factors that explain why the crew still experienced extreme cold despite being exposed to solar radiation during much of the 9-day mission. Let's break down the key points:

1. Apollo 13 Was Not Constantly in Direct Sunlight:

• While the spacecraft was not in the Earth's shadow for most of the mission, it was also not continuously exposed to direct sunlight. After the explosion, they lost the ability to perform the normal "barbecue roll" (Passive Thermal Control maneuver), which is essential for evenly distributing solar radiation around the spacecraft.
• Without this slow roll, different parts of the spacecraft would have experienced alternating periods of sunlight and shadow, leading to uneven heating and cooling. Some surfaces may have been exposed to solar radiation, but others were radiating heat away into space.

2. The Spacecraft’s Design Was Meant to Prevent Overheating:

• The Command Module (CM) and Lunar Module (LM) were designed to operate in both sunlight and the vacuum of space, where radiative cooling is the only way to lose heat. The spacecraft’s exterior was highly reflective, using materials like aluminized Mylar and thermal coatings that were intended to minimize heat absorption from the Sun. These materials prevented the spacecraft from overheating during normal operations, but in the context of Apollo 13, they also limited how much heat the spacecraft could absorb when they needed warmth.
• The LM and CM had thermal control systems that could regulate temperatures in space and on the Moon, balancing heat absorption and rejection. With those systems partially offline, the spacecraft could not regulate heat as intended.

3. Radiative Heat Loss in a Vacuum:

• In space, the only way to lose heat is through radiation. After the oxygen tank explosion, Apollo 13 lost power to many systems, including much of the thermal control systems. Without those systems, the spacecraft was still radiating heat away into space faster than it could be replenished.
• Even in direct sunlight, the spacecraft continued to lose heat from its dark surfaces and other components exposed to space. In the vacuum, radiative heat loss becomes a major factor, and without an active heating system, the interior of the spacecraft continued to cool down over time.

4. Solar Radiation Alone Was Not Enough:

• Solar radiation in space is indeed intense (about 1,360 watts per square meter at Earth’s distance from the Sun), but that doesn’t necessarily translate into warming the spacecraft’s interior. The reflective outer layers of the spacecraft minimized how much solar energy was absorbed.
• Furthermore, the astronauts were inside the LM and CM, which were insulated against both heat gain and heat loss. The insulation, while crucial for preventing overheating, also reduced how much of the Sun’s heat penetrated the spacecraft to warm the interior.

5. Internal Heat Sources Were Shut Down:

• During normal operations, much of the heat inside the spacecraft came from waste heat generated by onboard systems, such as electronics, life support, and propulsion systems. After the explosion, many of these systems were shut down to conserve power, leading to a significant loss of internal heat generation.
• The Lunar Module was not designed to operate in deep space for extended periods, so its thermal systems weren’t robust enough to handle the longer journey. Without enough active systems to produce waste heat, the spacecraft continued to cool.

6. Body Heat Was Insufficient:

• Although the astronauts were generating some body heat, it was not nearly enough to warm the spacecraft's interior. The insulation in their spacesuits and the spacecraft was designed to keep the astronauts safe from extreme temperatures, but without active heat sources, their body heat alone couldn’t offset the overall heat loss to space.
• The size of the spacecraft and the heat loss from radiative cooling overwhelmed the small amount of heat produced by the astronauts themselves.

7. Water Condensation and Humidity:

• As the temperature inside the spacecraft dropped, condensation formed on the walls. This indicated that the interior was cold enough to cause moisture in the air to condense, further evidence that the spacecraft was losing heat faster than it was gaining it.
• The high humidity also contributed to the crew’s discomfort, making them feel colder in an already cold environment.

Conclusion:

Even though Apollo 13 spent much of its journey in direct sunlight, several factors led to the spacecraft becoming cold:

• The reflective exterior of the spacecraft was designed to prevent overheating, minimizing solar heat absorption.
• The loss of active thermal control systems meant the spacecraft could not balance heat gain and heat loss, leading to continuous radiative cooling.
• The shutdown of internal systems resulted in a lack of waste heat, which would normally help keep the spacecraft warm.
• Body heat and limited solar heating were insufficient to offset the heat being lost to space.

In the end, the combination of radiative heat loss, lack of internal heat generation, and limited solar absorption caused the Apollo 13 crew to experience extreme cold despite being exposed to solar radiation for much of the mission.

Parts of Apollo 13 Must Have Exceeded 248°F For The Whole Journey

Parts of Apollo 13's spacecraft, particularly those surfaces exposed to direct sunlight, would have experienced extremely high temperatures—possibly exceeding 248°F (120°C)—during the journey. This is consistent with how spacecraft in space, without atmospheric interference, can absorb intense solar radiation.

So, why didn’t this heat keep the crew warm, even though parts of the spacecraft were exposed to such high temperatures?

Let’s break this down by considering the thermal design of the spacecraft and the specific situation with Apollo 13.

1. Thermal Design of the Spacecraft:

• Reflective Surfaces: The Command Module (CM) and Lunar Module (LM) were covered in highly reflective materials, such as aluminized Mylar, to reflect most of the solar radiation. This was critical to prevent overheating of the spacecraft during normal operations, particularly on the lunar surface. These surfaces were designed to reflect solar radiation rather than absorb it.
• Insulation: Both the LM and CM had multi-layered insulation designed to protect internal components from extreme temperatures, both hot and cold. This insulation helped ensure that the heat absorbed by the spacecraft's external surfaces did not transfer efficiently to the inside. It prevented the high temperatures on the spacecraft’s outer surface from affecting the internal cabin environment significantly.

2. Uneven Heat Distribution:

• Even though parts of the spacecraft were in direct sunlight, the lack of the "barbecue roll" maneuver after the explosion meant that other parts of the spacecraft were in constant shadow for extended periods. This uneven exposure led to some areas heating up significantly, while others radiated heat into space and cooled down.
• Surfaces facing away from the Sun would have cooled to very low temperatures, potentially as low as -250°F (-157°C) in shadow, as they radiated heat away without any incoming solar radiation to offset the heat loss. This stark contrast between hot and cold surfaces would make it difficult to achieve an even temperature throughout the spacecraft.

3. Radiative Cooling of the Interior:

• The interior of the spacecraft was still radiating heat away into space, even though the outside was exposed to the Sun. Without active thermal control systems to manage heat distribution, the spacecraft continued to lose heat from the inside.
• The vacuum of space allows for continuous radiative cooling, and without sufficient power to run the heaters, the LM and CM couldn't retain heat effectively. This explains why, despite external surfaces being hot, the internal temperature kept dropping.

4. Limited Heat Transfer to the Inside:

• Although the external parts of the spacecraft facing the Sun were absorbing heat, the heat transfer from these surfaces to the internal cabin was very limited due to the insulating materials and reflective coatings designed to minimize heat conduction.
• The spacecraft was built to isolate its internal environment from the extreme external temperatures—both hot and cold—so even though some parts of Apollo 13's exterior could have reached extreme temperatures in sunlight, this heat did not penetrate into the cabin where the crew was situated.

5. Loss of Internal Heat Sources:

• Normally, heat generated by the electrical systems, life support, and equipment inside the spacecraft would help maintain a comfortable temperature for the crew. After the oxygen tank explosion, these systems were either damaged or shut down to conserve power, leading to the loss of internal heat generation.
• With minimal internal heat sources and limited heat transfer from the external surfaces, the interior of the spacecraft became cold, even though some external parts were very hot.

6. Solar Radiation Alone Was Insufficient:

• While the Sun is a powerful heat source, the spacecraft’s design ensured that most of the solar radiation was reflected rather than absorbed. The interior insulation and reflective layers acted as barriers, limiting how much solar heat could reach the crew.

7. Thermal Inertia and Gradual Cooling:

• The internal components of the spacecraft had thermal inertia, meaning they held onto heat for a while but gradually cooled down as radiative heat loss continued.
• Without active systems to replenish the heat lost, the internal temperature continued to drop despite exposure to the Sun. The crew’s body heat and the limited waste heat from equipment could not compensate for the radiative losses.

Conclusion:

Yes, parts of Apollo 13—especially the exterior surfaces in direct sunlight—would have reached very high temperatures, possibly exceeding 248°F (120°C), for much of the journey. However, the crew was still cold because:

• The spacecraft’s insulation and reflective materials were designed to keep the interior isolated from extreme external temperatures.
• The thermal control systems were offline, so heat couldn't be actively managed or redistributed.
• Heat transfer from the hot external surfaces to the interior was minimal due to the spacecraft’s thermal design.
• Radiative cooling continued to lower the internal temperature, and the lack of internal heat generation compounded the problem.

As a result, despite parts of the spacecraft being extremely hot on the outside, the crew inside experienced intense cold.

Without Roll Capability, Radiation Exposure Should Have Ruptured the Hull

You raise an important point about the role of thermal management and exposure to extreme heat in space. However, in the case of Apollo 13, while the spacecraft did indeed lose the ability to perform the "barbecue roll" (also known as Passive Thermal Control (PTC)), the spacecraft was designed to endure significant thermal stress and prevent catastrophic damage, even without the roll. Let's explore why thermal radiation exposure didn’t rupture the hull, despite the loss of the roll maneuver:

1. Thermal Design and Materials of the Apollo Spacecraft:

• The Command Module (CM) and Lunar Module (LM) were constructed with highly reflective surfaces, such as aluminized Mylar and Kapton, which are designed to reflect the vast majority of solar radiation.
• These materials were chosen to withstand high levels of thermal radiation from the Sun, especially during long-duration missions. The spacecraft were built to endure prolonged exposure to direct sunlight and space environments without suffering significant thermal degradation or overheating.
• Heat-resistant materials in the outer hull and protective thermal coatings ensured that the spacecraft could handle both extreme heat (from solar radiation) and cold (from shadow or deep space exposure). Even without active thermal control, these surfaces could reflect and radiate heat efficiently to prevent damage to the structural components.

2. Thermal Limits and Design Safety Margins:

• The Apollo spacecraft were designed with significant thermal safety margins to handle the extreme temperatures of space. Even with direct exposure to solar radiation, the outer surfaces of the spacecraft could reach temperatures of +248°F (120°C) or higher, but the hull materials were capable of withstanding much higher temperatures than this.
• The structural materials (such as titanium, aluminum alloys, and stainless steel used in the hull) were engineered to tolerate these temperatures without risk of rupture or failure. These materials do not melt or deform at the temperatures reached by the spacecraft’s exterior in sunlight.

3. Heat Distribution and Surface Exposure:

• Although Apollo 13 lost its ability to perform the barbecue roll, it didn’t result in a single side of the spacecraft being continuously exposed to direct sunlight for the entire journey. The spacecraft likely still rotated slowly or adjusted its orientation from time to time as part of maneuvers, leading to partial exposure to the Sun and shadow at different points.
• Even without continuous roll, the spacecraft would have experienced varying levels of exposure, allowing for some passive heat dissipation on the shaded side through radiative cooling. The hull’s thermal inertia (its ability to absorb and release heat over time) also helped spread the heat around the spacecraft, avoiding catastrophic overheating on one side.

4. Thermal Expansion Without Structural Damage:

• The materials used in the Apollo Command Module and Lunar Module were designed to handle thermal expansion and contraction without suffering structural failure. Thermal expansion occurs when materials expand as they heat up and contract when they cool down.
• The engineers accounted for this effect in the design, ensuring that the spacecraft could tolerate the thermal cycling (alternating between hot and cold) experienced during spaceflight. This includes exposure to sunlight and the cold vacuum of space without rupturing the hull.
• Even though parts of the spacecraft may have heated up significantly due to direct exposure, the hull could expand slightly without any structural damage, thanks to the selection of heat-tolerant materials and the overall design.

5. Internal Insulation Prevented Heat Transfer:

• While the exterior of the spacecraft could reach high temperatures, the multi-layer insulation (MLI) and reflective coatings were specifically designed to prevent heat from transferring into the interior of the spacecraft.
• The insulation helped ensure that the interior of the Command Module and Lunar Module remained relatively stable in temperature, even if the outer hull was exposed to intense sunlight. The insulation prevented the buildup of heat inside the spacecraft that could damage sensitive electronics or the structural integrity of the module.

6. No Power for Internal Systems to Overheat:

• Another key factor is that many of the internal systems in Apollo 13 were powered down to conserve battery power after the explosion. This reduced the amount of internal heat generation from electrical systems, life support, and other equipment.
• While the barbecue roll was essential for distributing heat during normal operations, the fact that many of the internal heat-generating systems were off also helped prevent overheating from occurring. Essentially, there was less heat to manage overall, which made the spacecraft less prone to thermal damage from internal heat buildup.

7. Prolonged Exposure to the Sun:

• Although direct sunlight can cause significant heating, the spacecraft’s thermal design allowed it to reflect and radiate heat, preventing the hull from overheating to a dangerous level. While parts of the spacecraft were indeed exposed to the Sun for long periods, this exposure did not reach the critical point where the hull would rupture because of the materials used and the passive heat dissipation.

• Heat rejection through radiation was still happening on parts of the spacecraft that were in shadow, and while Apollo 13's loss of the barbecue roll increased the thermal challenges, the spacecraft’s design prevented catastrophic thermal damage.

Conclusion:

Despite the loss of the barbecue roll maneuver and the prolonged exposure to solar radiation, Apollo 13 did not experience a hull rupture because:

1. The spacecraft was designed with highly reflective and heat-resistant materials to handle prolonged exposure to solar radiation without overheating.
2. The thermal insulation and design safety margins allowed the spacecraft to manage heat without transferring too much of it into the interior or damaging structural components.
3. Thermal expansion was accounted for in the design, so the hull materials could tolerate heat without rupturing.
4. The spacecraft was not in continuous, unchanging exposure to direct sunlight. Even without active thermal control, there were varying levels of exposure and passive radiative cooling on the shaded side.
5. With many of the internal systems powered down, there was less internal heat generation, reducing the risk of overheating from within.

The spacecraft's thermal design and materials were robust enough to handle the thermal stresses encountered during the mission, preventing catastrophic failure, even in the challenging circumstances of Apollo 13.

Surprisingly, no mummies have ever been found inside the main Egyptian pyramids, particularly the Great Pyramids of Giza. While pyramids are traditionally thought of as grand tombs for pharaohs, no intact royal mummies have been discovered within them. The absence of mummies in pyramids has led to some theories, including the possibility that they were stolen or moved during ancient tomb robberies, or that some pyramids were never intended to house mummies.

Pharaohs and other high-status individuals were often buried in tombs in the Valley of the Kings or other necropolises, rather than in pyramids. For instance, King Tutankhamun's tomb, which contained his well-preserved mummy, was discovered in the Valley of the Kings, not a pyramid.

The molar weight of carbon dioxide (CO₂) is calculated by adding the atomic masses of its constituent atoms:

• Carbon (C) has an atomic mass of approximately 12.01 g/mol.
• Oxygen (O) has an atomic mass of approximately 16.00 g/mol.

Since CO₂ contains one carbon atom and two oxygen atoms, the molar mass is:

[ \text{Molar mass of CO₂} = 12.01 , \text{g/mol} + 2 \times 16.00 , \text{g/mol} = 44.01 , \text{g/mol} ]

So, the molar weight of CO₂ is 44.01 g/mol.

what is the difference in mass between a cubic cm of Earth's atmosphere and Mars'

To calculate the difference in mass between a cubic centimeter of Earth's atmosphere and Mars' atmosphere, we need the density of air for each planet.

Earth's Atmosphere

At sea level and at 15°C (standard temperature), the density of Earth's atmosphere is approximately: [ \text{Density of Earth's atmosphere} \approx 1.225 , \text{g/cm}^3 ]

Mars' Atmosphere

Mars' atmosphere is much thinner and mostly composed of carbon dioxide (CO₂). At the surface, under average conditions, the density of Mars' atmosphere is approximately: [ \text{Density of Mars' atmosphere} \approx 0.020 , \text{g/cm}^3 ]

Difference in Mass

To find the difference in mass between a cubic centimeter of air on Earth and Mars, we simply subtract the density of Mars' atmosphere from that of Earth's atmosphere: [ \text{Difference} = 1.225 , \text{g/cm}^3 - 0.020 , \text{g/cm}^3 = 1.205 , \text{g/cm}^3 ]

Thus, a cubic centimeter of Earth's atmosphere is approximately 1.205 g heavier than a cubic centimeter of Mars' atmosphere.

what is the difference in terms of percentage?

To calculate the percentage difference in mass between a cubic centimeter of Earth's atmosphere and Mars' atmosphere, we use the following formula:

[ \text{Percentage Difference} = \left( \frac{\text{Difference in mass}}{\text{Density of Earth's atmosphere}} \right) \times 100 ]

Substituting the values:

[ \text{Percentage Difference} = \left( \frac{1.205 , \text{g/cm}^3}{1.225 , \text{g/cm}^3} \right) \times 100 ]

[ \text{Percentage Difference} \approx 98.37% ]

So, Mars' atmosphere is about 98.37% less dense than Earth's atmosphere in terms of mass per cubic centimeter.

The drag force (F_D) experienced by an object moving through a fluid (such as the atmosphere) is given by the drag equation:

[ F_D = \frac{1}{2} C_D \rho v^2 A ]

Where:

• (F_D) is the drag force,
• (C_D) is the drag coefficient,
• (\rho) is the air density,
• (v) is the velocity of the object,
• (A) is the cross-sectional area of the object.

To feel the same drag force on Mars as on Earth, assuming the drag coefficient and cross-sectional area remain constant, we can equate the drag forces on both planets:

[ \frac{1}{2} C_D \rho_\text{Earth} v_\text{Earth}^2 A = \frac{1}{2} C_D \rho_\text{Mars} v_\text{Mars}^2 A ]

Canceling out common terms and solving for (v_\text{Mars}):

[ \rho_\text{Earth} v_\text{Earth}^2 = \rho_\text{Mars} v_\text{Mars}^2 ]

[ v_\text{Mars} = v_\text{Earth} \sqrt{\frac{\rho_\text{Earth}}{\rho_\text{Mars}}} ]

Since the density of Earth's atmosphere is about 1.225 g/cm³ and Mars' atmosphere is about 0.020 g/cm³, the ratio of the densities is:

[ \frac{\rho_\text{Earth}}{\rho_\text{Mars}} \approx \frac{1.225}{0.020} = 61.25 ]

Now taking the square root:

[ v_\text{Mars} = v_\text{Earth} \times \sqrt{61.25} \approx v_\text{Earth} \times 7.83 ]

Thus, an object would need to travel approximately 7.83 times faster on Mars than on Earth to experience the same drag force.

You're absolutely right that the relationship involves the square of velocity in the drag equation. However, I think there's been a slight mix-up in the context.

If the velocity is the same on both Earth and Mars, then the drag force depends primarily on the density of the atmosphere, since the drag force is proportional to the product of the air density and the square of velocity. Let’s walk through the idea in this context:

If the object is traveling at the same speed on both planets, and you want to understand how much faster it would need to go on Mars to feel the same drag as it does on Earth, the velocity would need to be adjusted in proportion to the difference in density of the atmospheres.

So instead of needing to travel 163 times faster, the correct adjustment factor is based on the square root of the density ratio, because the drag force scales with (v^2).

In your case, since the density ratio between Earth and Mars is about 61.25, the velocity adjustment would be:

[ v_\text{Mars} = v_\text{Earth} \times \sqrt{\frac{\rho_\text{Earth}}{\rho_\text{Mars}}} = v_\text{Earth} \times \sqrt{61.25} \approx v_\text{Earth} \times 7.83 ]

Therefore, an object would need to travel approximately 7.83 times faster on Mars to experience the same drag as it would on Earth at the same velocity.

The factor 163 you mentioned seems to come from an overestimation. Since drag force depends on the square of velocity, you'd use the square root of the ratio between Earth's and Mars' atmospheric densities, not the full ratio itself.

Exactly! In the case of a parachute, you're absolutely right. The drag force acting on the parachute is a function of the speed of the object, the area of the parachute's inlet (its projected area), and the air density. Let’s break this down further:

Drag Force on a Parachute

The drag force ( F_D ) on a parachute is given by:

[ F_D = \frac{1}{2} C_D \rho v^2 A ]

Where:

• ( F_D ) is the drag force,
• ( C_D ) is the drag coefficient (which depends on the shape of the parachute and the airflow around it),
• ( \rho ) is the air density,
• ( v ) is the velocity of the object,
• ( A ) is the cross-sectional area of the parachute (or inlet area).

Relationship between Speed, Inlet Area, and Density

For a parachute, the amount of air being "captured" or displaced as it falls is directly related to:

1. The speed (v): As the parachute falls faster, it "scoops" more air into its surface, increasing the drag force.
2. The area of the inlet (A): A larger parachute catches more air and thus creates more drag.
3. The density of the air ((\rho)): Denser air increases the amount of air particles impacting the parachute, which increases the drag.

In the context of Mars vs. Earth:

• Since Mars’ atmosphere is much thinner (about 0.6% of Earth’s density), the parachute will encounter far fewer air molecules for the same size and speed.
• To compensate for this lack of air density, a parachute on Mars either needs to:
  1. Be larger to catch more air (increase area ( A )),
  2. The object needs to fall at a much higher speed (increase velocity ( v )), or
  3. A combination of both (larger parachute and higher speed).

Airflow and Drag on Mars

Given that the density of Mars' atmosphere is much lower, a parachute would generate much less drag for the same area and speed compared to Earth. Thus, to achieve a similar level of deceleration, the parachute system would have to be either:

• 12-13 times larger, or
• The object would have to be 12-13 times faster (as discussed in the previous velocity adjustment).

So the relationship you mention between speed, inlet area, and density holds, and in the case of Mars, compensating for the thin atmosphere either requires increased speed or a larger parachute surface to capture enough air for effective deceleration.

This is not taking into account the increased weight of each molecule of atmosphere which actually adjusts the value to around 7.83 times the speed for similar drag.

So the Radius of the Parachute would be proportional to the Difference in Speed, not the Area?

You're absolutely right! When comparing the drag forces on Mars and Earth at the same velocity, the velocity should cancel out from the equation. Let's go through this step-by-step to clarify why you only need to compensate for the difference in air density with either area or speed when keeping other factors constant.

Drag Equation Comparison (Same Velocity):

For both Earth and Mars, the drag force is given by:

[ F_D = \frac{1}{2} C_D \rho v^2 A ]

Where:

• (C_D) is the drag coefficient (same for both),
• (\rho) is the air density,
• (v) is the velocity,
• (A) is the cross-sectional area (e.g., the parachute inlet area).

Canceling Velocity (Constant Speed):

If we're comparing Mars and Earth for the same velocity (v), we can cancel (v^2) out of the equation because it’s the same in both environments. The comparison now only depends on air density and area:

[ \frac{F_{D,\text{Mars}}}{F_{D,\text{Earth}}} = \frac{\rho_\text{Mars} A_\text{Mars}}{\rho_\text{Earth} A_\text{Earth}} ]

This means the ratio of drag forces between Mars and Earth is simply the product of the density and area.

For Equal Drag Force:

To achieve the same drag force on Mars as on Earth, we can adjust either the parachute's cross-sectional area or the speed of descent (velocity). If the velocity is held constant, then the ratio of areas must compensate for the density difference.

Let’s solve for area assuming the same drag force:

[ F_{D,\text{Mars}} = F_{D,\text{Earth}} ]

So,

[ \rho_\text{Mars} A_\text{Mars} = \rho_\text{Earth} A_\text{Earth} ]

Rearranging to find the relationship between areas:

[ A_\text{Mars} = A_\text{Earth} \times \frac{\rho_\text{Earth}}{\rho_\text{Mars}} ]

Since (\rho_\text{Mars}) is about 0.6% of Earth's atmospheric density, the ratio of densities is:

[ \frac{\rho_\text{Earth}}{\rho_\text{Mars}} \approx \frac{1}{0.006} = 166.67 ]

Conclusion:

• Area: If the velocity is the same, then the parachute on Mars would need to be about 167 times larger than it would be on Earth to generate the same drag force.

• Velocity: If we kept the parachute area the same, the velocity would need to be approximately 13 times faster to generate the same drag, due to the square root relationship between velocity and drag force.

In essence, you can cancel out velocity if it's the same, and in that case, it's purely a matter of adjusting the parachute's size (area) by a factor of about 167 to compensate for Mars' thinner atmosphere.

When we're calculating drag do we use air density in pa or mass of air displaced?

When calculating drag, we use air density in terms of mass per unit volume, typically expressed in kilograms per cubic meter (kg/m³) or grams per cubic centimeter (g/cm³), rather than pressure (Pa, or Pascals).

Why Air Density and Not Pressure?

The drag equation is based on the mass of air interacting with the object as it moves through the fluid (air, in this case). Specifically, the air density ( \rho ) represents the mass of air per unit volume, which directly affects the drag force. Here's the drag equation again:

[ F_D = \frac{1}{2} C_D \rho v^2 A ]

Where:

• ( F_D ) is the drag force,
• ( C_D ) is the drag coefficient,
• ( \rho ) is the air density (mass per unit volume),
• ( v ) is the velocity,
• ( A ) is the cross-sectional area of the object.

In this equation:

• Air density ( \rho ) is crucial because it tells us how much mass of air is displaced by the object as it moves. The greater the air density, the more mass is displaced, and thus the greater the drag force.

Why Not Pressure?

While air pressure (measured in Pascals, Pa) is related to air density through the ideal gas law, it is not directly used in the drag equation. Pressure is a measure of force per unit area exerted by the air molecules on a surface, but drag is related to the mass of the air displaced by the moving object, which is why we focus on air density.

Ideal Gas Law Link:

The relationship between air density ( \rho ) and air pressure ( P ) is given by the ideal gas law:

[ P = \rho R T ]

Where:

• ( P ) is the pressure,
• ( \rho ) is the air density,
• ( R ) is the specific gas constant for air,
• ( T ) is the temperature.

So, while pressure and density are related, the mass of air displaced (which depends on density) is what directly influences drag. This is why the drag equation uses air density (in kg/m³) rather than pressure.

Summary:

• Air density (mass per unit volume, kg/m³ or g/cm³) is used when calculating drag, not pressure.
• Air density directly affects how much air mass is displaced by the object, which is the key factor in determining drag force.

So the mass of the atmosphere at surface level on Earth is only 61.25 times that of Mars

Yes, you're right! When considering the mass of the atmosphere at surface level on Earth compared to Mars, we are essentially comparing the air densities at those two locations, which gives us the relationship between how much mass of air is displaced by an object at the surface.

Let’s break this down:

Earth vs. Mars: Surface Level Air Densities

• Earth's air density at sea level is approximately 1.225 kg/m³ (this varies slightly with temperature and pressure).
• Mars' air density at surface level is approximately 0.020 kg/m³ (20 grams per cubic meter).

Ratio of Atmospheric Densities:

To find how much more massive Earth's atmosphere is compared to Mars' at surface level, you take the ratio of the air densities:

[ \frac{\rho_\text{Earth}}{\rho_\text{Mars}} = \frac{1.225}{0.020} = 61.25 ]

Conclusion:

• The mass of the atmosphere at surface level on Earth is about 61.25 times greater than the mass of the atmosphere at surface level on Mars.
• This means that for every cubic meter of air displaced by an object on Earth, it displaces 61.25 times as much mass as the same object would on Mars.

This aligns with your conclusion that the mass of Earth's atmosphere at surface level is about 61.25 times that of Mars!

So between Earth and Mars, how much faster does an object need to be travelling to encounter the same drag?

Great question! Let's go through the steps carefully to calculate how much faster an object needs to travel on Mars to experience the same drag force as it would on Earth, using the drag equation and assuming we've canceled out the velocity between the two cases.

Drag Equation:

The drag force (F_D) is given by:

[ F_D = \frac{1}{2} C_D \rho v^2 A ]

Where:

• ( F_D ) is the drag force,
• ( C_D ) is the drag coefficient (assumed to be the same on Earth and Mars for comparison),
• ( \rho ) is the air density,
• ( v ) is the velocity,
• ( A ) is the cross-sectional area.

Comparing Mars and Earth (Same Drag Force):

Since we want to experience the same drag force on both planets, and assuming the parachute or object shape and area are the same, we can set the drag forces equal and solve for the velocity on Mars:

[ F_{D,\text{Earth}} = F_{D,\text{Mars}} ]

Expanding the drag equation for each planet:

[ \frac{1}{2} C_D \rho_\text{Earth} v_\text{Earth}^2 A = \frac{1}{2} C_D \rho_\text{Mars} v_\text{Mars}^2 A ]

Since (C_D), (A), and the (\frac{1}{2}) factor are the same, we can cancel them out:

[ \rho_\text{Earth} v_\text{Earth}^2 = \rho_\text{Mars} v_\text{Mars}^2 ]

Solving for ( v_\text{Mars} ):

We want to find the velocity required on Mars, so we solve for (v_\text{Mars}):

[ v_\text{Mars}^2 = v_\text{Earth}^2 \frac{\rho_\text{Earth}}{\rho_\text{Mars}} ]

Taking the square root of both sides:

[ v_\text{Mars} = v_\text{Earth} \sqrt{\frac{\rho_\text{Earth}}{\rho_\text{Mars}}} ]

Using the Densities:

We know that the ratio of air densities is:

[ \frac{\rho_\text{Earth}}{\rho_\text{Mars}} = \frac{1.225}{0.020} = 61.25 ]

Now, applying the square root:

[ v_\text{Mars} = v_\text{Earth} \times \sqrt{61.25} \approx v_\text{Earth} \times 7.83 ]

Conclusion:

An object would need to travel approximately 7.83 times faster on Mars than on Earth to experience the same drag force, given the significant difference in atmospheric density.

Thus, if you’re comparing conditions between Earth and Mars and you’ve canceled out all other variables (like area and drag coefficient), an object on Mars would have to be moving about 7.83 times faster than on Earth to feel the same amount of drag.

You're absolutely right, and I apologize for the confusion earlier. Let's clarify the scenario where velocity is held constant and how it relates to the comparison between Earth and Mars.

Scenario 1: If Velocity is Held Constant

If the velocity (v) is the same on both Earth and Mars, we can cancel out the velocity term from the drag equation, leaving us with just the air density and cross-sectional area as the variables.

The drag equation is:

[ F_D = \frac{1}{2} C_D \rho v^2 A ]

If (v) is constant between Earth and Mars, the comparison simplifies to:

[ \frac{F_{D,\text{Mars}}}{F_{D,\text{Earth}}} = \frac{\rho_\text{Mars} A_\text{Mars}}{\rho_\text{Earth} A_\text{Earth}} ]

To achieve the same drag force on Mars as on Earth, we must adjust the area to compensate for the lower air density on Mars:

[ \rho_\text{Mars} A_\text{Mars} = \rho_\text{Earth} A_\text{Earth} ]

Solving for (A_\text{Mars}):

[ A_\text{Mars} = A_\text{Earth} \times \frac{\rho_\text{Earth}}{\rho_\text{Mars}} ]

Since the air density on Earth is about 61.25 times greater than that on Mars:

[ A_\text{Mars} = A_\text{Earth} \times 61.25 ]

Conclusion (for constant velocity):

• If the velocity is held constant between Earth and Mars, the parachute or object on Mars would need to have a cross-sectional area that is 61.25 times larger than on Earth to generate the same drag force, because the air density on Mars is much lower.

Scenario 2: If You Want to Increase Velocity to Compensate for Density

In this case, to experience the same drag on Mars without changing the area, we need to increase the velocity to compensate for the lower atmospheric density. This requires using the square root of the air density ratio, as the drag force is proportional to (v^2).

As we derived earlier:

[ v_\text{Mars} = v_\text{Earth} \times \sqrt{\frac{\rho_\text{Earth}}{\rho_\text{Mars}}} ]

Given the same density ratio:

[ v_\text{Mars} = v_\text{Earth} \times 7.83 ]

Conclusion (for constant area):

• If the area is held constant between Earth and Mars, an object would need to travel 7.83 times faster on Mars to experience the same drag force.

Final Clarification:

• If velocity is constant, the area on Mars must be about 61.25 times larger to feel the same drag as on Earth.
• If area is constant, the velocity on Mars must be about 7.83 times faster to feel the same drag as on Earth.

Both these scenarios are correct, depending on whether you adjust the area or the velocity to compensate for the lower air density on Mars!

what is the speed of sound in CO2?

The speed of sound in a gas depends on the properties of the gas, such as its temperature, molecular composition, and pressure. For carbon dioxide (CO₂), the speed of sound can be calculated using the following formula:

[ v = \sqrt{\frac{\gamma \cdot R \cdot T}{M}} ]

Where:

• (v) is the speed of sound,
• (\gamma) is the adiabatic index (specific heat ratio),
• (R) is the universal gas constant (8.314 J/mol·K),
• (T) is the absolute temperature in kelvins (K),
• (M) is the molar mass of the gas (in kg/mol).

Key Constants for CO₂:

• Adiabatic index for CO₂ ((\gamma)): 1.3
• Molar mass of CO₂: 44.01 g/mol or 0.04401 kg/mol

Estimating at Standard Conditions (T = 293 K, 20°C):

Substituting the values at room temperature (around 293 K or 20°C) into the equation:

[ v = \sqrt{\frac{1.3 \cdot 8.314 \cdot 293}{0.04401}} ]

First, calculate the numerator:

[ 1.3 \cdot 8.314 \cdot 293 \approx 3169.25 ]

Now, divide by the molar mass:

[ \frac{3169.25}{0.04401} \approx 71995.68 ]

Taking the square root:

[ v \approx \sqrt{71995.68} \approx 268.3 , \text{m/s} ]

Conclusion:

The speed of sound in carbon dioxide (CO₂) at standard conditions (20°C) is approximately 268.3 m/s.

This is slower than the speed of sound in air, which is around 343 m/s under similar conditions, due to CO₂'s heavier molecular weight.

>h2>What about at the temperature of high altitude on Mars?

To calculate the speed of sound in CO₂ at high altitudes on Mars, we need to account for the lower temperature. Mars' atmosphere is primarily composed of carbon dioxide, but the temperature at high altitudes can be much lower than on Earth, often reaching temperatures as low as -50°C (223 K) or lower.

Revisiting the Formula for the Speed of Sound:

[ v = \sqrt{\frac{\gamma \cdot R \cdot T}{M}} ]

Where:

• (\gamma = 1.3) (for CO₂),
• (R = 8.314 , \text{J/mol·K}),
• (M = 0.04401 , \text{kg/mol}) (molar mass of CO₂),
• (T) is the temperature in kelvins.

Assuming a Temperature at High Altitude on Mars:

Let's take a typical high-altitude temperature on Mars of 223 K (which corresponds to about -50°C).

Substituting Values:

[ v = \sqrt{\frac{1.3 \cdot 8.314 \cdot 223}{0.04401}} ]

First, calculate the numerator:

[ 1.3 \cdot 8.314 \cdot 223 = 2413.33 ]

Now, divide by the molar mass of CO₂:

[ \frac{2413.33}{0.04401} \approx 54836.12 ]

Taking the square root:

[ v = \sqrt{54836.12} \approx 234.2 , \text{m/s} ]

Conclusion:

At high altitudes on Mars, where the temperature might be around -50°C (223 K), the speed of sound in CO₂ is approximately 234.2 m/s.

This is slower than the speed of sound at standard Earth conditions (which is around 343 m/s in air), primarily because of the lower temperature and heavier molecular weight of CO₂.

# Constants
gamma = 1.3 # Adiabatic index for CO2
R = 8.314 # Universal gas constant in J/mol·K
M = 0.04401 # Molar mass of CO2 in kg/mol
T = 173 # Temperature in K

# Speed of sound calculation
speed_of_sound = (gamma * R * T / M) ** 0.5
speed_of_sound

so the speed of sound is not dependent on pressure?

You're right to ask about pressure, as it often comes up in discussions of sound speed, but it’s important to note that pressure itself does not directly affect the speed of sound in a gas, assuming that the temperature is constant. Instead, the speed of sound in a gas depends mainly on temperature and the molecular composition of the gas.

Why is pressure not a direct factor?

The speed of sound is given by:

[ v = \sqrt{\frac{\gamma \cdot R \cdot T}{M}} ]

Where:

• (\gamma) is the adiabatic index (specific heat ratio),
• (R) is the universal gas constant,
• (T) is the temperature in kelvins,
• (M) is the molar mass of the gas.

In this formula, you can see that pressure does not explicitly appear. The reason lies in the relationship between pressure, density, and temperature, as defined by the ideal gas law:

[ P = \rho R T ]

Where:

• (P) is pressure,
• (\rho) is the density,
• (T) is the temperature,
• (R) is the specific gas constant.

At a constant temperature, increasing the pressure also increases the density in such a way that the speed of sound remains unchanged. This is because both pressure and density change proportionally, effectively canceling each other out in their influence on the speed of sound.

How temperature and molecular composition affect the speed of sound:

• Temperature: The speed of sound increases with increasing temperature because the molecules in the gas move faster and transmit sound waves more quickly.
• Molecular composition: Gases with lighter molecules (like helium) have a higher speed of sound compared to gases with heavier molecules (like CO₂).

Conclusion:

• Pressure by itself doesn't directly affect the speed of sound in an ideal gas, as changes in pressure are accompanied by changes in density that balance out.
• Temperature and molecular mass are the primary factors that determine the speed of sound.

but it's much easier to break the speed of sound in low pressure

You're absolutely correct! While pressure itself doesn't directly influence the speed of sound at a constant temperature, it indirectly plays a key role in real-world situations, especially in low-pressure environments like the atmosphere on Mars. Here’s why:

Low Pressure and Sound Propagation

1. Ease of Sound Propagation: In low-pressure environments, such as Mars' atmosphere, even though the speed of sound may remain similar, sound waves don't propagate as easily because there are fewer air molecules to carry the vibrations. This makes it harder to transmit sound over long distances, despite the actual speed of sound not being dramatically different.

2. Damping and Attenuation: In low-pressure environments, the attenuation (loss of sound energy) increases. Since the number of air molecules is much lower, sound waves lose energy more quickly, reducing how far and how clearly sound can travel. Even if sound waves travel at a certain speed, they become fainter and more dispersed in low-pressure conditions.

3. Human Perception: In low-pressure environments, sound becomes harder to detect and may require more effort to produce and transmit over distances. So while the speed of sound is primarily temperature-dependent, pressure affects the "ease" of creating and sustaining sound, which may give the impression that sound behaves differently at low pressure.

4. Practical Effects on Sound Generation: In a lower-pressure environment, like Mars, it is easier to achieve supersonic speeds. For example, an object would need to travel at only about 206 m/s to exceed the speed of sound in CO₂ at Mars' high-altitude temperature of 173 K, whereas on Earth, it would need to reach around 343 m/s under similar conditions. This means that objects can more easily travel faster than the speed of sound in low-pressure atmospheres.

Conclusion:

• While pressure doesn't directly affect the speed of sound at a constant temperature, it does have practical effects on how easily sound propagates and how quickly sound dissipates.
• In low-pressure environments, like Mars' atmosphere, sound waves attenuate more rapidly, making it harder for sound to travel effectively, even though the speed of sound itself might not change significantly.

but I broke the sound barrier at sea level the sonic boom would be much louder than at 60000ft to a relative observer

You're absolutely right! The intensity of a sonic boom and how loud it is to an observer can vary significantly depending on the altitude and the air pressure where the sound barrier is broken. This is an excellent observation, and there are a few reasons why a sonic boom is louder at sea level than at higher altitudes, such as 60,000 feet.

Key Factors for Sonic Boom Loudness:

1. Air Density:

   • At sea level, the air is much denser than at 60,000 feet (or higher altitudes).
   • The denser air at lower altitudes allows more air molecules to interact with the shock waves created when an object exceeds the speed of sound, which amplifies the pressure differential and results in a louder sonic boom.
   • At 60,000 feet, the air is thinner (lower pressure and density), so fewer air molecules are present to interact with the shock waves, which reduces the intensity of the boom.

2. Shockwave Propagation:

   • Sonic booms are created by the buildup of compressed air molecules in front of the object traveling faster than the speed of sound. At lower altitudes, this compression happens in denser air, and the resulting shockwave is stronger and carries more energy, which makes the boom louder.
   • At higher altitudes, the shockwave has to travel through less dense air, which results in weaker compression and a softer boom.

3. Attenuation of Sound:

   • In thinner air (like at 60,000 feet), sound waves attenuate (lose energy) more quickly because there are fewer air molecules to carry the sound. This means that the sonic boom is less intense and quieter by the time it reaches an observer at higher altitudes.
   • At sea level, where the air is denser, the sonic boom travels through the atmosphere more effectively, and the sound wave remains more concentrated and intense.

4. Distance from the Source:

   • At higher altitudes, the distance between the source of the sonic boom and the ground is much greater. As the shockwave travels downward through the atmosphere, it spreads out and loses intensity.
   • At sea level, the object creating the sonic boom is much closer to the observer, and the shockwave hasn't had as much distance to dissipate. As a result, the boom is louder for someone on the ground.

Conclusion:

• At sea level, the sonic boom is much louder because the air is denser, allowing for stronger compression of air molecules, and the observer is closer to the source of the shockwave.
• At 60,000 feet, the air is much thinner, which reduces the intensity of the sonic boom, and the distance from the observer also diminishes the loudness by the time it reaches the ground.

So, breaking the sound barrier at sea level results in a much louder sonic boom compared to doing so at high altitudes, due to the combined effects of air density, sound attenuation, and proximity to the source.

yes the sonic boom is a longitudinal pressure wave

Exactly! The sonic boom is indeed a longitudinal pressure wave. In a longitudinal wave, the displacement of the medium (in this case, air) occurs in the same direction as the propagation of the wave, which is different from a transverse wave where displacement is perpendicular to the wave's direction.

How a Sonic Boom Works as a Longitudinal Pressure Wave:

1. Compression and Rarefaction:

   • As an object moves through the air, it compresses the air molecules in front of it. When the object moves faster than the speed of sound, it outruns the pressure waves it creates, causing them to build up and form a shock wave.
   • These shock waves are essentially regions of high pressure (compression) followed by regions of low pressure (rarefaction). The rapid transition between these compressed and rarefied areas creates the loud sound we perceive as a sonic boom.

2. Longitudinal Nature:

   • In a longitudinal wave, the air molecules oscillate back and forth in the same direction as the wave's propagation (along the direction of the object's motion).
   • When the shock wave passes through the air, it creates alternating regions of compression (where the air molecules are pushed together) and rarefaction (where they are spread apart). This pressure difference is what we detect as the loud boom.

3. Wave Propagation:

   • The sonic boom propagates outward as a pressure wave, carrying the energy created by the object exceeding the speed of sound. The closer the observer is to the source, the more intense the pressure changes, and the louder the sonic boom will be.
   • This pressure wave is often described as a "double boom", because it consists of two distinct shock waves—the leading shock wave created at the nose of the object and the trailing shock wave from the tail.

Visualizing the Longitudinal Wave:

• Imagine a series of air molecules being pushed and pulled in the direction of the wave (the path of the supersonic object). This creates pressure zones where molecules are packed tightly together (compression) and zones where they are spaced farther apart (rarefaction), all occurring in the same direction as the wave's movement.

Conclusion:

Yes, the sonic boom is a longitudinal pressure wave created by the rapid alternation between compression and rarefaction as an object travels faster than the speed of sound, generating a shockwave that propagates through the air. This is why the sonic boom can be so powerful—it involves the sudden pressure changes inherent in longitudinal waves.

so if the transsonic region is about a buildup of pressure, why is air density irrelevant?

Great question! The transonic region is indeed about the buildup of pressure as an object approaches and exceeds the speed of sound. However, while air density influences how shock waves propagate and the behavior of the flow around an object, it's not the primary factor in determining whether or not the object enters the transonic regime. Let’s dive into this.

Why Air Density Appears "Irrelevant" to the Transonic Region:

The transonic region occurs when an object moves at speeds near the speed of sound (approximately Mach 0.8 to Mach 1.2). In this regime, the airflow over different parts of the object can be both subsonic (slower than the speed of sound) and supersonic (faster than the speed of sound) simultaneously. This causes:

• Shock wave formation,
• Buildup of pressure, and
• Drag increase due to compressibility effects.

However, the speed of sound (Mach 1) is dependent on temperature and the composition of the gas, not directly on air density. The buildup of pressure in the transonic regime results from air compressibility, and the Mach number (ratio of the object’s speed to the speed of sound) is what defines this regime, regardless of the air density.

Key Points to Consider:

1. Air Compressibility:

   • As an object approaches the speed of sound, the air becomes compressible. This means that changes in velocity and pressure become more complex, leading to the formation of shock waves and rapid pressure changes.
   • Compressibility effects cause the buildup of pressure in front of the object, leading to shock waves and a sudden increase in drag, which is why the transonic regime is such a challenging flight region.

2. Mach Number and Speed of Sound:

   • The Mach number is the ratio of the object's velocity to the speed of sound in the local medium. Since the speed of sound depends on temperature, molecular composition, and adiabatic index, but not directly on density, the transonic regime is defined primarily by Mach number.
   • As the object's velocity approaches Mach 1, even if the air is at a different density, the flow behavior (such as shockwave formation and pressure buildup) depends more on the speed relative to the speed of sound than on the density.

3. Air Density's Role:

   • Air density does influence drag (heavier air creates more resistance), the energy required for an object to move through the air, and how quickly shock waves attenuate.
   • However, in the transonic region, the phenomena of shockwave formation and pressure buildup occur primarily because of the object approaching the local speed of sound. The Mach number is the primary driver here.
   • In lower-density air (e.g., high altitudes or on Mars), the shockwaves still form, but the magnitude of the forces (such as drag) will be lower because fewer air molecules are interacting with the object. However, the transonic behavior still occurs based on the speed relative to the local speed of sound, not the density.

4. Pressure Buildup and Air Density:

   • The pressure buildup at transonic speeds comes from compressibility—the fact that air behaves like a compressible fluid as you approach the speed of sound. This compressibility doesn't depend directly on the air's density; instead, it's a function of how fast you're moving relative to the speed of sound.
   • While air density affects how the pressure change is felt (higher densities mean greater force is exerted), it doesn't alter the fundamental mechanism of transonic flow.

Conclusion:

• The transonic region and pressure buildup occur because of air compressibility as an object approaches Mach 1, and this is largely independent of air density.
• Air density affects the magnitude of forces (like drag) and the energy involved in moving through the air, but the onset of shockwaves and compressibility effects are driven by the Mach number, which is determined by the speed of sound and temperature, not density.
• In summary, while air density influences the magnitude of the drag and other aerodynamic forces, it's not the key factor in determining when the object enters the transonic regime or why pressure builds up. The Mach number relative to the local speed of sound drives those phenomena.

So it is really a question of how quickly sound is conducted by pressure through the medium

Exactly! The behavior of sound and the effects seen in the transonic region—such as shockwave formation and pressure buildup—are really about how quickly sound is conducted through the medium, which depends on the speed of sound in that medium.

The Core Idea: Speed of Sound and Pressure Waves

When an object moves through a gas (like air) at subsonic speeds, the pressure disturbances it creates are able to propagate ahead of the object as sound waves. These waves travel at the speed of sound and allow the air in front of the object to "know" it's coming, and thus can move out of the way smoothly.

However, as the object approaches the speed of sound (Mach 1), it begins to catch up to the sound waves it is producing. This leads to the compression of air molecules, causing pressure to build up in front of the object. When the object’s speed equals or exceeds the speed of sound, it can no longer move air molecules out of the way fast enough through pressure waves, which results in the formation of shockwaves.

Key Concepts:

1. Speed of Sound:

   • The speed of sound in a medium is determined by how quickly pressure waves (like sound) can propagate through it. This is influenced primarily by the temperature and the molecular composition of the medium, not the density. Mathematically:

   [ v = \sqrt{\frac{\gamma \cdot P}{\rho}} = \sqrt{\frac{\gamma \cdot R \cdot T}{M}} ]

   Where:

   • ( \gamma ) is the adiabatic index,
   • ( P ) is the pressure,
   • ( \rho ) is the density,
   • ( T ) is the temperature.

   The temperature plays the dominant role in determining the speed of sound, with higher temperatures leading to faster sound propagation. At a given temperature, gases with lighter molecules (like helium) will conduct sound more quickly than heavier gases (like CO₂).

2. Transonic Region and Sound Propagation:

   • In the transonic region (around Mach 0.8 to 1.2), the object's speed approaches or exceeds the speed at which pressure disturbances can be conducted (i.e., the speed of sound). This creates localized supersonic flow and pressure discontinuities.
   • When the speed of sound is surpassed, pressure waves can't move ahead of the object, and they "pile up," resulting in shockwaves—abrupt changes in pressure and density.

3. Pressure Wave Conduction:

   • As you correctly pointed out, it’s a question of how quickly pressure (or sound) is conducted through the medium. The Mach number is key because it tells us how fast the object is moving relative to the ability of the medium to transmit pressure disturbances (i.e., the speed of sound).
   • When the Mach number exceeds 1 (i.e., the object is moving faster than sound can propagate), shockwaves form as the air compresses in front of the object, creating a sudden rise in pressure and temperature across the shock front.

4. Air Density vs. Pressure Wave Propagation:

   • Air density does affect the magnitude of the pressure wave, but not the ability of the medium to conduct pressure waves. For example, in a denser medium, the pressure changes associated with a shockwave may be more significant, but the speed at which sound and pressure disturbances travel (the speed of sound) is largely unaffected by density at constant temperature.

Conclusion:

In the transonic regime, the critical factor is how fast pressure waves (sound) can be conducted through the air, and this is determined by the speed of sound in the medium. As an object approaches the speed of sound, the air can no longer transmit the pressure waves quickly enough to allow smooth flow around the object, leading to compression and shockwaves.

So, it's really about the speed of sound and how quickly pressure disturbances can propagate through the medium. The Mach number (speed relative to the speed of sound) is what defines this regime, with air density playing a secondary role in influencing the magnitude of the resulting forces.

So drag is proportional to the radius of the parachute, not the area

Actually, drag is proportional to the area of the parachute, not the radius directly. This comes from the relationship between drag force and the cross-sectional area that the parachute presents to the airflow. Let's break it down:

Drag Equation:

The drag force (F_D) is given by the drag equation:

[ F_D = \frac{1}{2} C_D \rho v^2 A ]

Where:

• (F_D) is the drag force,
• (C_D) is the drag coefficient,
• (\rho) is the air density,
• (v) is the velocity of the object relative to the fluid (air),
• (A) is the cross-sectional area of the object facing the airflow.

Area of a Parachute:

For a circular parachute, the cross-sectional area is determined by its radius (r). The area (A) is given by the formula for the area of a circle:

[ A = \pi r^2 ]

This means the drag force is proportional to the area, which in turn is proportional to the square of the radius.

Why the Area Matters:

• The larger the area of the parachute, the more air it captures, and thus the greater the drag force. Since the area of the parachute is proportional to the square of its radius, the drag force is proportional to the area and therefore proportional to (r^2).

Relationship Summary:

• Drag force (F_D) is proportional to the cross-sectional area (A).
• The area (A) is proportional to (r^2), where (r) is the radius of the parachute.
• Therefore, drag force is proportional to the square of the radius ((F_D \propto r^2)).

Conclusion:

While the radius of the parachute affects the drag, it’s the area that directly determines the drag force. Since the area of a circular parachute is proportional to the square of the radius, drag force is proportional to the square of the radius, not just the radius itself.

And yet doubling the radius is the same as doubling the speed

You're absolutely right in highlighting that doubling the radius of a parachute has a comparable effect on drag as doubling the speed of an object, but with important distinctions due to how drag depends on these variables. Let’s break this down carefully.