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No, the escape velocity for lunar injection and Martian injection is not the same, as they depend on the gravitational forces of the Earth, Moon, and Mars, as well as the specifics of the injection orbit.

1. Escape Velocity from Earth for Lunar Injection:

   • This refers to the velocity required to leave Earth's gravity and enter a trajectory that will take a spacecraft toward the Moon.
   • The escape velocity from Earth's surface is approximately 11.2 km/s. However, for a lunar injection, the spacecraft typically doesn't need to reach full escape velocity but instead enters a transfer orbit (called a trans-lunar injection, TLI). The velocity required for TLI from low Earth orbit (LEO) is about 2.38 km/s in addition to the orbital velocity of the spacecraft in LEO (about 7.8 km/s).

2. Escape Velocity for Martian Injection:

   • Similarly, a trans-Martian injection (TMI) refers to the velocity needed to send a spacecraft from Earth toward Mars. This is slightly higher than the velocity needed for a lunar injection, as Mars is farther away from Earth and requires a greater amount of energy.
   • The velocity needed for TMI from low Earth orbit is around 3.6 km/s beyond the orbital velocity in LEO.

In summary:

• The lunar injection requires a lower velocity due to the Moon being closer to Earth.
• The Martian injection requires a higher velocity because of the greater distance to Mars.

The specifics of these velocities depend on mission design, spacecraft mass, and the particular injection trajectory chosen for the mission.

Why is the martian craft not slowed by Earth's gravity like Apollo?

The Martian-bound spacecraft, like the Apollo missions, is indeed influenced by Earth's gravity. However, the key difference lies in the trajectory, energy requirements, and target destination. Here's a detailed breakdown:

1. Apollo's Trajectory to the Moon (Trans-Lunar Injection):

• The Apollo spacecraft used a Trans-Lunar Injection (TLI) to move from Earth orbit toward the Moon. After leaving low Earth orbit (LEO), it was still under the influence of Earth’s gravity but had enough velocity to enter a transfer orbit toward the Moon.
• The Moon is relatively close (about 384,000 km), so the spacecraft does not completely escape Earth's gravity and is still within the Earth's gravitational sphere of influence. Earth's gravity slows down the spacecraft after the initial boost, but the spacecraft is aimed to "fall" toward the Moon's gravity and be captured by it, allowing a relatively gentle approach.

2. Martian-Bound Spacecraft (Trans-Martian Injection):

• A spacecraft headed to Mars uses a Trans-Martian Injection (TMI), which sends it on a much longer trajectory. After this burn, the spacecraft is still influenced by Earth’s gravity initially, but it is given sufficient velocity to leave Earth’s gravity well entirely.
• The key difference here is that Mars is much farther away than the Moon, so the spacecraft needs to achieve escape velocity from Earth’s gravity, and this velocity is higher than that needed for lunar missions.

3. Escape Velocity and Influence of Gravity:

• Once a Martian-bound spacecraft reaches escape velocity (about 11.2 km/s at Earth's surface, slightly less from LEO), it overcomes Earth's gravitational pull. This means Earth's gravity no longer "slows it down" beyond a certain point, as the spacecraft is on an unbound trajectory. While the spacecraft will initially be slowed by Earth's gravity, it has enough velocity to escape entirely.
• The Apollo spacecraft, on the other hand, did not need to reach full escape velocity; it only needed enough energy to reach the Moon, which meant it remained under Earth’s gravitational influence longer. As a result, its speed dropped after the TLI until it got closer to the Moon.

4. The Role of Earth's Gravity:

• Both the Apollo missions and Martian-bound spacecraft experience a gravitational deceleration from Earth after their burns (TLI or TMI), but the difference is:
  • The Apollo mission was designed to fall into the gravitational influence of the Moon, so it did not need to completely escape Earth's gravity.
  • The Mars-bound spacecraft is designed to fully escape Earth’s gravity, which requires achieving escape velocity. Once it escapes, Earth's gravity no longer plays a significant role.

5. Why Mars-Bound Craft Keeps Its Speed:

• For the Martian mission, the spacecraft achieves enough speed to escape Earth's gravitational influence, so while it slows down a bit during its initial departure, the spacecraft still retains sufficient velocity to keep moving through interplanetary space. Once it escapes Earth’s gravity, it continues its journey toward Mars, where it will eventually be influenced by Mars' gravity for capture or orbital insertion.

In summary, the key difference is that the Martian-bound craft reaches escape velocity to leave Earth's gravitational influence entirely, while Apollo missions only needed to enter a transfer orbit to the Moon, staying within Earth's gravitational well longer. Thus, after escaping, the Martian-bound craft is not significantly slowed by Earth’s gravity, unlike the Apollo missions that were bound for the nearby Moon.

What altitude is the TLI orbit?

The Trans-Lunar Injection (TLI) typically begins from a low Earth orbit (LEO), which is generally between 160 km and 2,000 km above Earth's surface. For the Apollo missions, the TLI was initiated from an orbit at around 185 km (115 miles) altitude.

Key Details:

• Apollo Missions' Orbit Altitude: The Apollo spacecraft entered a circular parking orbit around Earth at about 185 km altitude. After completing one or two orbits, the spacecraft executed a TLI burn to increase its velocity and begin its trajectory toward the Moon.
• Other Missions: In general, TLI burns can be initiated from different altitudes, depending on mission design. The key factor is that the spacecraft needs to be in a stable low Earth orbit to conduct the burn and gain the necessary velocity to reach the Moon.

Thus, the 185 km altitude is the historical precedent from the Apollo missions, but it can vary slightly based on the mission's launch and trajectory requirements.

Both Craft Obtain LEO then Perform a Second Burn

Yes, both lunar-bound and Mars-bound spacecraft typically first achieve low Earth orbit (LEO) and then perform a second burn to begin their respective journeys—either toward the Moon (for lunar missions) or Mars (for interplanetary missions). Here's a breakdown of the process for each:

1. Lunar Missions (e.g., Apollo) - Trans-Lunar Injection (TLI):

• First Burn: The spacecraft is launched into low Earth orbit (LEO), typically around 160 to 200 km in altitude.
• Parking Orbit: Once in LEO, the spacecraft remains in this orbit temporarily (a "parking orbit"), which allows mission controllers to check the spacecraft systems and precisely time the second burn.
• Second Burn (TLI): The Trans-Lunar Injection (TLI) burn is then performed, which boosts the spacecraft's velocity, sending it into a transfer orbit toward the Moon. This burn increases the spacecraft's speed to about 10.8 to 11 km/s, just under Earth's escape velocity, allowing it to coast toward the Moon while remaining under the influence of Earth's gravity.

2. Mars Missions - Trans-Martian Injection (TMI):

• First Burn: Similar to lunar missions, Mars-bound spacecraft are first launched into LEO. The altitude may vary, but it's usually within the range of 200 to 400 km.
• Parking Orbit: The spacecraft orbits Earth for a short period in LEO. This parking orbit allows for system checks and precise timing for when to initiate the second burn.
• Second Burn (TMI): The spacecraft performs a Trans-Martian Injection (TMI) burn, which increases its velocity beyond Earth's escape velocity (about 11.2 km/s) to achieve a trajectory that will carry it away from Earth's gravitational influence and set it on course for Mars. This burn typically requires more energy than a TLI burn due to the greater distance to Mars.

Why Two Burns?

• Fuel Efficiency: Launching directly from Earth’s surface into a trajectory toward the Moon or Mars is inefficient due to the high gravitational forces at low altitudes. Instead, spacecraft achieve LEO first, which reduces the initial energy needed and allows the spacecraft to be in a stable orbit where further manoeuvres can be more efficient.
• Timing and Precision: The second burn (TLI or TMI) is performed at a precise point in the orbit, known as the perigee (lowest point), to optimize the spacecraft's trajectory toward its destination. This requires careful alignment with the Moon's or Mars' position, which is easier to manage from LEO.

Summary:

• Both lunar-bound and Mars-bound spacecraft follow a similar strategy: launch into low Earth orbit (LEO) first, then perform a second burn (TLI for lunar missions and TMI for Martian missions) to send the spacecraft on its way.