Hohmann transfer ellipse to Mars orbit from LEO

[Raphael30]

Homework Statement

We want to send a satellite from a low Earth orbit of 320 km to mars. Calculate the change in velocity required to join the transfer ellipse.

Homework Equations

Earth velocity: (μ_S/R_EarthRev)^1/2
Transfer velocity at perihelion: (2μ_SR_MarsRev/(R_EarthRev(R_EarthRev+R_MarsRev))^1/2
Orbital velocity at LEO: (μ_E/R_LEO)^1/2
Escape velocity at LEO: (2μ_S/R_EarthRev)^1/2

The Attempt at a Solution

Ok so the Earth velocity is about 29780 m/s, the transfer velocity has to be 32730 m/s at perihelion (when it meets the Earth's revolution orbit). The orbital velocity is 7722 m/s and the minimal escape velocity is 10920 m/s. At minimal escape velocity, the maximal total departure velocity of the satellite from the sun's referential would be 40 700 m/s. However, this velocity drops as the satellite moves away from Earth, the escape velocity tending towards 0 and the total velocity tending towards 32780, all this because of the gain in potential energy. Counting the sun's attraction, this means the satellite would follow an orbit around the sun fairly similar to the Earth's. We could then suppose that the velocity needed for the satellite to escape the Earth's orbit has to be added somehow to the velocity needed to join the transfer ellipse. However, I have absolutely no idea how to combine the kinetic and potential energy of a system within another system and therefore I don't know when to sum kinetic energies, when to sum velocities directly, etc. I know that the final answer should be a gain in velocity of about 3600 m/s, far less than the 6138 obtained by summing the missing velocities directly and that this has something to do with some kind of Oberth effect, but I really can't seem to figure it out.

 

[Raphael30]

My attempt so far: The difference in energy between the Earth's revolution orbit and the transfer orbit is μ_S(R_MarsRev-(R_EarthRev)/(2R_EarthRev(R_MarsRev+R_EarthRev)). If we express the satellite's energy as the sum of the Earth's and the satellite's, the difference between the satellite's energy and the transfer orbit energy becomes μ_S(R_MarsRev-(R_EarthRev)/(2R_EarthRev(R_MarsRev+R_EarthRev)) + μ_S/2R_LEO. Also, knowing that (v+Δv)²/2=v²/2 + v Δv/2+ Δv²/2, I obtained ΔKE=v Δv/2+ Δv²/2. The problem is v is relative to the referential and I'm mixing up referentials. What I'm trying now is finding the ΔKE at LOE for any Δv, using v=v_LOE, then substracting the ΔKE required to escape the Earth's gravity and obtain ΔKE after the escape, for which we know v is the Earth's speed around the Sun and we want Δv=2950 m/s. It could definitely be made simpler but it seems to make sense...

 

[gneill]

I know that the final answer should be a gain in velocity of about 3600 m/s, far less than the 6138 obtained by summing the missing velocities directly...

Not sure how that's possible given that the ΔV for escaping LEO is already 3198 m/s. That would only leave a budget of 402 m/s to go from Earth orbit to Mars orbit.

 

[Raphael30]

That's what Wikipedia says at least: https://en.wikipedia.org/wiki/Hohmann_transfer_orbit. Too bad they don't explain how to get such low Δv.

 

[gneill]

I don't have a terrible lot of faith in Wikipedia content. I'll plug away at the ΔV's for a bit to see if it's plausible, but I don't hold much hope.

 

[Raphael30]

From what I see everywhere, the way people solve this is: 1) use conservation of energy to find an equation for the remaining relative velocity as the distance from the Earth tends towards infinity for any burnout velocity greater then escape velocity 2) invert this to have the burnout velocity as a function of the remaining velocity: v₀=(v_rem² + v_esc²)^0.5 3) use v_p-v_E as the remaining velocity 4) substract the orbital velocity from the burnout velocity to find the necessary push. With our values and this method, the result is slightly below 3600 m/s. The method clearly requires corrections but it sounds like a pretty decent approximation.

 

[gneill]

Yes, I came across similar workings when I went exploring the web.

Once I realized that the "patching conic" going from LEO to the Hohmann transfer orbit is a hyperbolic trajectory, the method becomes clear.

For any hyperbolic orbit,
$v^2 = v_{esc}^2 + v_\infty^2$
Where:

$v$ is the velocity on orbit
$v_{esc}^2$ is the local escape velocity
$v_\infty^2$ is the hyperbolic excess velocity (remaining speed at infinity)
​

Now, we want $v_\infty$ to be the difference between the transfer orbit speed and the Earth's orbital speed (so when the object has "escaped" from the Earth it will have the speed required to be on the Hohmann transfer orbit). Find $v$ for the LEO position and then the required $\Delta V$ is the difference between it and the LEO orbit speed.