Help calculating interplanetary transfers

[Mad Rocket Scientist]

I've been trying to calculate a transfer from earth to venus for my blog, but I'm stuck.

Here's what I have so far:

Source for constants:

https://nssdc.gsfc.nasa.gov/planetary/factsheet/earthfact.html
https://en.wikipedia.org/wiki/Standard_gravitational_parameter

A transfer orbit between Earth and Venus depends on the position of the planets at departure and arrival. For this example, the position of Venus does not matter, as aerobraking will work at either of those speeds.

Earth should be at apoapsis, because in this case we want to lose, rather than gain, velocity as Venus is in a lower orbit than Earth.

Let's begin. First, the velocity of Earth around the Sun at apoapsis. We don't really need to calculate this, as it is well known: 29290 m/s.

The transfer orbit is an elliptical orbit with its apoapsis at Earth's apoapsis, and its periapsis at Venus' apoapsis (I think this gives the lowest delta-v). This gives a semi-major axis of:

(Radius of Earth's orbit at departure + Radius of Venus' orbit at arrival)/2.

Or (152.1 * 10^9 m + 108.94 * 10^9 m)/2 = 130.52 * 10^9 m.

Velocity at apoapsis:

v^2 = 1.327 * 10^20 (2/152.1 * 10^9 - 1/130.52 * 10^9)

v = 26985.2 m/s

Velocity at periapsis:

v^2 = 1.327 * 10^20 (2/108.94 * 10^9 - 1/130.52 * 10^9)

v = 37676.3 m/s

The velocity of the spaceship in Earth orbit can be calculated from the altitude, and the altitude is as low as possible to take maximum advantage of the Oberth effect. I'm going to guess around 200 km.

v^2 = 3.986 * 10^14 (2/6578000 - 1/6578000) 

v = 7784.34 m/s.

Up until here everything seems fine.

Now we subtract the spacecraft's orbital velocity from the earth's orbital velocity: 21505.66. We subtract because we need to lose velocity.

But now it seems like we have too slow an orbit even before the escape burn, which seems unlikely to me. I can't find anything that could help me here anywhere else. Any ideas on where I messed up?

 

[Bill Phil]

I'm pretty sure you shouldn't subtract orbital velocity of the craft from the apoapsis velocity... For one thing you have to get to escape velocity, and C3=0 basically puts you into the same orbit as Earth, give or take a bit. You have to escape Earth first, then subtract leftover velocity.

So you have to be going about 11 km/s. Subtract 7784 m/s, and you get the delta v to just get to a solar orbit that is close to Earth's. Now you have to add the remainder. At transfer apoapsis, velocity is about 27 km/s, but you have a velocity (Earth's) of 29.29 km/s. Subtract these and you get the next amount.... assuming that it's that simple.

[Gaarst]

Hyperbolic excess velocity is what's killing you.

Basically you can't add your orbital velocity around a body to the body's orbital velocity around its star to obtain your velocity around the star, because you need to escape the gravity well of the planet first. What you need to do is add your orbital velocity when you have escaped the planet's gravitational influence, ie: at infinity, which is exactly what the hyperbolic excess velocity (HEV) is.

For example: if you reach exactly escape velocity around your planet, your HEV will be equal to zero: you will escape the planet's influence but you will have exactly zero energy left to alter your interplanetary trajectory.

Computing the HEV is quite simple, it is given by:

HEV² = v² - v_e²

Where v is your orbital velocity and v_e the escape velocity.

For your example the velocity of the Earth around the Sun is 29290 m/s and the velocity at the apoapsis of your transfer orbit is 26985 m/s, this means that your HEV will be equal to 2305 m/s.

A circular orbit at 200km has an orbital velocity of 7784 m/s. By definition, the escape velocity at that altitude is the circular orbital velocity times √2, then v_e = 11008 m/s.

From the equation above, your orbital velocity at 200km after your escape burn will be:

v² = HEV² + v_e² = 2305² + 11008²
v = 11246 m/s

This means that your escape burn will be about 3462 m/s, or about only 250 m/s over the escape velocity.

Thanks to the Earth's gravity well giving you tons of gravitational energy, a 3462 m/s burn in Earth orbit is equivalent to a ejection burn of about 3200 m/s in orbit plus a 2300 m/s burn once you've escaped Earth.

 

In your original reasoning, besides neglecting HEV, you've also assumed that the 7784 m/s of orbital velocity you have at 200 km would be translated into a 7784 m/s difference in orbital velocity once you've escaped the Earth. Thing is you forgot to account for the fact that you are loosing speed as you get away from the Earth because the latter is pulling you towards it: you're fighting against gravity so you change kinetic energy (speed) to potential energy (distance to Earth). Eventually you're only left with the HEV which is the important number to consider for interplanetary transfers.

 

(In reality things will be a bit off because you're never getting infinitely far away from the Earth: you remain at a given distance from it, but its gravitational effect on you becomes smaller as you get far away because the Sun has a greater influence on you.)

[Mad Rocket Scientist]

Ah, that makes sense. Thanks!