9
$\begingroup$

I'm particularly interested in understanding this beautiful solar system delta-v map and concepts on its Wikipedia page.

Solar System delta-V Map

In contrast to other planets there is no Earth Transfer only an Earth Capture/Escape.

Is there a difference between Transfer and Capture/Escape for the Earth? Is it the perspective of the map is entirely from the Earth POV and a different map would need to be calculated for Mars?

It would be helpful to include equations in your explanation.

TIA

$\endgroup$

1 Answer 1

10
$\begingroup$

To begin with, you are correct that the perspective of the map is from the Earth's point of view. I also believe that a different map would be needed for people interested in traveling from Mars to other places in the solar system besides the Earth, although lots of the values from the Earth-centric map could probably be reused for the Mars-centric map.

(I'll encourage you to ask your question about C3 = 0 in a separate post since SE's policy is to strive for one question per post.)

The values along the central spine of the map represent the amount of additional delta-v that is required to place the spacecraft on the elliptical Hohmann transfer orbit that will take it to another planetary body on the map. The additional delta-v values on the map for our nearest neighbors, Venus and Mars, are 280 m/s and 388 m/s respectively.

If the spacecraft is traveling to planets that are closer to the sun, then after expending the specified total delta-v to escape the Earth's gravity well and slow down (relative to Earth traveling around the sun), a spacecraft will be at roughly the aphelion of an elliptical orbit that will take it to the destination planet. The perihelion of that elliptical orbit will be the same as the orbital radius of the destination planet.

If the spacecraft is traveling to planetary bodies that are further away from the sun, then, after expending the delta-v to escape from Earth and speed up (relative to Earth traveling around the sun), the spacecraft will be at the perihelion of an elliptical orbit that will take it to the destination planet. The aphelion of that elliptical orbit will be the same as the orbital radius of the destination planetary body.

To determine the "transfer" values for planets that are further away, continue adding up the values along the map's central spine between Earth and the destination planet. For example, the additional delta-v to initiate a Hohmann transfer from Earth to Jupiter will be 388+923+379+3+397+1099=3189 m/s. Of course, it's not necessary (or even possible) to stop at all the planets along the way. The spine of the map is just providing you with the incremental differences in delta-v between Earth and the Hohmann transfer orbit that will get you to the next destination further away from Earth.

Once you reach the elliptical Hohmann transfer orbit's perihelion (for Venus and Mercury) or aphelion (for outer planetary bodies) then the spacecraft will need to expend more delta-v to circularize its orbit so that it will remain in the proximity of the destination planetary body. This is represented by the value between the " transfer" dot and the "escape/capture" tickmark on the map's branches. So, for Mars, this would be 673 m/s.

Now, you may have assumed that we can use basic orbital mechanics to directly calculate some of the delta-v values on the map if we know the Sun's gravitational parameter, μ, (which is 1.327 × 10^20 m3/s2), the orbital radius of the Earth (1.5 × 10^11 m), and the orbital radius the destination planet. For example, Mars' orbital radius is 2.28 × 10^11 m. (Let's also assume that the planet's orbits are circular for now). Let's try this...

The speed that the planets are traveling around the sun can be calculated by using

$$v = \sqrt{\mu/r}$$

The speed of the elliptical Hohmann transfer orbit can be calculated using the vis-viva equation

$$v = \sqrt{\mu(2/r-1/a)}$$

where 'a' is the semimajor axis which is the average of the orbit's aphelion and perihelion. Set 'r' equal to the aphelion if you want the speed at the aphelion and set 'r' equal to the perihelion if you want the speed at the perihelion.

You can get a set of delta-v values by calculating the absolute value of the difference between the planet's orbital speed and the speed on the elliptical orbit.

Earth's Orbital Velocity is 29743 m/s and Mars' orbital velocity is 24125 m/s. The semimajor axis of the elliptical Hohmann transfer orbit between them is 1.89E+11 m. The velocity at the perihelion of that orbit (obtained with the vis-viva equation) is 32668 m/s. A spacecraft orbiting the sun at Earth's orbital radius will need to accelerate from 29743 m/s to 32668 m/s. This requires a delta-v of 2925 m/s.

When the spacecraft drifts around to the aphelion of the Hohmann transfer orbit, it will be traveling at 21493 m/s but Mars will be traveling at 24126 m/s. So, to circularize its orbit to Mars's orbital radius, a change in velocity of 2633 m/s is needed.

But these numbers do not match up with the numbers on the map. We just calculated 2925 m/s for Mars transfer but the map says it only takes 388 m/s. Similarly, we calculated 2633 m/s for Mars capture/escape but the map says that it takes 673 m/s.

The reason for the discrepancy is that the map assumes that the burns will be executed when the spacecraft is deep within the nearest planet's gravity well, and thus will take advantage of the Oberth effect. The Oberth effect significantly lowers the amount of delta-v that the departing rocket needs to place itself on the Hohmann transfer orbit. It also lowers the amount of delta-v that the spacecraft needs for capture at the destination planet.

You can learn more about how to directly calculate the values on the map from the orbital mechanics wiki page. Here's an excerpt...

The Hohmann transfer orbit alone is a poor approximation for interplanetary trajectories because it neglects the planets' own gravity. Planetary gravity dominates the behavior of the spacecraft in the vicinity of a planet and in most cases Hohmann severely overestimates delta-v, and produces highly inaccurate prescriptions for burn timings. A relatively simple way to get a first-order approximation of delta-v is based on the 'Patched Conic Approximation' technique.

$\endgroup$
2
  • $\begingroup$ Thanks for the answer. The legend of the map specifically states, "burns at periapsis; gravity assist and inclination changes ignored", which would seem to exclude the Oberth effect. Also thanks for the link. $\endgroup$
    – Galerita
    Apr 1 at 5:20
  • 6
    $\begingroup$ Yes, that confirms what I was saying. I think the intent here (see semi-colon) is that "Burns at Periapsis" and "Gravity assist and inclination changes ignored" are separate statements. The legend is not saying "Burns at periapsis ... ignored". $\endgroup$
    – phil1008
    Apr 1 at 5:33

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.