Need help with Elliptical Orbits, Velocity Change, and general Astrophysics.

[-ctn-]

Okay, I tried searching for an answer but didn't find anything close to what I'm looking for.

To back up and explain a little - I'm learning some basic Astrophysics and dynamics because spaceflight is awesome and learning is fun. I was never any good at math in college or highschool, so if I ask a dumb question or have a dumb answer, don't get too frustrated.

Anyways, here's what my problem is. A spacecraft in an elliptical orbit has a change in velocity. The closer to the parent body you are, the faster you go. The farther away, the slower you go. In a circular orbit, your velocity is constant because you are the same altitude (or distance) from the parent body for the entire orbit.

How would I calculate this change in velocity in an elliptical orbit?

As an example, I'll use KSP. Let's say we have a spacecraft traveling at 6 km/s on a transfer trajectory from Kerbin to Duna. When leaving Kerbin's Sphere of Influence, the craft is going at 6 km/s. By the time it reaches Duna's Sphere of Influence, the velocity will have slowed quite a bit - because of the gravitational pull (I assume) of being in the Sun's Sphere of Influence for the majority of the transfer.

Is there a way to calculate:

A) What the final speed will be when approaching Duna's SoI

Is there a set rate at which the speed decreases (assuming it takes into account distance from the Sun and the Sun's gravitational parameter)

C) If I pick a point in an elliptical orbit, and I know - the original velocity, distance from and gravitational parameter of the parent body - is it possible to determine the exact velocity of the craft? (i.e. to plot a graph with time intervals)

Thanks in advance for any help!

[Bill Phil]

If you know your semi major axis, you can determine speed at any altitude along your orbit.

Look up the Vis-viva equation.

Just keep in mind that vertical velocity is zero at any Apse(apo, peri), and that it increases when you approach Apo and decreases as you go away. That's vertical alone, though.

Edited March 6, 2015 by Bill Phil

[K^2]

Just to clarify this a bit. Total mechanical energy of a satellite in a n elliptical orbit is -GMm/(2a), where G is gravitational constant, M and m are masses of the two relevant bodies, and a is the semi-major axis. (This assumes M >> m.) The potential energy of the satellite at some distance r from the center is -GMm/r. The difference between these two energies is kinetic energy of the craft, which is mv²/2. Solve that for v and you will have the speed of the craft.

Getting orientation is equally straight forward, but you use conservation of angular momentum instead. The quantity v_tr, where v_t is the tangent velocity, is conserved. Furthermore, precisely at apsides v = v_t. You can then get radial velocity from speed and tangential velocity using Pythagoras' theorem.

[PB666]

You can remove m and convert this to e/m for the satellite, you can also solve for thermodynamic energy change by taking the integral of the mgh equation from r₁ to r₂ (altitudes + CBs radius) to get velocity. So it can be done either way depending on what information one has. I think it is GM/r₂ - GM/r₁ (been 35 years so I might have gotten partially wrong)

[Superfluous J]

You can quickly determine your velocity at any point in your orbit in-game by dropping a maneuver node and pulling the retrograde marker until your predicted path is falling directly into the thing your're orbiting. However much dV that maneuver node takes, is how fast you were going.

I hate calculating anything

[PB666]

You can quickly determine your velocity at any point in your orbit in-game by dropping a maneuver node and pulling the retrograde marker until your predicted path is falling directly into the thing your're orbiting. However much dV that maneuver node takes, is how fast you were going.

I hate calculating anything

We need a thread on how to calculate stuff.

[maltesh]

Okay, I tried searching for an answer but didn't find anything close to what I'm looking for.

To back up and explain a little - I'm learning some basic Astrophysics and dynamics because spaceflight is awesome and learning is fun. I was never any good at math in college or highschool, so if I ask a dumb question or have a dumb answer, don't get too frustrated.

Anyways, here's what my problem is. A spacecraft in an elliptical orbit has a change in velocity. The closer to the parent body you are, the faster you go. The farther away, the slower you go. In a circular orbit, your velocity is constant because you are the same altitude (or distance) from the parent body for the entire orbit.

How would I calculate this change in velocity in an elliptical orbit?

As an example, I'll use KSP. Let's say we have a spacecraft traveling at 6 km/s on a transfer trajectory from Kerbin to Duna. When leaving Kerbin's Sphere of Influence, the craft is going at 6 km/s. By the time it reaches Duna's Sphere of Influence, the velocity will have slowed quite a bit - because of the gravitational pull (I assume) of being in the Sun's Sphere of Influence for the majority of the transfer.

Is there a way to calculate:

A) What the final speed will be when approaching Duna's SoI

Is there a set rate at which the speed decreases (assuming it takes into account distance from the Sun and the Sun's gravitational parameter)

C) If I pick a point in an elliptical orbit, and I know - the original velocity, distance from and gravitational parameter of the parent body - is it possible to determine the exact velocity of the craft? (i.e. to plot a graph with time intervals)

Thanks in advance for any help!

Specific positions, yes. Specific time intervals.... that gets a bit hinky. Note: Use radians with all the below equations, not degrees.

Given a specific point on your orbit, determining your time to or since periapsis is plug and and play. Solve for True Anomaly (theta) from current radial distance from the sun's center®, semimajor axis (a), and eccentricity(e).

Calculate Eccentric Anomaly (E) from True Anomaly. (Fixed equation, 3/11)

Calculate Mean Anomaly (M) from Eccentric Anomaly.

And from Mean Anomaly and Orbital Period (p), you can find the time since Periapsis (t_p).

Position along an elliptical orbit as a function of time runs into the issue that there's no closed-form equation for going from Eccentric Anomaly to Mean Anomaly. I tend to use Newton's Method when going in the opposite direction.

Edited March 11, 2015 by maltesh
Had the wrong equation for Eccentric Anomaly

[Rosco P. Coltrane]

BTW, just wanted to add something. Keep in mind that these are point masses. So if you want to know the speed you will have at 80 km above Duna in your orbit, you don't plug 80000 as a value for r into the calculations, you plug 80000 + Duna's radius. At least when you're close to it, if you are very far away from the body you can get away with ignoring it because the error introduced will me small.