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I know that thanks to the Vis-Viva equation, knowing the elliptic parameter of a celestial orbiting object we can get its speed on any point of the orbit.

But what about the acceleration? Is there an equivalent famous formula?

Thanks for your help

Vincent ISOZ
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    where is the “t” parameter in the Vis-Viva equation? I don’t see it to make the derivative…. – Vincent ISOZ Dec 07 '23 at 16:57
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    Vis-viva gives velocity from distance, v = sqrt(GM(2/r - 1/a), so with no further clarification, it seem you want acceleration from distance? But that's just Newton's law of gravity... – SE - stop firing the good guys Dec 07 '23 at 18:32
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    Newton law of gravity doesn’t just give the centripetal acceleration? – Vincent ISOZ Dec 07 '23 at 20:30
  • Yeah, I guess I should've thought more about what acceleration means in this context -- I was thinking derive with respect to distance, but if you do that you're going to end up with just gravity. And that makes sense -- naturally, if you're in an orbit, then the only acceleration is gravitational. – Darth Pseudonym Dec 07 '23 at 20:31
  • Then is there a formula to get the tangential acceleration to the orbit? – Vincent ISOZ Dec 07 '23 at 20:33
  • @VincentISOZ At any given point in the orbit, the only thing accelerating the body is the gravitational attraction (leaving aside tiny forces like light pressure and drag from stray gas molecules). Remember, an orbit is just falling down while moving forward fast enough to keep missing the planet. – Darth Pseudonym Dec 07 '23 at 20:36
  • The tangential acceleration at any point is just a trig function based on the gravitational attraction and the angle between the ship's direction of travel and the center of the planet it's orbiting. – Darth Pseudonym Dec 07 '23 at 20:39

1 Answers1

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Equivalent Vis-Viva equation but for acceleration

Comments are already mostly conclusive, I'll stick my neck out and answer a bit further.

Let's see what "equivalent" might mean to the OP:

I know that thanks to the Vis-Viva equation, knowing the elliptic parameter of a celestial orbiting object we can get its speed on any point of the orbit.

But what about the acceleration? Is there an equivalent famous formula?

Well for equivalently famous I'd say no, otherwise either the question wouldn't be asked or it would have been insta-answered.

And for equivalently useful/powerful I'd say it would have then become equivalently famous, so refer to the no above.

Below I will talk about the intimate connection between the vis-viva equation and fundamental conservation laws. (for example, $v^2$ is just a way to get at kinetic energy per unit mass)

To my knowledge there are no equivalent fundamental conservation laws associated with acceleration. So like I said, I'm going to stick my neck out and say a flat no to an equivalent acceleration-based equation.

There's one total energy that sloshes back and forth between two types; kinetic and potential. That's the basis of the vis-viva equation. But there's always just one forces on an orbiting particle. So no sloshing, and no dice!

As @ comments (rhetorically I assume)

Newton law of gravity doesn’t just give the centripetal acceleration?

Yes it does.

$$\mathbf{a} = \frac{\mathbf{F}}{m} = \frac{-GM}{m} \ \frac{\mathbf{r}}{|r|^3}$$

But let's look more closely at the famous vis-viva equation. We could just say it gives us velocity, or more accurately, it gives us $v^2$ so actually speed rather than a velocity vector proper.

But what is this equation actually? It's about conservation of energy and momentum.

If we use specific orbital energy (total kinetic + potential energy per unit mass, or just set mass=1). The $v^2$ term is the instantaneous kinetic energy, the $-GM/r$ term is the instantaneous potential energy, and the $2GM/a$ is the constant total energy.

So it's really just saying $\text{KE} = \text{E}_{tot} - \text{PE}$.

The particular derivation given in the current version of the Wikipedia article requires conservation of angular momentum to show that $\text{E}_{tot} = -GM/a$ and the expression works for bound elliptical orbits.

$$v^2 = GM\left(\frac{2}{r} - \frac{1}{a} \right)$$

Ellipses and hyperbolae are real-world orbits with variable eccentricities, while circles and parabolas are mathematical curiosities with eccentricities of exactly 0 and 1 which don't really happen (probability is zero).

For hyperbolae there's just a sign change for the total energy (since for the unbound orbit it's now positive), thus $\text{E}_{tot} = +GM/a$ and

$$v^2 = GM\left(\frac{2}{r} + \frac{1}{a} \right)$$

Note the sign change for hyperbolae.

For a circle it's just $$v^2 = GM\frac{1}{a}$$ and for a parabola it's just $$v^2 = GM\frac{1}{r}$$

For more on this see

uhoh
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