I would like to calculate the velocity of an asteroid orbiting around a star (Sun) at the perihelion of its orbit. I know the excentricity of the ellipse and its semimajor axis.

I have found that the vis-viva equation is used to calculate the velocity of an object on an elliptical orbit and that the perihelion is at distance r = a(1-e). However I (simply enough) cannot see how to mathematically combine these two pieces of information in order to get the velocity at the perihelion.

(I am not so much looking for just a formula but rather a proof/intuition regarding how to get from the vis-visa equation for velocity to a perihelion velocity equation)

  • $\begingroup$ Generally, it is impossible without knowing the mass of the star. If you know that, just substitute r for (1-e)a in the formula ... after you do that, you'll notice that you know all parameters of the equation and can calculate the speed. $\endgroup$ – Tosic Nov 22 '19 at 8:59
  • $\begingroup$ @Tosic i kinda feel a little dumb now tbh haha, I know the mass of the star, I had everything this whole time... If you put that as an answer I'll accept it ! $\endgroup$ – A.D Nov 22 '19 at 9:16
  • 1
    $\begingroup$ BTW, when doing calculations involving GM it's more accurate to use the standard gravitational parameter than to use separate values of G & M. $\endgroup$ – PM 2Ring Nov 22 '19 at 10:12
  • 1
    $\begingroup$ @PM2Ring alright, glad to gain some knowledge ! Just a question though: how can the sgp be known to a greater accuracy than GM ? (I'm guessing it has to do with how it's calculated but...) $\endgroup$ – A.D Nov 22 '19 at 10:29
  • 1
    $\begingroup$ @A.D That Wikipedia article briefly explains: "Conversely, measurements of the smaller body's orbit only provide information on the product, μ, not G and M separately." Also see space.stackexchange.com/a/39930 It's really hard to measure G to high precision. It's generally done using some variation of the Cavendish torsion balance experiment. Also see Bending Spacetime in the Basement. $\endgroup$ – PM 2Ring Nov 22 '19 at 11:35

The vis-viva equation is commonly written like this:

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

For $r=a(1-e)$:

$$v = \sqrt{GM\left(\frac{2}{a(1-e)} - \frac{1}{a}\right)} = \sqrt{GM\frac{1}{a}\left(\frac{2}{1-e}-1\right)} = \sqrt{GM\frac{1}{a}\left(\frac{1+e}{1-e}\right)}$$.

The derivation of the vis-viva equation is not at all trivial and can be found here.

The product $GM$ is also called the standard gravitational parameter and for solar system bodies is often known more accurately than $G$ and $M$ separately. For the Sun $GM_☉$ is about 1.327E+20 m³ s⁻² which in different units is 1.327E+11 km³ s⁻² or about 1.0 AU³ year⁻².


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

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