Calculating specific dates of reaching apses given orbital elements?

Given a specific date and heliocentric orbital elements ($$a, e, i, \Omega, \omega$$, mean anomaly) of a spacecraft in a heliocentric orbit, how can we calculate when it will reach perihelion and how far that will be from the Sun?

• I've made a small edit to your question mostly for formatting, but added "heliocentric" to the orbit since you've asked about distance to the Sun. Please feel free to edit further. If you have any example of orbital elements that you'd like to see converted and can add them to your question, that might make the question and its answers more interesting. Just a thought. – uhoh Nov 10 '20 at 0:12

You're in luck that you have the mean anomaly, that makes the calculations much easier. The mean anomaly is the angle expressing the fraction of the orbital period that has been covered since perihelion (and not the "real" angle between perihelion, the Sun and the object. That's the true anomaly and harder to calculate). So given a mean anomaly in radians, the time since perihelion is:

$$T_{since\ perihelion} = \frac{M}{2\pi} \cdot T$$

Where $$M$$ is the mean anomaly and $$T$$ is the orbital period. You want the time the opposite way around, so some tiny modification is needed:

$$T_{until\ perihelion} = \left(1 - \frac{M}{2\pi}\right) \cdot T$$

This still requires the orbital period though, which you can get from:

$$T = 2\pi \sqrt{\frac{a^3}{\mu}}$$

Where we have yet another dependency, $$\mu$$, the gravitational paramater of the Sun, which is the mass of the Sun multiplied by the Newtonian gravitational constant. $$\mu = M_\odot G$$

That should be everything required for the time part. Getting the perihelion distance is much easier:

$$r_{perihelion} = a\cdot (1-e)$$

And if you happen to need the aphelion too:

$$r_{aphelion} = a\cdot (1+e)$$

None of these calculations should require $$i$$, $$\Omega$$ and $$\omega$$ since they only affect the orientation of the orbit.

• What if instead of the perihelion distance, we want to find a specific date when it is as a specific distance from the Sun? – user38230 Nov 11 '20 at 5:49
• @pmatt9356 Then you would need to to true anomaly calculations. – SE - stop firing the good guys Nov 11 '20 at 7:51