How to find equatorial orbit pericenter position if I know $\mu$, $a$, $e$, excentricity anomaly (observed at current moment), force focus position (but don't know empty focus position) and body position and velocity at current moment?

Using kepler's laws. I see how to do it for orbit with inclination but in my case orbit lays in the ecliptic plane.

  • $\begingroup$ What is "excentricity anomaly"? Do you mean eccentric anomaly? $\endgroup$
    – PM 2Ring
    Oct 7, 2021 at 10:32

1 Answer 1

Given parameters Symbol
radial distance vector $\vec{r}$
veloicity vector $\vec{v}$
standard gravitational parameter $\mu$
semi-major axis a

1. Calculate the specific angular momentum vector, $\vec{h}$

Specific angular momentum is the cross product of the radial distance and velocity vectors, and is useful for finding many of the other Keplerian paramenters, both when modeling 2d and 3d orbits.

$$\vec{h} = \vec{r} \times \vec{v}$$

2. Calculate the eccentricity vector, $\vec{e}$

The eccentricity vector is a vector with the magnitude of the orbital eccentricity, that points from the center of the object being orbited, towards the pericenter.

$$\vec{e}=\frac{\vec{v} \times\vec{h}}{\mu} - \frac{\vec{r}}{|\vec{r}|}$$

It can also be calculated without using the angular momentum vector: $$\vec{e} = \left(\frac{|\vec{v}|^2}{\mu} - \frac{1}{|\vec{r}|}\right)\vec{r} - \frac{\vec{r} \cdot\vec{v}}{\mu} \vec{v}$$

3. Calculate the pericenter distance, $q$

For elliptical orbits ($e<=1 ,a > 0$), and hyperbolic trajectories ($ e > 1 ,a < 0$) $$q = a(1-|\vec{e}|)$$

For parabolic trajectories ($e = 1, a = \pm \infty$): $$q = \frac{|\vec{h}|^2}{2\mu}$$

Circular orbits don't have a well-defined pericenter.

4. Calculate the pericenter position vector, $\vec{q}$

The pericenter position vector, $\vec{q}$, has magnitude of the pericenter distance $q$, in the direction of the eccentricity vector.

$$\vec{q} = q \frac{\vec{e}}{|\vec{e}|}$$

  • $\begingroup$ What is about a 2D case where cross product is a scalar value? I don't have an h vector, just a scalar. Or I can use 3D vectors with z = 0, in this case h vector will be (0; 0; h_scalar) $\endgroup$
    – Robotex
    Oct 7, 2021 at 10:10
  • $\begingroup$ I've added a calculation for eccentricity vector that does not use the angular momentum vector, from the linked wikipedia article. $\endgroup$
    – notovny
    Oct 7, 2021 at 10:23
  • $\begingroup$ Thank you very much. I will try to use this calculations when I will be near the laptop $\endgroup$
    – Robotex
    Oct 7, 2021 at 10:28
  • $\begingroup$ Thank you, this formula is working. But the vector points from pericenter to apocenter. I don't know why $\endgroup$
    – Robotex
    Oct 13, 2021 at 9:26
  • $\begingroup$ Oh, looks like it's just a bug in visualisation $\endgroup$
    – Robotex
    Oct 13, 2021 at 9:35

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