As a simple project to introduce myself to orbital mechanics I'm attempting to calculate impact times of an object close to Earth. Right now I am using only Keplerian mechanics (no air resistance or other perturbing forces). From Fundamentals of Astrodynamics, given an object with eccentric anomaly $E_0$ at $t_0$ and $E_1$ at $t_1$:

$$t_1 = t_0 + \sqrt{\frac{a^3}{\mu}}\left( E_1 - e\sin E_1 - (E_0 - e\sin E_0)\right)$$

where $a$ is the semi-major axis, $e$ is the eccentricity, and $\mu$ is the gravitational parameter. We can then calculate the radius of the periapsis. Assuming a spherical Earth, if this is less than the radius of the Earth the object will impact. If this is the case, letting $r_1 = R = $ Earth radius, we can calculate the eccentric anomalies at $t_0$ and $t_1$ using:

$$ \cos\nu_i =\frac{a(1 - e^2) - r_i}{e r_i}$$ $$ \cos E_i = \frac{e + cos\nu_i}{1 + e\cos\nu_i}$$

where $\nu_i$ is the true anomaly, and $r_i$ is the distance from the centre of the Earth at $t_i$. This method works well for non-parabolic trajectories (I'm open to modifications for parabolic trajectories).

The next step is to loosen the restriction of a spherical Earth and allow Earth to be an ellipsoid, in other words let $R = R(z)$ (symmetric around rotational axis). Specifically I'd like to allow for a bulge at the equator by prescribing an equatorial radius, a polar radius and an eccentricity. Is it possible to solve this problem analytically or does it have to be done numerically?

This is my first post on this stack exchange; I felt it was better suited here than scicomp for example. If that's not the case please let me know.

  • $\begingroup$ A near-Earth object will be on a hyperbolic trajectory with respect to Earth, so you can't use these equations, which are for something in orbit around Earth. $\endgroup$
    – Mark Adler
    Oct 8, 2015 at 3:45
  • $\begingroup$ Not necessarily, you could have something like an ICBM for example. $\endgroup$ Oct 8, 2015 at 6:30
  • $\begingroup$ As an aside, do you feel that book is worth picking up? I focused on history, law, management, etc. in my space related degree (with little focus on else), but I'm always interested in branching out. $\endgroup$
    – spacer
    Oct 8, 2015 at 11:44
  • $\begingroup$ Yeah definitely! It's really well written and it covers a wide range of topics from basic orbits to interplanetary trajectories. Each section goes over the history of the methods being used; something I found really interesting. The only downside is that it was written in the 1970s, so it's somewhat out of date, but hey, it was good enough to get us to the moon. $\endgroup$ Oct 8, 2015 at 12:25
  • 1
    $\begingroup$ I had assumed that by near-Earth object, you meant a Near-Earth Object. $\endgroup$
    – Mark Adler
    Oct 8, 2015 at 20:33

1 Answer 1


In the general case you will need to solve numerically for the radii at which the orbit intersects the ellipsoid. Then you can solve analytically for the times at which the orbit is at those radii. That is, if you ignore $J_2$. Since you made it an ellipsoid, you have introduced a $J_2$, so you will no longer be in a Keplerian orbit. To take $J_2$ into account, the times would then also need to be solved for numerically.


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