Is software available for writing two line element files from Keplerian elements?

Question

I am interested in writing a TLE file from existing Keplerian elements. Is software available on the web to do this?

Answer 1

Keplerian orbital elements and the mean orbital elements used in the simplified perturbation models are rather different beasts. Keplerian elements assume a spherically symmetric central body and no perturbations from other bodies, atmosphere, etc. The simplified perturbation models account for those perturbations (non-spherical Earth, third body accelerations, atmospheric drag, radiation pressure) in a simplified manner. The one thing the two concepts do have in common is that the computational cost of computing position and velocity at some point in time is independent of the time difference between epoch and the point in time; neither approach uses numerical integration.

The way two line elements are created is via a number of measurements of some indicator of satellite state that are spread over time. The numerical values in the TLE are adjusted via an orbit determination process so as to minimize in a least squares sense a weighted sum of the squared errors between measurement and simplified perturbation model prediction. An inverse mapping from measurement (or from Cartesian states, or from Keplerian elements) to mean elements is not needed.

That said, people have tried to do this. An outline of an algorithm:

1. Compute Cartesian state (position and velocity) from your Keplerian orbital elements.
2. Transform that Cartesian state to the True Equator Mean Equinox frame. As far as I can tell, the US Air Force is the only entity that uses TEME. Your Keplerian elements are almost certainly in some other ECI frame (there are a bunch of them).
3. Form an initial guess of the TLE set by
   • Setting the epoch time to the time of the Keplerian elements.
   • Setting the TLE orbital elements to the TEME Keplerian elements computed from the TEME position and velocity.
   • Setting B* to zero or to some informed guess.
4. Repeat until converged:
   1. Compute Cartesian state from the TLE set.
   2. Compute the differences between the target position and velocity versus the position and velocity computed from the TLE set.
   3. Combine the error vectors to form a scalar error value. For example, $||\Delta \boldsymbol{x}||^2/||\boldsymbol{x}||^2 + ||\Delta \boldsymbol{v}||^2/||\boldsymbol{v}||^2$.
   4. Exit the loop if the error is sufficiently small.
   5. Estimate the Jacobian between the TLE elements and the Cartesian state.
   6. Use the Jacobian from step 4.5 and the error vectors from step 4.2 to compute an updated TLE set. Care might be needed here, particularly for nearly circular or nearly equatorial orbits. You might want to use singular value decomposition or ANOVA so as to only attack the statistically significant elements.