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I have a time series of satellite position, $\mathbf{r}(t_i) = (r_i,\theta_i,\phi_i)$ representing geocentric radius, colatitude and longtiude at universal times $t_i$. I want to compute the Local Time of Ascending Node (LTAN) as a function of universal time, $$ LTAN(t_i) $$ My approach so far has been to simply analyze the half-portion of each orbit in which the satellite is ascending (e.g. $\theta_{i+1} < \theta_i$) and find when the satellite crosses the geographic equator ($\theta_i > 90^{\circ}$ and $\theta_{i+1} < 90^{\circ}$), and then calculate the local time using the longitude $\phi_i$ and universal time $t_i$.

This approaches works well enough, but has the disadvantage that it produces the same LTAN for the entire orbit, while the real LTAN is likely changing slightly throughout the orbit.

What would be a more correct way to compute $LTAN(t)$ given an ephemeris $(r_i,\theta_i,\phi_i)$?

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    $\begingroup$ For a generic satellite, I don't think there is a better answer than you already have, because every ascending node will be at a different local time. If you're talking only about a sun-synchronous satellite, then the local time at the ascending node is (nearly) constant, so it is a well-defined number that continues to apply to the same object even after many revolutions. In that case, you want to convert to Keplerian elements, and adapt the RAAN parts of Lagrange's planetary equations to work in LTAN, to monitor and station-keep your sun-synch to maintain the desired zero LTAN-dot. $\endgroup$
    – Ryan C
    Oct 10, 2022 at 21:29

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