# Is the amount of accurate information in a TLE equal to that of a single state vector?

My question is basically regarding whether there is orbital information "lost" when converting from a Two Line Element set (TLE) to a state vector.

Given that the following are true:

• I have a TLE, which stores the orbital elements of an earth-orbiting object
• I can take that TLE and determine a specific orbital position for any given time (a state vector)
• A state vector can be used to determine orbital elements.

Is it true to say that:

I can take a TLE for my spacecraft, generate a single state vector for a particular time (perhaps the TLE epoch), and hand it to my friend. My friend can take that state vector and generate a TLE and have it be just as "good" as my TLE.

If not, why? Do I need multiple state vectors? How many are equivalent to the amount of accuracy given in a TLE?

• Could you please edit your question to make it even more excellent by translating, just once, TLE into human language? Many people who have an interest, and the mathematical knowledge to follow question and answer, are going to be prevented from doing so by not knowing what a Trans-Lunar Experience is. Aug 14 '19 at 7:04
• @MartinKochanski I've made an edit since I'm here at the moment. Has it improved your Trans-Learning Exchange experience? ;-)
– uhoh
Aug 14 '19 at 8:23
• btw, in TLE, mean motion(revolution per day). Here day is 24 Hrs or sidereal day ? Aug 14 '19 at 11:08
• @Prakhar I don't have a source for you at the moment, but I have always assumed it is 24 hours because that's based on the second which is well defined. A sidereal day drifts because the rotation of the Earth isn't steady. update: Aha! I do have a source for you! See the last sentence in the block quote in How to obtain UTC of the epoch time in a satellite TLE (two line element)? which links to Celestrak's FAQ #3
– uhoh
Aug 14 '19 at 16:41
• Oh, that was a wierd question and even for that you got an answer ! Amazing. Aug 14 '19 at 17:28

You definitely have the right idea and understand some of the various ways to propagate orbits:

• Keplerian orbits
• TLEs with SGP4
• numerical state vector propagation

But all of these turn out to be approximations.

• Keplerian orbits are conic sections and assume two bodies (point masses or spherically symmetric mass distributions) and nothing else. An elliptical orbit will execute over and over without any changes.

• Propagating TLEs with SGP4 is way more accurate because SGP4 contains a pretty good model of the main bumps and wiggles of Earth's gravity field, and a pretty good approximation of aerodynamic drag. If you are in a high orbit (period greater than 225 minutes) then the "deep space correction (SDP4)" will also implement a pretty good approximation of the average effects of the Sun and Moon's gravity on the orbit as well. For more on that see the (currently unanswered) question How do “Deep space” corrections in SGP4 account for the Sun's and Moon's gravity?

However all of the pretty good approximations are just that, and no more. So while SGP4 can spit out state vectors with a dozen decimal places, none of those are "correct" so starting with one of them and propagating with a fancy orbit propagator will always have some significant amount of error.

• "fancy" numerical state vector propagation is what is done when the best possible accuracy is required. It will have a finely detailed model of Earth's gravity field, up-to-the-minute space weather predictions to model Earth's atmosphere (and the spacecraft's aerodynamic characteristics) for a more detailed estimation of drag, and have the exact positions of the Sun and Moon and calculate their gravitational influences in detail as well.

But "garbage-in, garbage-out" still applies. If your initial state vector is of limited accuracy, then integrating from there will not be more accurate, and over time the differences can diverge in some cases.

Is it true to say that:

I can take a TLE for my spacecraft, generate a single state vector for a particular time (perhaps the TLE epoch), and hand it to my friend. My friend can take that state vector and generate a TLE and have it be just as "good" as my TLE.

This particular point is wrong because one single state vector contains no information on drag. You could have a 1000 kg ball of gold and a giant balloon with the same state vectors, but the balloon would de-orbit quickly while the solid gold ball would loose altitude extremely slowly due to its high mass/area ratio.

The way TLEs are made in the real world is a bit complicated. Roughly speaking, predictions of orbits from "fancy" models are first fit to all the observational data, probably meany radar and optical measurements over time. Then TLEs are generated by varying their input parameters in order to best match the fancy predictions.

You can just keep varying each parameter in your home-made TLE, then propagating each time with SGP4, in order to converge on a good fit to the prediction of your fancy model.

But they will never match completely because of all of the "pretty good" approximations in SGP4 that are not in your fancy model, and if you don't use a sufficiently long timespan in the fitting procedures for one model to see how the other model is loosing altitude due to drag, then your fitted TLE will be wrong.

• TLDR; a TLE contains more information than a state vector.
– user20636
Aug 14 '19 at 18:50
• @JCRM and a simultaneously less, since they are designed to produce "pretty good" trajectories using SGP4 rather than "pretty accurate" trajectories using a "fancy" propagator.
– uhoh
Aug 15 '19 at 0:59

You should be able to convert a TLE (element set) to a state vector (Cartesian XYZ and velocity) and back to TLE without loosing possitions information, provided you also provide bstar (drag coefficient). Going forward is easy/standard, going back a little bit less common/might require running an optimiser if you can't find a tool.

Uhoh gave a good summary on the subtle differences and losses, but missed a big one: the mass of the central body. The same state vector applied to a craft in orbit around a different body will result in vastly different TLE.

This is rarely of significance because what body is being orbited (and by extension, it's mass) is hardly ever an unknown parameter - but since TLE describe the geometry of the orbit (independent of underlying physics that cause it), and state vector is all about initial conditions in a physical model, the latter must know the mass of the planet while in the former it's indirectly encoded in the geometry - physics model correspondence.

• +1 This is totally right. TLEs proper never apply outside of bound orbits to Earth for just this reason since they are issued only for use by SGP4 and SGP4 has so many specific details of Earth hard-wired into the code in so many places that it would never work around another planet. The emphasis of Has a TLE ever been issued for a spacecraft trajectory not bound to Earth orbit? was on the "bound" part, but the answer to the "to Earth" part is just as much "no".
– uhoh
Jul 23 at 8:29