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The Celestrak website provides information on reading the TLE ( Two-Line Element Set ) format. In Line 1, Column 34-43 & 45-52 give information about First Time Derivative of the Mean Motion and Second Time Derivative of Mean Motion.

Is there any way to calculate or estimate these two parameters for a spacecraft, if you don't have its TLE at disposal to simply read them from there ? And what do they actually mean ?

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    $\begingroup$ Could you elaborate a little bit - what are you really after. Time derivative of motion = velocity? And second derivative = acceleration? You would need the forces acting on the body - propulsion, gravity. But is sense this is not what you are asking? $\endgroup$
    – Floris
    Commented Aug 16, 2014 at 1:35
  • $\begingroup$ @Floris - This may be just what I am asking, and it was my first idea ( 1st der. of MM = velocity, 2nd der of MM = acceleration , as you said). But it somehow sounded too simple to me, because SGP orbit propagator, that uses TLE, uses a rather complex model. Anyway, with this particular velocity and acceleration, we are speaking about degress or radians per time unit, right ? $\endgroup$
    – James C
    Commented Aug 16, 2014 at 1:46
  • $\begingroup$ It seems to me that the units are "orbits" - 2 pi radians per unit time (day?). The real motion involves lots of small corrections due to the moon, atmospheric drag, etc - which is why it is hard to give a "good" answer without knowing how you intend to use this (and why you can't use the published data...)? $\endgroup$
    – Floris
    Commented Aug 16, 2014 at 2:05
  • $\begingroup$ @Floris - Ah yes, "orbits" per unit time make sense. How I want to use it is simple, yet not obvious from my question. I want to write simple satellite tracking and visualization software and use the SGP propagator, and I will also do "experiments" with not actually existing satellites. That means I will have to produce some TLE data. $\endgroup$
    – James C
    Commented Aug 16, 2014 at 9:45

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The mean motion $n$ of a satellite is its angular velocity, averaged over one period. In other words, if the satellite rotates around the Earth with period $P$, its mean motion $n$ is $$ n = \frac{2\pi}{P} $$ If the Earth were a perfectly spherical symmetric object, and there were no other perturbing agents in the Universe (in other words, if Earth + satellite were a closed system), then $n$ would be an exact constant of motion.

However, several effects are not included in a two-body, idealized problem, and they make $n$ time-variable. One such effect is the Earth's oblateness, which makes the gravitational potential differ from a simple $1/r$ law. Another effect is the presence of the Moon, which also perturbs the satellite motion. If, instead of a human-built satellite, we were discussing the motion of moons around their planets, other perturbing agents would the gravitational influences of nearby planets and of the Sun.

The fact that $n$ is not constant in time is clearly seen in accurate positional data extending over many orbits. Fitting the ensemble of these data yields both a first and a second derivative, to allow more precise predictions of the satellite position in the nearby future. Since however the variation of $n$ is not subject to any simple law, it needs to be recomputed all the time.

So, in short, the answer is no, you cannot derive those parameters from data in your possession. In order to be able to do that, you should own the whole set of position measurements over very many orbits which is surely in the hands of both civilian and military space agencies.

If you wanted instead to compute from first principles what those parameters ought to be, you will have to study the importance of the Earth's oblateness, and of the Moon, in a restricted three-body problem.

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  • $\begingroup$ Indeed. These parameters are "ephemeral" - they keep changing. It ought to be possible to write down an approximate expression for an "ideal" oblate earth, and the lunar and solar effects on gravity are well modeled (tidal variation of gravity). But what you end up with is "something that looks like the motion" you could get for an artificial satellite - not the actual motion. So whether this is good enough depends on your intended application ... Is it a science demonstration, a video game, or something more critical? $\endgroup$
    – Floris
    Commented Aug 16, 2014 at 12:13
  • $\begingroup$ @Floris - Actually something between a science demonstration and a video game, so we can say it's both, and definitely not critical. Formulae for estimation of these parameters ( possibly with information about included and not included effects ) will do. $\endgroup$
    – James C
    Commented Aug 16, 2014 at 12:37
  • $\begingroup$ @MariusMatutiae - In your last paragraph you said there are principles that make it possible to compute the parameters. Could you point me to a reference that explains the procedure or at least has the expressions written for it ? I understand it will only mean a crude estimation of parameters, but that is ok for my application. $\endgroup$
    – James C
    Commented Aug 17, 2014 at 12:56
  • $\begingroup$ @JamesC If you have first and second derivative, what else do you need? $\endgroup$
    – MariusMatutiae
    Commented Aug 17, 2014 at 13:06
  • $\begingroup$ @MariusMatutiae - That is exactly what I need - first and second derivative, if I don't have TLE for a space object available, or I am making up an object of my own. Therefore, how to compute / estimate them ? I am still referring to my original question and the last paragraph of your answer. $\endgroup$
    – James C
    Commented Aug 17, 2014 at 13:10

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