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I have TLE and RINEX navigation file for GPS satellite PRN 20, for 01.04.17r. 19:56:56 and 20:00:00 respectively. See also The Receiver Independent Exchange Format Version 3.03.

It seems that the value of longitude of the ascending node (Ω0) differs for both while all the other parameters are relatively similar.

RINEX file:

M0 = -0.41907076212790001
e  =  0.0044262305600569997
ω  =  1.5754052384999999
i  =  0.92697366533970005
Ω0 = -0.4446281969834

TLE file:

M0 =  5.8389169820721856
e  =  0.0042077
ω  =  1.5770882387483363
i  =  0.92682044735729674
Ω0 =  2.7589292150900362

What can be the reason for this?

While the values for M0 are close modulo 2π, the values for Ω0 seem to differ by only π, or about 180°.

Checking also other datasets the result is similar, although the difference between values differs.

TLE:

GPS BIIR-4  (PRN 20)    
1 26360U 00025A   17091.83120691 -.00000002  00000-0  00000-0 0  9995
2 26360  53.1029 158.0750 0042077  90.3605 334.5453  2.00565359123815

RINEX:

20 17  4  1 20  0  0.0 4.665344022214D-04 1.591615728103D-12 0.000000000000D+00
5.600000000000D+01-1.231250000000D+01 5.538802141913D-09-4.190707621279D-01
-5.606561899185D-07 4.426230560057D-03 6.150454282761D-06 5.153646507263D+03
 5.904000000000D+05-6.705522537231D-08-4.446281969834D-01 3.725290298462D-08
 9.269736653397D-01 2.442500000000D+02 1.575405238500D+00-8.669646839876D-09
 2.714398779890D-11 0.000000000000D+00 1.942000000000D+03 0.000000000000D+00
 2.000000000000D+00 0.000000000000D+00-8.381903171539D-09 5.600000000000D+01
 5.899500000000D+05 4.000000000000D+00
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    $\begingroup$ As a suggestion, have you checked that PRN20 of TLE and RINEX are the same satellite?. The offset of $M_0$ and $\Omega_0$ that you are reporting are typical between different satellites of a constellation. Also, your shortcut of TLE and RINEX shoes PRN20 while you started the question asking about PRN1 (a typo error maybe?) $\endgroup$
    – Julio
    Nov 3 '17 at 10:48
  • $\begingroup$ Where did you get the RINEX file? $\endgroup$ Nov 3 '17 at 19:36
  • $\begingroup$ The rinex file is from garner.ucsd.edu, TLE from celestrak.com/NORAD/elements/gps-ops.txt. @Julio how can I check whether PRN20 is the same satellite for both files? I know the satellite numbers are different for glonass rinex and tle files but I thought for GPS it's the same. Yes PRN1 is a typo. $\endgroup$
    – kas
    Nov 13 '17 at 20:34
  • $\begingroup$ @MartynaKasyr, if I were you I will search for another satellite in your TLE that matches the current orbital elements values of your RINEX file to barely confirm (or not) the discrepancy of PRN numbers between TLE and RINEX. There is also an open question about this topic here space.stackexchange.com/questions/20887/… $\endgroup$
    – Julio
    Nov 15 '17 at 10:56
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They are in different reference frames. The TLE element set is defined relative to the True Equator, Mean Equinox (TEME) reference frame, and the elements in the RINEX Nav message are defined relative to Earth Centered, Earth Fixed (ECEF). Since both frames use the true equator, other elements (such as inclination) don't vary as much as the location of the node.

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They are in different reference frames, but there is more to it than that. Even once you have converted TEME to ECEF, there will still be differences, because things that have the same name actually have very different meanings. RINEX is just a file format; the burden is placed on users to understand that GPS nav messages and TLEs use the same field names for different purposes.

The NAVSTAR GPS Space Segment/Navigation User Interface Specification IS-GPS-200 (Revision M, dated May 2021), Section 20.3.3.4, "Subframes 2 and 3", gives 12 pages of instructions of how the user segment is required to perform ephemeris calculations, which do not quite line up with the way anyone else uses those terms. In particular, note these statements (emphasis mine):

The ephemeris parameters describe the orbit during the curve fit intervals described in section 20.3.4. Table 20-II gives the definition of the orbital parameters using terminology typical of Keplerian orbital parameters; it shall be noted, however, that the transmitted parameter values are such that they provide the best trajectory fit in Earth-Centered, Earth-Fixed (ECEF) coordinates for each specific fit interval. The user shall not interpret intermediate coordinate values as pertaining to any conventional coordinate system. The user shall compute the ECEF coordinates of position for the phase center of the SVs’ antennas utilizing a variation of the equations shown in Table 20-IV. Subframes 2 and 3 parameters are Keplerian in appearance; the values of these parameters, however, are produced... via a least squares curve fit of the propagated ephemeris of the phase center of the SVs’ antennas

Fit intervals are three or four hours long. The numbers in GPS nav messages give the values which provide the best fit over that entire interval, so they are not the same as osculating elements for the GPS orbits at any specific epoch. You have to read the docs for how to handle them, and you have to use the equations they tell you to (including, for example, a specific way of iterating Kepler's equation towards a solution), or the answer you get will not have the same error statistics that these tables exist to let you achieve.

These warnings are even more true about Two-Line Element sets. Those numbers are even farther from osculating. They are called mean elements. It could be argued that word should apply to the way GPS does it, but that's not the way the word is used in orbital mechanics. "Mean" usually just means "average", but there are many ways to average something, and the term's technical use in astrodynamics is both more specific and more general than you might expect. In the words of H. G. Walter (1967), "by mean elements we understand osculating elements from which short-periodic and long-periodic perturbations of the earth’s potential have been subtracted." In order to convert from mean elements to osculating elements, you have to know exactly which perturbations were chosen for subtraction, and you need to calculate those corrections and add them back in, or you will get the wrong answer.

Orbit elements in TLEs are specified in "Kozai form", which means using the exact list of perturbations chosen for removal in Yoshihide Kozai, The motion of a close earth satellite, Astronomical Journal 64 (367-377) 1959. For example, the value to be placed in the field labeled "semimajor axis" is not the usual $a$ (which the linked paper calls $a_0$), but rather the "mean" $\bar{a}$, which equals $a_0$ times the factor

$$1-\frac{A_2}{p^2}\left(1-\frac{3}{2}\sin^2 i\right)\sqrt{(1-e^2)}$$

where $p=a(1-e^2)$ and $A_2$ is a gravity expansion coefficient that equals $\frac{3}{2}J_2 R^2$, where $J_2$ is the familiar thing and $R$ is the earth's equatorial radius.

This is by far the simplest such formula. The equivalent expressions for argument of perigee ($\bar{\omega}$) and right ascension of the ascending node ($\bar{\Omega}$) run to several pages. Note that the conversion from Kozai's $A_2$ to the now common $J_2$ I obtained from Brouwer (1959), which describes a different and much more complicated set of mean elements, and has a table showing how to translate among nine different forms of gravity expansion coefficients. Brouwer's article ends with the statement,

I now regret that I introduced $k_2$, $k_4$ in my paper in 1946. The principal reason was that they give a particularly simple form for the expression of the potential in the equatorial plane. If I could have foreseen the increase in interest in the subject and the confusion to which I was contributing, I would have chosen ... the alternative form which was used by Vinti (1959). I intend to revert to this form and recommend this to other authors.

Brouwer's plea appears to have worked, as astrodynamics seems to have settled firmly on the form Vinti credited to R. H . Merson and D. G. King-Hele, Use of Artificial Satellites to Explore the Earth's Gravitational Field: Results from Sputnik 2 (1957β), Nature 182 (640) 1958, the familiar

$$V(r,\theta)=-\frac{GM}{r}\left[1-\sum_{p=2}^{\infty}J_p\left(\frac{R}{r}\right)^p P_p(\sin\theta)\right]$$

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