Calculations for the Earth's artificial satellites

Index


- TLE practice

Unless otherwise stated, all data are obtained from the TLEs downloaded from www.space-track.org.
According to 18 SPCS, their TLE must always be used with the SGP4/SDP4 propagator; see this link (login required): "SGP4 is an analytic method based on a general perturbation theory for generating ephemerides for satellites in earth-centered orbits. It is the proper means for correctly propagating a USSPACECOM 18th Space Control Squadron (18 SPCS) Two Line Element (TLE).".

Well, but what's the error when we use TLEs without SGP4? Let's do some calculations for the well-known IRNSS-1H / PSLV-C39 satellite.
All the numbers are presented with an unrealistic accuracy and \(\mu = 398600.8\ km^3\cdot s^{-2}\) (with the TLEs, WGS-72 ellipsoid must be used).

Suppose that we want to calculate the osculating semi-major axis, perigee and apogee of that satellite from the following TLE:
1 42928U 17051A   18272.08593412  .00449873 -94661-5  36430-3 0  9997
2 42928  19.1496 178.1014 1756625 192.5230 162.5249 12.31022939 41414

The wrong way

The easiest (and the most wrong) way is to use the mean motion (columns 53 to 63 in the line 2) to calculate the semi-major axis with the formula \(a=\sqrt[3]{\mu(\frac{T}{2\pi})^2}\), then we use the eccentricity (columns 27 to 33 in the line 2, decimal point assumed) to calculate the perigee radius vector with \(Rp=a \cdot (1-e)\) and the apogee radius vector with \(Ra=a \cdot (1+e)\).

Since the mean motion is expressed in revolutions per day, we first need to convert 12.31... in seconds: T = 86400 / 12.31022939 = 7018.55320992 s.
Now we can calculate \(a=\sqrt[3]{398600.8\cdot(\frac{7018.55320992}{2\pi})^2} = 7923.02830819\ km\) , \(Rp=7923.02830819 \cdot (1-0.1756625) =6531.24934800\ km\) and \(Ra=7923.02830819 \cdot (1+0.1756625) =9314.80726838\ km\).

The correct procedure

Now let's do the calculations by following the correct procedure. We first calculate the satellite state (km, km/s) for the TLE epoch with the SGP4 propagator; this can be easily done with the CSpOC's library Sgp4PropMse() function:
Rx= -9260.27177034, Ry= 306.968939716, Rz= -0.000210324947856
Vx= -0.458859515256, Vy= -5.63117994141, Vz= 1.96073394119
Now we use PosVelToKep() to obtain a = 7928.27076977 km and e = 0.1745066005, from which we obtain Rp = 6544.7351899 km and Ra = 9311.80634965 km.

To summarize, for that TLE, the wrong way gives the following errors: semi-major axis: 5.2 km, perigee: 13.5 km and apogee: 3 km.

The graph shows the result for the perigee obtained from 421 TLEs.
The red plot is the wrong way, while the green plot is the correct procedure.
The error is the blue plot calculated as wrong_perigee - correct_perigee.

If we use the wrong way, not only we can get very big errors, but we also totally lose the wide perturbation on the perigee.

For a detailed analysis of the errors, please take a look here.

- Interactive graphs

Some pages use plotly.js, an open-source JavaScript graphing library to generate dynamic and interactive graphs. Clicking and holding with your mouse allows you to zoom and pan. The graph usage is very intuitive, but if you need detailed explanations, please take a look at the pages An Introduction to Chart Studio Modebar and Zoom, Pan, and Hover Controls.
When plotly.js is used, the page could take a few seconds to load.

- Air density

For the calculation of the air density, the NRLMSISE-00 atmosphere model is used along with an updated data file for the solar and geomagnetic indices.
The average air density is calculated by numerical integration of the air density at the satellite position as a function of time.

- Time

Since the TLE epoch is given in UTC, also all the other times are given in UTC.

- Satellite altitude and radius vector

A widely adopted notation used to describe the size of an orbit is something like 222 x 333 km, but it's a rather ambiguous notation because it's usually not specified what those two numbers exactly represent; they could be: altitudes above an unknown ellipsoid or radius vectors scaled to an unknown sphere. Let's see the differences with the help of a graph:

it shows the magnitude of the Tiangong-2 radius vector scaled to a sphere with a radius of 6371 km (blue plot) and its altitude above the WGS-72 ellipsoid (used by CSpOC). Also shown is the latitude of the station (dashed line). We see big differences between the magnitude of the radius vector and the altitude; if we consider the former, the orbit is 303.7 x 295.8 km, while if we consider the altitude, the orbit is 305.2 x 291.4 km.

All the radius vectors shown in this site are usually scaled to a surface of a sphere with a radius of 6371 km, the mean Earth's radius used in geodesy, which is obtained from: (2 · Re + Rp) / 3.
For example, if the graph shows: "Radius vector - 6371 [km]", it means that a value equal to 411 km (the approximate ISS altitude) represents a radius vector of 6371 + 411 = 6782 km.

- Mean radius vector and semi-major axis

Often the graphs show the unusual wording "mean radius vector". That value is equivalent to the semi-major axis and it represents the average distance between the Earth's barycenter and the satellite.
I use "mean radius vector" instead of "semi-major axis" just to avoid the confusion with the osculating semi-major axis.
The mean radius vector is calculated by numerical integration of the inverse of the radius vector as a function of the time for 1 orbit.

- Orbital inclination

The graph of the orbital inclination of the LEO satellites usually shows three plots: average, minimum and maximum inclination, but why are there three inclinations?
Because the Earth is not a perfect sphere; its flattening (mainly the J2 zonal harmonic) generates a perturbation that slightly changes the satellite velocity vector in a way that also the orbital inclination changes. The bigger the radius vector, the smaller the inclination change.

The graph shows the ISS and LAGEOS-2 orbital inclination (in the TEME reference frame).
The radius vector of the former is about 6782 km, while the one of the latter is about 12170 km; notice the much smaller inclination amplitude of the LAGEOS-2 satellite.

- Mean orbital speed and eccentricity

The mean orbital speed is calculated as the length of 1 orbit divided by the orbital period and it is obtained from the numerical integration of the orbital speed as a function of time.
The eccentricity is calculated as: e= (Ra - Rp) / (Ra + Rp), where Ra is the apogee radius vector and Rp is the perigee radius vector.

- TLE noise

All the graphs are shown "as is", without any artificial smoothing or scale compression to look better. As a consequence, noise spikes may be present.