Regression analysis

Here's a basic tool for the regression analysis of some orbital parameters. For example, if the mean radius vector (Rave) is selected, the regression could show the decay rate of a satellite like the HST or, more generally, the regression shows the average rate of change of the parameter for the selected date interval. The interval is represented by the two red dots.
In order to calculate the regression, at least two points are needed; if the date interval is not sufficiently wide, "Error: n < 2" appears below the "Regression" label.
The sample Pearson correlation coefficient r is also shown below the rate of change (r should go from -1 to 1, but occasionally the result may be slightly higher than 1 due to truncation errors).


Initial date

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Final date

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days
Period
Parameter
Regression
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Lines
Dots

The graph shows the derivative of the semi-major axis (da/dt or \(\dot{a}\)) and the derivative of the mean motion (dn/dt or \(\dot{n}\)).
According to the paper "Thermospheric densities derived from spacecraft orbits: Accurate processing of two-line element sets": "B* is a fitting parameter alongside the mean elements, whereas the corresponding value of \(\dot{n}/2\) is computed a posteriori." (see the left column on page 4). The value of \(\dot{n}\) is taken from the TLE (line 1, columns 34-43) and \(\dot{a}\) is obtained from \(\dot{n}\) and \( \mu = a^{3} \cdot n^{2} \) by \( \dot{a} = -2\mu \cdot \dot{n} / (n^{3} \cdot 3a^{2}) \).

The graph shows the average, minimum, and maximum radius vector and the eccentricity (click a plot legend to enable/disable the related plot).

The graph shows the average, minimum and maximum orbital speed.
When a reboost or a deboost is shown, the deltaV used for that manoeuvre can be easily calculated as the difference between the mean speed before the manoeuvre and the mean speed after the same manoeuvre, but beware of the spikes.

The graph shows the average, minimum and maximum orbital inclination with respect to the TEME reference frame.
The variation of the inclination during one orbit is caused by the Earth's flattening.

The graph shows the right ascension of the the ascending node and its rate of precession.
The blue plot represents the actual TEME equator crossing (it's not the osculating ascending node). It is calculated by propagating the satellite both forward in time and backward; the smallest time since the TLE epoch is taken to calculate the angular distance between the ascending node and the mean equinox of date.
The orange plot represents the rate of precession.

Here's the long-periodic perturbations on the argument of perigee.
The blue plot represents the actual AoP (it's not the osculating AoP). The calculation starts from the ascending node found as explained in the previous graph, then the satellite is propagated forward (always forward) until the perigee is found. The AoP is calculated as the angular distance between the ascending node and the perigee in the direction of motion on the orbital plane.
The orange plot represents the rate of precession (in the presence of very noisy data, zooming can help a bit).

Here's the average, minimum and maximum air density at the satellite position (logarithmic scale) and the time averaged radius vector (note that it's not the mean radius vector or semi-major axis, but here the time averaged radius vector does make sense because the air density is numerically integrated against the time). Click a plot legend to enable/disable the related plot.

Here's the average rate of change of the mean radius vector calculated as the angular coefficient of the linear regression of the mean radius vector (the regression starts 15 days before the plotted points). The dashed horizontal line represents the average rate of change of all the points.