sdp4 {asteRisk}

Propagate an orbital state vector with the SDP4 model

Description

Given an orbital state vector of a satellite, applies the SDP4 model to propagate its orbit to the desired time point. This allows the calculation of the position and velocity of the satellite at different times, both before and after the time corresponding to the known state vector (referred to as "epoch"). Kepler's equation is solved through fixed-point integration. The SDP4 model is a modified version of the SGP4 model that takes into account the secular and periodic perturbations of the Moon and the Sun on the orbit of the satellite. It also considers the Earth resonance effects on 24-hour geostationary and 12-hour Molniya orbits. Thanks to this, the SDP4 model can correctly propagate the orbit of objects in deep space (with orbital periods larger than 225 minutes, corresponding to altitudes higher than 5877.5 km). However, it should be noted that SDP4 employs only simplified drag equations, and lacks corrections for low-perigee orbits. Therefore, it is recommended to apply the standard SGP4 model (available through the sgp4 function) for satellites that are not in deep space. This implementation is based on a previous SDP4 implementation in Julia (SatelliteToolbox).

Usage

sdp4(n0, e0, i0, M0, omega0, OMEGA0, Bstar, initialDateTime, targetTime,
     keplerAccuracy=10e-12, maxKeplerIterations=10)

Arguments

n0

Mean motion of the satellite at epoch in radians/min.

e0

Mean eccentricity of the orbit of the satellite at epoch. Eccentricity ranges from 0 (perfectly circular orbit) to 1 (parabolic trajectory).

i0

Mean orbital inclination of the satellite at epoch in radians.

M0

Mean anomaly of the satellite at epoch.

omega0

Mean argument of perigee of the satellite at epoch.

OMEGA0

Mean longitude of the ascending node of the satellite at epoch. Also known as right ascension of the ascending node.

Bstar

Drag coefficient of the satellite in units of (earth radii)^-1^. Bstar is an adjusted value of the ballistic coefficient of the satellite, and it indicates how susceptible it is to atmospheric drag.

initialDateTime

Date-time string in UTC indicating the time corresponding to the known state vector of the satellite. Unlike for SGP4, it must be provided in all cases since it is required to calculate Moon and Sun perturbations.

targetTime

Time at which the position and velocity of the satellite should be calculated. It can be provided in two different ways: either as a number corresponding to the time in minutes counting from epoch at which the orbit should be propagated, or as a date-time string in UTC.

keplerAccuracy

Accuracy to consider Kepler´s equation solved. If two consecutive solutions differ by a value lower than this accuracy, integration is considered to have converged.

maxKeplerIterations

Maximum number of iterations after which fixed-point integration of Kepler's equation will stop, even if convergence according to the accuracy criterion has not been reached.

Value

A list with three elements. The first two elements represent the position and velocity of the satellite at the target time, in the TEME (True Equator, Mean Equinox) frame of reference. Position values are in km, and velocity values are in km/s. Each of these two elements contains three values, corresponding to the X, Y and Z components of position and velocity in this order. The third element indicates the algorithm used to propagate the orbit (sdp4).

References

https://juliapackages.com/p/satellitetoolbox https://celestrak.org/NORAD/documentation/spacetrk.pdf http://www.celestrak.org/publications/aiaa/2006-6753/AIAA-2006-6753.pdf

Examples

# The following orbital parameters correspond to an object with NORAD catalogue
# number 24208 (Italsat 2) the 26th of June, 2006 at 00:58:29.34 UTC.

n0 <- 1.007781*((2*pi)/(1440))  # Multiplication by 2pi/1440 to convert to radians/min
e0 <- 0.002664 # mean eccentricity at epoch
i0 <- 3.8536*pi/180 # mean inclination at epoch in radians
M0 <- 48.3*pi/180 # mean anomaly at epoch in radians
omega0 <- 311.0977*pi/180 # mean argument of perigee at epoch in radians
OMEGA0 <- 80.0121*pi/180 # mean longitude of ascending node at epoch in radians
Bstar <- 1e-04 # drag coefficient
epochDateTime <- "2006-06-26 00:58:29.34"

# Calculation of the orbital period

2*pi/n0

# The period is higher than 225 min, and therefore the SDP4 model should be
# applied. Furthermore, from the mean motion in revolutions/day, it can be
# seen that it is a geostarionary satellite with a 24-hour period. Let´s
# calculate and compare the position and velocity of the satellite at epoch
# and 1 day later.

state_0 <- sdp4(n0=n0, e0=e0, i0=i0, M0=M0, omega0=omega0, OMEGA0=OMEGA0,
                Bstar=Bstar, initialDateTime=epochDateTime, targetTime=0)
state_1day <- sdp4(n0=n0, e0=e0, i0=i0, M0=M0, omega0=omega0, OMEGA0=OMEGA0,
                   Bstar=Bstar, initialDateTime=epochDateTime, targetTime=1440)
state_0
state_1day

# The position and velocity are very similar after a full day, in accordance
# with the geostationary orbit