Introduction¶

Nowadays, a huge need for LEO is rising. With the spreading of the Internet of Things (IoT) we need more and more Machine To Machine (M2M) communications. However this is not conceivable with GEO satellites because the Time Round Trip is far too high. In this context, the aim of this project is to implement an orbit and link budget calculator for a ground station of a LEO satellite. The program is written with LabView as it is compatible with the antenna's processor owned by the Telecom Bretagne Lab in ISAE campus.

This report explain how it is possible to calculate the satellite’s position and how to perform the link budget if the satellite is in sight. Thus, the state of art of orbit propagators will be presented followed by a complete description of the Two-Lines Element (TLE) and then the calculation of the link budget will be explained.

State of Art of the Orbit Propagators
h2. State of Art of the Orbit Propagators

Orbit propagators are mathematical models made to estimate the satellite’s position and velocity. These models take into account the effects which perturbate the satellite from his ideal orbit. These perturbations are mostly the result of the non-spherical earth mass distribution, the atmospheric resistance and gravitation effects from the sun or the moon.
Five of these propagators exist : SGP, SGP4, SDP4, SGP8 and SDP8. SGP means Simplified General Perturbations models and SDP means Simplified Deep Space Perturbations. Here is a short description for each of these propagators :

• SGP is the first orbit propagator. It has been developed by Hilton and Kuhlman in 1966 thanks to Kozai research work made in 1959. It is made for satellite orbiting near the Earth which considers satellite with an orbital period lower than 225 minutes.
• SGP4 has been developed by Ken Cranford in 1970. It is a improvement of the previous propogator in order to track the growing number of satellites in orbit at this time. It is also used for near Earth satellites.
• SDP4, developed by Hujsak in 1979, is the SGP4 propagator adpated for deep space objects. This consider satellites with an orbital period greater than 225 minutes.
• SGP8, also used for near Earth satellites, is almost like the SGP4 propogator but the calculation methods are different. However it follows the same models for the atmospheric and gravitational effects.
• SDP8 is the SGP8 propagator adpated to deep-space effects. Moreover, SGP8 and SDP8 are better to manage the orbital decay.

In this project, we chose to use the SGP4 orbit propagator. On one hand because we are dealing with low orbit satellites and on the other hand because this propagator is the most commonly use to develop satellite stracking software. Besides, the NORAD (North American Aerospace Defence Command) element sets are provided using SGP4 or SDP4. Thus it is more accurate to implement one of these propagator.

The Two-Lines Element¶

Once the orbit propagator is chosen, some datas are required at the input in order to calculate the satellite’s position. These datas are stored within the Two Lines Element (TLE). They can be found on the internet but it is important to check if they are up to date whereas precision will be lost in the calculation. TLE are provided by NORAD (North American Aerospace Defence Command).
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Here is the TLE format (2 lines of 69 characters) :

AAAAAAAAAAAAAAA  b.b  c.c  d.d  e.e  f  RRR         KM x km
1 gggggU hhiiijjj kklll.llllllll +mmmmmmmm +nnnnn-n  ooooo-o p qqqqr
2 ggggg sss.ssss ttt.tttt uuuuuuu vvv.vvvv ddd.dddd xx.xxxxxxxxyyyyyz

Line 0 description :

AAAAAAAAAAAAAAA : name of the satellite
b.b : length (meters)
c.c : width (meters)
d.d : height (meters)
e.e : standard magnitude
f : standard magnitude determination method (d = calculation ; v = observation)
RRR : equivalent radar section (meter square)
KM : altitude at apogee
km : altitude at perigee
Remark : Most of the time there is no line 0.

Line 1 description :

ggggg : Number within the U. S. Space Command (NORAD)
U : Classification (here U means Unclassified = not secret)
hh : Last two numbers of the lunching year
iii : lunch number of the year
jjj : one to letters pointing a piece of the lunch
kk : two last numbers of the year when these elements have been estimated
lll.llllllll : day and fraction of the day when these elements have been estimated
+ mmmmmmmm : half of the first derivative of the mean movement (revolution per day square), this stands for the acceleration and deceleration of the satellite
+ nnnnn-n : sixth of the second derivative of the mean movement (revolution per day cube)
ooooo-o : pseudo balistic coefficient, used by the SGP4 orbit propagator (1/terrestrial radius)
p : type of ephemeris
qqqq : number of this set of elements
r : cheksum (modulo 10)

Line 2 description :

ggggg : Number within the U. S. Space Command (NORAD)
sss.ssss : orbit inclination with respect to the terrestrial equator (degrees)
ttt.tttt : right ascension of the orbit ascending node (degrees)
uuuuuuu : excentricity
vvv.vvvv : perigee argument (degrees)
ddd.dddd : mean anomaly (degrees)
xx.xxxxxxxx : mean movement (revolution per day)
yyyyy : number of revolution when these elements have been estimated
z : cheksum (modulo 10)

Here is an example of a TLE for the NOAA-19 :

1 33591U 09005A   15310.52866608  .00000161  00000-0  11260-3 0  9997
2 33591  99.0081 260.8643 0014724 126.2184 234.0350 14.11998019347577

Here is some helpful remarks to interpret the TLE’s values :

• concerning the day and fraction of the day, also called epoch, when these elements have been estimated at line 1, the first two numbers are the last numbers of the year (15 stands for 2015 in the example). The next three numbers designate the number of the day in the year (310 represents November the 6th). Then the last numbers are the fraction of the day. By multiplying this decimal number by 1440 (the number of minutes in one day), we obtain the time of the day when these elements have been determined. This timeStamp reprensents the epoch. It is the moment where the satellite arrives at the ascending node.
• When there is a « + » before the values it means that these values can be either positive or negative
• Concerning the excentricity at line 2, the decimal is assumed (0014724 means 0.0014724 in the example)
• Concerning the pseudo-balistic coefficent at line 1, the decimal is assumed and the last integer represents the minus values at the power of ten (11260-3 stands for 0.11260 x 10^-3)

It is even more important to have updated TLEs for satellite with orbit lower than 350 km and if it is often subject to manover such as the ISS (International Spatial Station)

Distance elevation and azimuth calculation¶

ECI coordinates¶

The coordinates of the satellite given by the propagator are given in the Earth-Centered Inertial (ECI) coordinate system . This system is a cartesian coordinate system whose origin is located at the center of the earth (at the center of mass to be precise). The z axis is orthogonal to the equatorial plane pointing north, the x axis is pointing towards the vernal equinox, and the y axis is such that the system remains a direct orthogonal system The x and y axis are located in the equatorial plane as shown in the following figure.

This system is convenient to represent the positions and velocities of space objects rotating around the earth considering firstly that the origin of the system is the center of mass of the earth, and that the system does not rotate with the earth. Indeed, "inertial" means that the coordinate system is not accelerating, therefore not rotating: considering the way the three axis are defined, the system is fixed in space regarding the stars.
The problem here is that our ground is station is located on the surface of the earth, so its coordinates are given in the geodetic coordinate system, which is obviously a system that is rotating with the earth.
We had therefore two options: either calculate the coordinates of the ground station in the ECI coordinate system, or calculate the coordinates of the satellite in the geodetic coordinate system. We chose the first option .

θ(τ)calculation¶

The main problem to go from the geodetic system to the ECI system is to calculate the angle between the observer meridian (longitude) and the vernal equinox direction. This angle, also called the local sidereal time depends on the star and not on the sun and that is why it is a bit touchy to calculate. For our implementation and tests, we relied on the algorithms and example given in the magazine Satellite Time ,in the column Orbital Coordinate Systems, Part II. θ(τ) is actually define as the sum between the antenna east longitude and the greenwich sidereal time GST, which is the angle between the greenwich meridian and the vernal equinox that we will note  θg(τ).
θg(τ)can be calculated using the formula:

θg(τ) = θg(0h) + ωe·Δτ (1)

where Δτ is the number of seconds elapsed since 0h at the time of the calculation and ωe = 7.29211510 × 10-5 radians/second is the Earth's rotation rate. θg(0h) is the GST calculated at 0h that day and it is given as

θg(0h) = 24110.54841 + 864018.812866 Tu + 0.093104 Tu2 - 6.2 × 10-6 Tu3 (2)
where Tu = du/36525 and du is the number of days of Universal Time elapsed since the julian date 2451545.0 (2000 January 1, 12h UT1). Therefore, to calculate θg(τ), the first thing we need is the julian date of the day, which we can deduce from the julian date of the year as follow:

JD = Julian_Date_of_Year() + Number of day since the first of January;

This calculation is done in the VI julian date.vi where the calculation of the Julian date of the year is done using the Meeus' approach.

Once we had the Julian date, we could calculate du and therefore Tu, and that way we calculated θg(0h) using (2). From there, we were able to calculate θg(τ) using (1) and since we know the east longitude of the antenna we obtained θ(τ). Those calculations are done in the VI tetag.vi.

Geodetic to ECI conversion¶

Once we had the local sidereal time, we were able to make the change of system using the following formulas:

x=Rcos θ
y=Rsin θ
z=Re sin φ

where Re = 6378.135 km, φ is the latitude of the antenna, θ is the local sidereal time and R = Re cos φ
This is done in the VI, Antenna coordinates ECI.vi.

Distance¶

Elevation and azimuth¶

Link budget¶