# Satellite orbit element

Satellite orbit elements

The satellite orbit elements set the parameters for the orbits of objects that orbit a celestial body according to Kepler's laws . They are used to determine the orbit and include the six orbit elements of an undisturbed system and additional correction parameters that take account of orbital disturbances, for example due to friction with the atmosphere, inhomogeneous gravitational fields, solar storms or radiation pressure .

The orbit elements for most satellites are published by the American Air Force Space Command as so-called Two Line Elements (TLE) . The data of a calculated forecast is compared with the actual observation by tracking stations on earth and updated orbit elements derived from this are published.

## The satellite orbit elements

### 6 track elements

The orbit of a satellite in space in an undisturbed gravitational field of a planet is clearly determined by six orbit elements : two for the shape of the orbit, three for the position in space, one for the reference to time.

The six determinants can be determined by different sizes, which is why there are a large number of different path element tuples. An example:

### Railway disruptions

Satellites experience orbital disruptions caused, among other things, by:

These interferences cause satellites:

An excellent trajectory is the solar synchronous orbit . The disturbance of the right ascension of the ascending node is just so great that a satellite always flies over the earth at the same local time. Which is as a new orbital parameters local time ascending node (Engl. Local Time of Ascending Node , LTAN) defined that specifies the local time of the overflight. The local time of the descending node ( Local Time of Descending Node LTDN) is offset by 12 hours from the LTAN.

Propagation models add correction values ​​to improve the accuracy of the orbit prediction.

## TLE definition of the satellite elements

The Two Line Elements (TLE) differ from the classic parameter set:

Instead of the major semi-axis a , they indicate the mean angular velocity  n . You define the time-dependent position of an object using the time and the mean anomaly .

1. Numerical eccentricity (Eccentricity)${\ displaystyle \ varepsilon}$
2. Mean motion (Mean Motion)${\ displaystyle n}$
3. Inclination (Inclination)${\ displaystyle i}$
4. Right ascension of the ascending node (Right Ascension of Ascending Node)${\ displaystyle \ Omega}$
5. Argument of Periapsis (Argument of Perigee)${\ displaystyle \ omega}$
6. Mean anomaly (Mean Anomaly)${\ displaystyle \ mathrm {M}}$
7. Epoch (epoch) with time${\ displaystyle t}$

optional:

## Drag coefficient

Friction slows down a satellite. In near-earth orbits less than 800 km, it is so large that a satellite falls to earth in a spiral within a few years or decades. Therefore, in addition to the classic 6 Kepler elements (and the epoch for small-scale, precise computing) an additional orbit element is necessary for this special astrophysical complex of problems.

The propagation models follow different approaches:

In the simplest case, the Simplified General Perturbations (SGP) model, the drag coefficient is either a ballistic factor or the first derivative of the mean movement over time, divided by two.

Definition 1

The drag coefficient is a measure of the rate of descent per unit of time at which the satellite is heading towards earth. Without its own formula symbol, it is simply called (read: n-point halves ) and has the unit cycles per day squared (1 / d 2 ). ${\ displaystyle {\ dot {n}} / 2}$

The SGP model developed in 1966 for satellites in near-earth orbit is based on a greatly simplified analytical perturbation theory and is therefore only used for approximate calculations.

The SGP4 model developed in 1970 is used most frequently for near-earth satellites. Its algorithm is also used by NASA for all satellites with an orbital time of less than 225 minutes (corresponds to an orbit height of about 6,000 km).

Definition 2

The drag coefficient in the SGP4 model is called (pronounced: B-Star or B-Star ) and is specified as follows: ${\ displaystyle B ^ {*}}$

In aerodynamic theory, every object has a ballistic coefficient , which is calculated from its mass divided by the product of its (air) drag coefficient (usually a value between 2 and 4) and its cross-sectional area (see also the service life of satellite orbits ): ${\ displaystyle B}$${\ displaystyle m}$${\ displaystyle c_ {w}}$${\ displaystyle A}$

${\ displaystyle B = {\ frac {m} {c_ {w} A}}}$ (10)

The ballistic coefficient indicates how strongly an object is braked: the higher the value, the lower the braking effect. (NOTE: In the publication "Models for Propagation of NORAD Element Sets" is defined differently, see Chapter 12.) ${\ displaystyle B}$

${\ displaystyle B ^ {*}}$is an expanded value of and uses the density of the atmosphere at the reference altitude as a reference value . has the unit earth radii −1 . ${\ displaystyle B}$${\ displaystyle \ rho _ {0}}$${\ displaystyle h_ {0}}$${\ displaystyle B ^ {*}}$

${\ displaystyle B ^ {*} = {\ frac {1} {2}} \ cdot B \ cdot \ rho _ {0}}$ (11)

In flow the air density of the atmosphere and the drag of satellites with one. These are highly variable due to the changing solar activity and the resulting changing composition of the atmosphere. The useful duration of use of SGP4 for LEO satellites is limited to a few days or a few weeks, since it only applies to the atmosphere at the epoch. ${\ displaystyle B ^ {*}}$${\ displaystyle B ^ {*}}$

## The Two Line Elements Format TLE

Satellite orbit elements can be encoded in a format commonly known as the NASA / NORAD Two Line Elements Format , or TLE for short . As the common English expression suggests, the elements are represented as number blocks in two lines. The representation is historically justified, as it was originally developed for 80-column punched cards and further processed with FORTRAN programs. The parameters of the orbit and the satellite position can then be calculated in advance with one of the propagation models for a desired point in time. For reasons of accuracy, the orbital elements should not be older than a few days, especially for satellites with low orbit. The position and shape of the orbit as well as the position of the satellite can be calculated at least at the time of the epoch, as the example in Section 2.2 shows, without the great expense of the sometimes very complex numerical procedures .

### epoch

A data set of orbital elements is a “snapshot” of the satellite orbit at a specific point in time called an epoch. With this snapshot, the numerical values ​​of all satellite orbit elements are recorded.

Definition : The epoch is a number that specifies the point in time when the “snapshot” was taken. ${\ displaystyle t}$

### application

Groundtrack of the International Space Station
Azimuth and elevation for the ISS at a given observation location

With so-called tracking programs, a satellite can be followed in real time ( Fig. 3 ) or the time of the overflight over a certain point on earth can be calculated. Because under certain conditions one can observe the overflights from the earth with the naked eye. This is especially true - because of its size - for the International Space Station .

### example

#### Two Line Elements of the International Space Station ISS

Epoch: Feb. 9, 2006, 20: 26: 00.0 h UTC (= CET - 1 h)

NORAD original format (two lines, 69 characters per line including spaces):

ISS(ZARYA)
1 25544U 98067A   06040.85138889  .00012260  00000-0  86027-4 0  3194
2 25544  51.6448 122.3522 0008835 257.3473 251.7436 15.74622749413094


Edited format: For better understanding, missing spaces, exponents, leading zeros and decimal points have been added (changes marked in red). Commas also replace the decimal points. Prepared in this way, the elements can e.g. For example, it can be used with a spreadsheet , provided it uses a comma as the decimal separator.

1 25544_U 98067A 2006_040,85138889 0,00012260 0,0000e-0 0,86027e-4 0 319_4
2 25544 51,6448 122,3522 0,0008835 257,3473 251,7436 15,74622749_41309_4


#### Explanation of the groups of numbers

The groups of numbers are explained below using the prepared format:

1st line:

1 Line No. 1
U Classification (U = unclassified)
98067A International designation, d. H. Start year (2 digits), start number in the year (3 digits), object of the start (max. 3 characters)
2006 Period: 2006
040.85138889 Epoch: day no. 40 = February 9th, fraction of the day 0.85138889 = 20h 26min 00.0s
0.00012260 Drag coefficient in the SGP model: = 0.00012260 d −2${\ displaystyle {\ dot {n}} / 2}$
0.0000e-0 Negligible drag coefficient in the SGP model (mostly zero): = 0 · 10 −0 d −3${\ displaystyle {\ ddot {n}} / 6}$
0.86027e-4 Drag coefficient in the SGP4 model: = 8.6027 · 10 −5 earth radii −1${\ displaystyle B *}$
0 Ephemeris type (0 = SGP4 model)
319 consecutive data record number
4th Check sum modulo 10

2nd line:

2 Row No. 2
51.6448 Inclination = 51.6448 ° ${\ displaystyle i}$
122.3522 Right ascension of the ascending node = 122.3522 ° ${\ displaystyle \ Omega}$
0.0008835 numerical eccentricity of orbit = 0.0008835 ${\ displaystyle \ varepsilon}$
257.3473 Argument of perigee = 257.3473 ° ${\ displaystyle \ omega}$
251.7436 Mean anomaly = 251.7436 ° ${\ displaystyle \ mathrm {M}}$
15.74622749 Mean movement: = 15.74622749 d −1${\ displaystyle n}$
41309 Circulation No. 41309 since launch
4th Check sum modulo 10

#### Explanation of the representation of the epoch

The presentation of the date and time in the usual format ( year-month-day ) and ( hours: minutes: seconds ) is too unwieldy for calculation programs. Therefore a decimal format is used for satellite orbit elements instead of the usual format.

In the example above, the time of the epoch is represented in TLE format by the number sequence 06040.85138889.

In the 5-digit group in front of the decimal point, the first two digits 06 stand for the year of the epoch, in this case 2006.

The next three digits 040 stand for the current day number in the year. The numbers 001 stand for January 1st, and the numbers 365 stand for December 31st (366 in a leap year). So the digits 040 stand for February 9th.

The 8-digit group of digits after the decimal point stands for the fraction of a day, in this case 0.85138889 times a day. This in turn can be converted into a time and results in 20h 26min 00.0s coordinated universal time UTC :

0.85138889 days · 86 400 seconds / day = 73 560 seconds = 20h 26min 0.0s

## Calculation example

### Orbit and position

The orbit data results in the following values ​​for the position and orientation of the orbit, the position and the drag coefficient of the International Space Station ISS from the "NORAD Two Line Elements" at the time of the epoch:

 Epoch: ${\ displaystyle t}$= February 9, 2006; 20: 26: 00,0h UTC Inclination: ${\ displaystyle i}$ = 51.6448 ° Right ascension of the ascending node: ${\ displaystyle \ Omega}$ = 122.3522 ° Argument of perigee: ${\ displaystyle \ omega}$ = 257.3473 ° medium movement: ${\ displaystyle n}$= 1.15 · 10 −3 s −1 Rotation time: ${\ displaystyle T}$ = 5,487.029 s major semi-axis: ${\ displaystyle a}$ = 6,723,842.235 m numerical eccentricity: ${\ displaystyle \ varepsilon}$ = 0.0008835 small semi-axis: ${\ displaystyle b}$ = 6,723,839.610 m Distance of the perigee from Center of the earth: ${\ displaystyle r _ {\ text {Peri}}}$ = 6,717,901,720 m Distance of apogee v. Center of the earth: ${\ displaystyle r _ {\ text {Apo}}}$ = 6,729,782,750 m medium anomaly: ${\ displaystyle \ mathrm {M}}$ = 251.7436 ° eccentric anomaly: ${\ displaystyle \ mathrm {E}}$ = 251.6955 ° true anomaly: ${\ displaystyle \ nu}$ = 251.6475 ° Radius vector: ${\ displaystyle r (\ nu)}$ = 6,725,707,950 m Drag coefficient in the SGP model: ${\ displaystyle {\ dot {n}} / 2}$= 0.00012260 d −2 Drag coefficient in the SGP4 model: ${\ displaystyle B ^ {*}}$= 8.6027 · 10 −5 earth radii −1

The numerical values ​​listed above can now be used in one of the propagation models for forecast calculations. Because of the near-earth orbit of the International Space Station, either the SGP or the SGP4 model comes into question, which differ mainly in the perturbation theories used and thus the computational effort. Precise predictions can only be made with the SGP4 model.

Notes on the results :

• To calculate the semi-major axis with equation (2) for the product of gravitational constant of the earth and earth ( ), the value of the Geodetic Reference System 1980 adopted: .${\ displaystyle a}$${\ displaystyle G}$${\ displaystyle M}$${\ displaystyle \ mu = G \ cdot M}$${\ displaystyle \ mu = 3 {,} 986005 \ cdot 10 ^ {14} \, \ mathrm {m} ^ {3} \ mathrm {s} ^ {- 2}}$
• One can see that the orbit by the small value of the numerical eccentricity only differs from the radius of the ideal circular path. For approximate calculations, a circular path with a radius can therefore be assumed in such a case .${\ displaystyle \ varepsilon \ sim 0 {,} 09 \, \%}$${\ displaystyle from \ leq 2 {,} 625 \, \ mathrm {m}}$${\ displaystyle r = a}$
• In many tracking programs, instead of the distance from the center of the earth, the height above the earth's surface in perigee, apogee or the current position is determined, although it is not always clear which reference the programs use for the calculation. Usually this is the earth's radius at the equator. Since the earth is not a sphere, but an ellipsoid (see WGS84 ), this is only true if the corresponding points are exactly above the equator. Strictly speaking, one would have to determine the base point ( nadir ) on the earth's surface for the corresponding satellite position and calculate the current, actual height above ground from there. Some programs also use a mean radius of the earth's ellipsoid of revolution to determine the orbit.
• The eccentric anomaly was determined from the Kepler equation with the help of the Newton-Raphson method for the iterative calculation of zeros and finally the true anomaly . Since the orbit corresponds almost to a circular orbit, the difference between the mean and true anomaly is just 0.096 °. The following also applies here: If the numerical eccentricity is very small, the true anomaly can be set equal to the mean anomaly for approximate calculations .${\ displaystyle \ mathrm {E}}$${\ displaystyle \ nu}$${\ displaystyle \ varepsilon}$${\ displaystyle \ nu}$${\ displaystyle \ mathrm {M}}$

### Effects of the railway disruptions

For simple calculations it is sufficient to only consider the orbital disturbances due to the flattened shape of the earth and the braking effect due to the high atmosphere.

#### Gravitational influences

The irregular gravitational field of the earth exerts a “tipping moment” on the orbital plane of the orbit of a satellite close to the earth, which the orbital plane evades by a precession movement according to the gyroscopic laws . This evasive movement means that the ascending node or the nodal line is not fixed, but rotates slowly in the equatorial plane and the right ascension of the ascending node changes continuously. The orbit plane rotates around the z-axis of the astronomical coordinate system (Fig. 1) . This temporal change in degrees per day (° / d) can be calculated using the following relationship ( = earth radius): ${\ displaystyle \ Omega}$${\ displaystyle r_ {A}}$

${\ displaystyle {\ dot {\ Omega}} = - 9 {,} 9641 \ cdot \ left ({\ frac {r_ {A}} {a}} \ right) ^ {\ frac {7} {2}} \ cdot {\ frac {\ cos i} {\ left ({1- \ varepsilon ^ {2}} \ right) ^ {2}}}}$. (12)

At the same time, the apsidal line rotates around the center of the earth in the plane of the orbit - also due to the influence of gravity. This also changes the argument of perigee over time, which can be calculated in degrees per day (° / d): ${\ displaystyle \ omega}$

${\ displaystyle {\ dot {\ omega}} = 4 {,} 98 \ cdot \ left ({\ frac {r_ {A}} {a}} \ right) ^ {\ frac {7} {2}} \ cdot {\ frac {5 \ cos ^ {2} i-1} {\ left ({1- \ varepsilon ^ {2}} \ right) ^ {2}}}}$. (13)

If the corresponding values ​​from the TLE example are inserted into both equations, the result is that the right ascension of the ascending node decreases by 5.1401 ° / d and the argument of the perigee increases by 3.8308 ° / d. In this calculation, however, it is assumed that the values ​​of the semi-major axis , the inclination and the numerical eccentricity remain constant, which is not the case in reality (see Section 4: Changes over a longer period of time). For a very short-term forecast calculation, this is nevertheless sufficiently accurate. ${\ displaystyle a}$${\ displaystyle i}$${\ displaystyle \ varepsilon}$

The effect of this path disruption can also be used positively. A sun-synchronous orbit can be generated by appropriate selection of the inclination or the perigee can be held over a fixed earth point, which is used for Molnija orbits . This can be calculated as follows:

 ${\ displaystyle {\ dot {\ Omega}} = {\ frac {360 ^ {\ circ}} {365 \, {\ text {days}}}}}$ ${\ displaystyle \ quad \ Rightarrow i = 96 ^ {\ circ} {-} 99 ^ {\ circ}}$ ${\ displaystyle {\ dot {\ omega}} = 0}$ ${\ displaystyle \ quad \ Rightarrow i = 63 {,} 43 ^ {\ circ}}$

#### Braking effect

In the simplest case, the resistance coefficient is the first derivative of the mean motion with respect to time and in the TLE of the above example for the ISS , that is given. ${\ displaystyle {\ dot {n}} / 2 = 0 {,} 00012260d ^ {- 2}}$${\ displaystyle {\ dot {n}} = 0 {,} 00024520d ^ {- 2}}$

Based on the mean movement from the TLE, the number of revolutions per day increases . Inserted into equations (1) and (2), this results in a decrease in the major semi-axis of 67.177 meters per day. ${\ displaystyle n = 15 {,} 74647269d ^ {- 1}}$${\ displaystyle a}$

## Diagrams

A recording over a longer period of time shows how the satellite orbit elements actually change over time. In the following, the course of the two line elements (244 data sets) and the values ​​derived from them for the period from June 11, 2005 to February 11, 2006 are graphically prepared for the International Space Station ISS. In the diagrams, the x-axis represents the time axis, the y-axis the associated values. In the darkened period (July 27 - August 6, 2005) the Space Shuttle Discovery was docked during the STS-114 mission.

### Course of the mean movement

Diagram 1: Course of the mean movement

The middle movement is the satellite orbit element in which the change due to the braking effect is one of the most noticeable. The closer a satellite comes to the earth, the higher its orbital speed and thus the number of orbits per day (diagram 1). ${\ displaystyle n}$

So that the ISS does not burn up in the atmosphere at some point, the orbit is raised from time to time. These orbit corrective maneuvers, known as "orbit reboost", are carried out by igniting the on-board engines or the engines of the docked space shuttle or progress spaceship. The red dots in the adjacent diagrams mark the point in time of a reboost. Depending on the burn time of the engines, the track is raised to a greater or lesser extent and the number of revolutions is reduced again. Further effects are explained in the following sections.

### Course of the major semi-axis

Diagram 2: Course of the major semi-axis

The change in the semi-major axis is inversely proportional to the change in the mean motion (see equations (1) and (2)). This is the best way to understand the rise and fall of the orbit (diagram 2). ${\ displaystyle a}$${\ displaystyle n}$

It can be seen that during the STS-114 mission the orbit was corrected a total of six times by igniting the shuttle engines. In addition to other smaller ones, there was a significant change on November 11, 2005, when the major semi-axis increased by 7,731.5 m.

You can also see that the decrease does not take place evenly. This is due to changes in the density of the “high” atmosphere caused by the irregular activity of the sun. In order to determine a longer-term trend, one can still calculate a linear trend . This is shown in the diagram from the last orbit reboost. The slope or rate of descent of the regression line ( y = m · x + b ) is an average of -81.7 meters per day.

### Course of the inclination

Diagram 3: Course of the inclination

The inclination of the orbit plane fluctuates slightly around an average value of . The main reason for this are the gravitational influences of the moon. Significant jumps usually result from the orbit correction maneuvers (diagram 3). ${\ displaystyle i = 51 {,} 6433 ^ {\ circ}}$

### Course of the numerical eccentricity

Diagram 4: Course of the numerical eccentricity

The numerical eccentricity is mainly influenced by the radiation pressure of the sun and by the solar winds, which cause an acceleration away from the sun. In addition to solar activity, this also depends on the reflection factor and the size of the solar panels of a satellite, which at the ISS are relatively large at 74 meters wide (expansion stage 2005). ${\ displaystyle \ varepsilon}$

Although the values ​​of the numerical eccentricity are very small - deviation from the ideal circular path between 0.0685 ‰ and 1.1033 ‰ - a clear jump at the time of the last orbit reboost can be seen with the appropriate scaling (diagram 4). This is due to the fact that the acceleration when the orbit was raised mainly worked in the direction of the apogee and the orbital ellipse was stretched somewhat, which in turn increases the eccentricity (see also the next section, orbital level diagram). If such orbit correction maneuvers are planned, NASA announces this in its bulletins, from which one can see the change in direction and speed in a vector representation ( attention : the information there is in feet per seconds and nautical miles ).

### Course of the train stations

Diagram 5: Course of the train levels

In most tracking programs, the height in kilometers above the surface of the earth is given for the perigee and apogee instead of the distance from the center of the earth. However, there is often no information about the radius of the earth used. The reference here is the equatorial radius of the WGS84 ellipsoid with 6,378.137 km. This value is subtracted from the results obtained from equations (4) and (5), and the orbital heights in perigee and apogee above the earth are obtained.

The actual course of the path is only shown here, since the fluctuations in the numerical eccentricity are directly incorporated. Therefore, in Diagram 5, the course of the numerical eccentricity is shown again to make the influence clear. ${\ displaystyle \ varepsilon}$${\ displaystyle \ varepsilon}$

Furthermore, the effect mentioned in the previous section becomes clear here, namely that the last orbit reboost had an effect more in the direction of the apogee. The increase in train stations in apogee is 12.172 km in contrast to 3.292 km in perigee.

### Course of the right ascension of the ascending node

Diagram 6a: Course of the right ascension

According to the gyroscopic laws, the ascending node rotates around the z-axis of the astronomical coordinate system. Shown as a function of the sine (diagram 6a), the right ascension of the ascending node results in an almost harmonic course, i.e. H. the change over time is almost linear. ${\ displaystyle \ Omega}$

Diagram 6b: Course of the mean rotation of the right ascension

Only when you determine the values ​​of the mean rotation per day (diagram 6b) can you see them deviate slightly from an average value. In addition, a tendency can be seen that between two orbit reboosts - the time interval just has to be large enough - the rotation rate increases by around 0.00025 ° per day. According to equation (12), this must also be the case, since the major semi-axis is included there as a decreasing quantity over time. The small fluctuations in the rotation around the regression line result from the fluctuations in the inclination and the numerical eccentricity . ${\ displaystyle a}$${\ displaystyle i}$${\ displaystyle \ varepsilon}$

### Course of the argument of perigee

Diagram 7: Course of the argument of perigee

The rotation of the apsidal line and thus of the perigee is anything but stable. If you compare the curve in Diagram 7 with that of the numerical eccentricity in Diagram 4, it becomes apparent that the rotation is more harmonious only with a significant increase in the eccentricity. ${\ displaystyle \ varepsilon}$

If the average change is calculated for this area (from November 11, 2005) using a regression analysis , the argument of perigee increases by 3.7669 ° per day.