Orbital Mechanics
- To achieve a stable orbit around the earth, a spacecraft must first be beyond the bulk of the earth’s atmosphere, i.e., in what is popularly called space.
- According to Newton's law of motion F = ma. Where a = acceleration, F = force acting on the object and m = mass of the object. It helps us understand the motion of satellite in a stable orbit.(neglecting any drag or other perturbing forces).
- (F = ma) states that the force acting on a body is equal to the mass of the body multiplied by the resulting acceleration of the body.
- Thus, for a given force, the lighter the mass of the body, the higher the acceleration will be.
- When in a stable orbit, there are two main forces acting on a satellite: a centrifugal force due to the kinetic energy of the satellite, which attempts to fling the satellite into a higher orbit, and a centripetal force due to gravitational attraction of the planet about which the satellite is orbiting, which attempts to pull the satellite towards the planet.
- If these two forces are equal the satellite remains in a stable orbit.
Forces involved in orbital mechanics
1.Gravitational force : Attraction between any two objects.
2.Centrifugal force : An outward-directed force that normally balances the inward-directed centripetal force.
The standard acceleration due to gravity at the earth surface is \(981 cm/s^2\). The value decreases with height above the earth’s surface. The acceleration, a, due to gravity at a distance r from the centre of the earth is $$a = μ/r^2 km/s^2$$
Where the constant \(μ\) is the product of the universal gravitational constant \(G\) and the mass of the earth \(M_E\).
The product \(GM_E\) is called kepler’s constant and has the value \(3.98 x 10^5 km^3/s^2\). The universal gravitational constant is \(G = 6.672x 10^{-11} Nm^2/kg^2\).
The mass of the earth \(M_E = 5.97 × 10^{24} kg\).
Since force= mass × acceleration, the centripetal force acting on the satellite, \(F_{in}\) is given by $$F_{in} = m × (μ/r^2)$$ $$= m × (GM_E/r^2)$$
In a similar fashion, the centrifugal acceleration is given by \(a = v^2/r\).
Which will give the centrifugal force, \(F_{out}\) as \(F_{out} = m × (v^2/r)\)
If the forces of the satellite are balanced \(F_{in} = F_{out}\) $$m × (μ/r^2) = m × (v^2/r)$$
Hence the velocity v of the satellite in a circular orbit is given by \(v = (μ/r)^{1/2}\)
If the orbit is circular, the distance traveled by a satellite in one orbit around a planet is 2πr , where r is the radius of the orbit from the satellite to the center of the planet. Since distance divided by velocity equals time to travel the distance, the period of satellite’s orbit, T, will be $$T = \frac{2πr}{v} = \frac{2πr}{(μ/r)^{1/2}}$$ $$T^2 = \frac{2πr^3}{μ}$$
Using standard mathematical procedures we can develop an equation for the radius of the satellite’s orbit, r, namely.
N.B.: For elliptical orbit, to calculate time period of satellite, you have to replace r by a. where a = semi major axis of the elliptical orbit
Kepler’s Laws
Kepler’s laws of planetary motion apply to any two bodies in space that interact through gravitation. The laws of motion are described through three fundamental principles.
Kepler’s First Law, as it applies to artificial satellite orbits, can be simply stated as follows: ‘The path followed by a satellite around the earth will be an ellipse, with the center of mass of earth as one of the two foci of the ellipse.’ This is shown in Figure:
If no other forces are acting on the satellite, either intentionally by orbit control or unintentionally as in gravity forces from other bodies, the satellite will eventually settle in an elliptical orbit, with the earth as one of the foci of the ellipse. The ‘size’ of the ellipse will depend on satellite mass and its angular velocity.
Kepler’s Second Law can likewise be simply stated as follows: ‘For equal time intervals, the satellite sweeps out equal areas in the orbital plane.’ The following figure demonstrates this concept.
The shaded area \(A_1\) shows the area swept out in the orbital plane by the orbiting satellite in a one hour time period at a location near the earth. Kepler’s second law states that the area swept out by any other one hour time period in the orbit will also sweep out an area equal to \(A_1\). For example, the area swept out by the satellite in a one hour period around the point farthest from the earth (the orbit’s apogee), labeled \(A_2\) on the figure, will be equal to \(A_1\), i.e.: \(A_1 = A_2\).
This result also shows that the satellite orbital velocity is not constant; the satellite is moving much faster at locations near the earth, and slows down as it approaches apogee. This factor will be discussed in more detail later when specific satellite orbit types are introduced.
Kepler’s Third Law is as follows: ‘The square of the periodic time of orbit is proportional to the cube of the mean distance between the two bodies.’ This is quantified as follows:$$T^2 = \left[\frac{4π^2}{μ}\right]a^3$$
Where T = orbital period in s; a = distance between the two bodies, in km; \(μ = Kepler’s Constant = 3.986004×105 km^3/s^2\). If the orbit is circular, then a = r, and $$r = \left[\frac{μ}{4π^2}\right]^{\frac{1}{3}}T^{\frac{2}{3}}$$
This demonstrates an important result: Orbit Radius = [Constant] × (Orbit Period)\(^{2/3}\)
Under this condition, a specific orbit period is determined only by proper selection of the orbit radius. This allows the satellite designer to select orbit periods that best meet particular application requirements by locating the satellite at the proper orbit altitude. The altitudes required to obtain a specific number of repeatable ground traces with a circular orbit are listed in the following table 1.
| Revolutions / day | nominal period (hours) | Nominal altitude (km) |
|---|---|---|
| 1 | 24 | 36000 |
| 2 | 12 | 20200 |
| 3 | 8 | 13900 |
| 4 | 6 | 10400 |
| 6 | 4 | 6400 |
| 8 | 3 | 4200 |
Orbital Elements:
Apogee: A point for a satellite farthest from the Earth. It is denoted as ha.Perigee: A point for a satellite closest from the Earth. It is denoted as hp.
Line of Apsides: Line joining perigee and apogee through center of the Earth. It is the major axis of the orbit. One-half of this line’s length is the semi-major axis equivalents to satellite’s mean distance from the Earth.
Ascending Node: The point where the orbit crosses the equatorial plane going from north to south.
Descending Node: The point where the orbit crosses the equatorial plane going from south to north.
Inclination: The angle between the orbital plane and the Earth’s equatorial plane. Its measured at the ascending node from the equator to the orbit, going from East to North. Also, this angle is commonly denoted as i.
Line of Nodes: the line joining the ascending and descending nodes through the center of Earth.
Prograde Orbit: an orbit in which satellite moves in the same direction as the Earth’s rotation. Its inclination is always between 00 to 900. Many satellites follow this path as Earth’s velocity makes it easier to lunch these satellites.
Retrograde Orbit: an orbit in which satellite moves in the same direction counter to the Earth’s rotation.
Argument of Perigee: An angle from the point of perigee measure in the orbital plane at the Earth’s center, in the direction of the satellite motion.
Right ascension of ascending node: The definition of an orbit in space, the position of ascending node is specified. But as the Earth spins, the longitude of ascending node changes and cannot be used for reference. Thus for practical determination of an orbit, the longitude and time of crossing the ascending node is used. For absolute measurement, a fixed reference point in space is required. It could also be defined as “right ascension of the ascending node; right ascension is the angular position measured eastward along the celestial equator from the vernal equinox vector to the hour circle of the object”.
Mean anamoly: It gives the average value to the angular position of the satellite with reference to the perigee.
True anamoly: It is the angle from point of perigee to the satellite’s position, measure at the Earth’s center.
i Inclination, Ω Right Ascension of ascending node, ω Argument of perigee v True anomaly.
Orbital Elements Following are the 6 elements of the Keplerian Element set commonly known as orbital elements.
1. Semi-Major axis (a)2. Eccentricity (e)
They give the shape (of ellipse) to the satellite’s orbit.
3. Mean anomaly (M0)
It denotes the position of a satellite in its orbit at a given reference time.
4. Argument of Perigee
It gives the rotation of the orbit’s perigee point relative to the orbit‟s nodes in the earth’s equatorial plane.
5. Inclination
6. Right ascension of ascending node
They relate the orbital plane’s position to the Earth. As the equatorial bulge causes a slow variation in argument of perigee and right ascension of ascending node, and because other perturbing forces may alter the orbital elements slightly, the values are specified for the reference time or epoch.
Look angle determination
The look angles for the ground station antenna are Azimuth and Elevation angles. They are required at the antenna so that it points directly at the satellite. Look angles are calculated by considering the elliptical orbit. These angles change in order to track the satellite.
For geostationary orbit, these angels values does not change as the satellites are stationary with respect to earth. Thus large earth stations are used for commercial communications, these antennas beamwidth is very narrow and the tracking mechanism is required to compensate for the movement of the satellite about the nominal geostationary position.
For home antennas, antenna beamwidth is quite broad and hence no tracking is essential. This leads to a fixed position for these antennas.
Sub satellite point: The point, on the earth’s surface of intersection between a line from the earth’s center to the satellite.
Orbital perturbation
- Theoretically, an orbit described by Kepler is ideal as Earth is considered to be a perfect sphere and the force acting around the Earth is the centrifugal force. This force is supposed to balance the gravitational pull of the earth.
- In reality, other forces also play an important role and affect the motion of the satellite. These forces are the gravitational forces of Sun and Moon along with the atmospheric drag.
- Effect of Sun and Moon is more pronounced on geostationary earth satellites where as the atmospheric drag effect is more pronounced for low earth orbit satellites.
- As the shape of Earth is not a perfect sphere, it causes some variations in the path followed by the satellites around the primary. As the Earth is bulging from the equatorial belt, and keeping in mind that an orbit is not a physical entity, and it is the forces resulting from an oblate Earth which act on the satellite produce a change in the orbital parameters.
- This causes the satellite to drift as a result of regression of the nodes and the latitude of the point of perigee (point closest to the Earth). This leads to rotation of the line of apsides. As the orbit itself is moving with respect to the Earth, the resultant changes are seen in the values of argument of perigee and right ascension of ascending node.
- Due to the non-spherical shape of Earth, one more effect called as the “Satellite Graveyard” is seen. The non-spherical shape leads to the small value of eccentricity at the equatorial plane. This causes a gravity gradient on GEO satellite and makes them drift to one of the two stable points which coincide with minor axis of the equatorial ellipse.
- Working satellites are made to drift back to their position but out-of-service satellites are eventually drifted to these points, and making that point a Satellite Graveyard.
Atmospheric Drag
- For Low Earth orbiting satellites, the effect of atmospheric drag is more pronounces. The impact of this drag is maximum at the point of perigee. Drag (pull towards the Earth) has an effect on velocity of Satellite (velocity reduces).
- This causes the satellite to not reach the apogee height successive revolutions. This leads to a change in value of semi-major axis and eccentricity. Satellites in service are maneuvered by the earth station back to their original orbital position.
Orbit determination
Orbit determination requires that sufficient measurements be made to determine uniquely the six orbital elements needed to calculate the future of the satellite, and hence calculate the required changes that need to be made to the orbit to keep it within the nominal orbital location. The control earth stations used to measure the angular position of the satellites also carryout range measurements using unique time stamps in the telemetry stream or communication carrier. These earth stations generally referred to as the TTC&M(telemetry tracking command and monitoring) stations of the satellite network.
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