|Lecture 2, Space Segment (http://www.satcom.co.uk/article.asp?article=29)|
|Welcome to the second lecture in the RPC Satcom Tutorial
At any point in the lecture, you can skip to a latter section, or back to earlier sections using the guide on the left.
Topics contained in this lecture:
Laws Of Motion
Kepler's laws were originally developed to describe the motion of planetary bodies, but they are just as relevant for theoretical satellite orbits around the earth.
The geometry of an ellipse:
Eccentricity (e) is defined as:
Newton's universal law of gravitation tells us that:
From Newton's Second Law of Motion we get:
Centripetal acceleration (assume circular motion for simplicity):
Taking these three expressions, and simplifying we get the following equation:
|The "Keplerian Elements" are a set of 6 parameters
which define the position of a satellite at a given epoch (date and time) as well as
defining the shape and position of its elliptical orbit around the Earth.
The 6 parameters are outlined below.
With these elements at a known epoch, it is possible to fix the position of the satellite in space and predict its theoretical future movement.
Note: The theoretical equations can fully determine the relative orbital motion of two bodies, but the case of three bodies has no analytical solution.
Kepler And Satellite Ephemeris Formulae
|Semi-Major Axis of satellite orbit:
Smaj = ((T/(2*π))2*Mu)(1/3)
Semi-Minor Axis of satellite orbit:
Smin = (Smaj2-C2)0.5 metres
M = Ecc - E*sin(Ecc)
V = tan-1(tan(Ecc/2)*((1+E)/(1-E)))
Radius of Satellite Orbit
Radius = Smaj*(1-E2)/(1-E*cos(V)) metres
Satellite Linear Speed
Speed = (Mu*(2/Radius - 1/Smaj))0.5
Q = sin-1(sin(I)*sin(V+Per))
U = cos-1(cos(V+Per)/cos(Q)) + Asc
|Great circle angle between an earth station and a satellite
Ngc = cos-1(sin(P)*sin(I)+cos(P)*cos(I)*cos(Us-Ue))
Earth station elevation angle to the satellite
Ee = tan-1((cos(Ngc)-R/(H+R))/sin(Ngc))
Earth station azimuth angle to the satellite
Ae = tan-1(sin(Usat-Ues)/(sin(P)*cos(Us-Ue)-tan(I)*cos(P))
Earth station off-axis angle between the principal axis of the antenna and the direction of a second satellite
Ne = cos-1(sin(Ee1)*sin(Ee2)+cos(Ee1)*cos(Ee2)*cos(Ae1-Ae2))
Slant range between an earth station and satellite
S = (-)R*sin(Ee)+((R*sin(Ee))2+H*(H+2*R))0.5 metres
S = R*sin(Ngc)/cos(Ee+Ngc) metres
Geostationary satellite azimuth angle where the earth's axis of rotation defines zero degrees azimuth
As = sin-1(R*cos(P)/Z*sin(Us-Ue))
As = tan-1(R*cos(P)*sin(Us-Ue)/(R+H-R*cos(Us-Ue)*cos(P)))
Geostationary satellite elevation angle where the earth's equatorial plane defines zero degrees elevation
Es = sin-1(R*sin(P)/S)
Es = tan-1(sin(As)*tan(P)/sin(Us-Ue))
Satellite off-axis angle between the principal axis of the satellite antenna and the direction of the earth station
Ns = cos-1(sin(Ese)*sin(Esb)+cos(Ese)*cos(Esb)*cos(Ase-Asb))
Beamwidth of a satellite antenna with an elliptical beam in the direction of an earth station
Phio = (A2*B2/A2(sin(Q))2+B2(cos(Q))2))0.5
Earth station latitude expressed in terms of satellite azimuth and elevation angles
P = sin-1(tan(Es)/R/cos(As)*(R+H-R*cos(Ngco)))
Earth station longitude expressed in terms of satellite azimuth and elevation angles
Us-Ue = sin-1(sin(As)*tan(P)/tan(Es))
|The Real World|
|(Refer to simplifying assumptions from section 1)
The stationary point is the centre of mass, not the centre of the Earth - but the satellite is so small as to make this negligible in virtually all cases.
The earth cannot be treated as a point mass since it is not a sphere:
Other forces will act on the satellite to perturb the orbit:
The effects of the "real world" forces are to cause movements of the satellites orbit relative to the Earth which would not be predicted by Kepler:
This means that not all orbits are useful!
Approximate magnitudes of relative forces acting on a satellite at specific heights above the Earth's surface:
[Relative force = (Average force exerted by perturbation) / (Force exerted by a symmetrical Earth)]
Major perturbations of a general satellite orbit:
Relatively complex equations govern rate of precession of argument of perigee and plane of orbit.
Stable or near stable conditions exist which lead to useful orbits:
Perturbations mean that real orbits cannot be predicted over extended period, however prediction is reliable over short periods of a few days for most orbits.
So predictions are used based on regularly updated orbital element sets to determine the real world satellite path and position.
Types of Satellite Orbit
Low Earth Orbit
Highly Elliptical Orbit
|Satellite Orbit Implications On Design|
Period & Visibility
Environment & Power
Number of satellites required?
Number of Earth stations required?
|A satellite is comprised of a payload supported by a platform.
The platform takes up the largest proportion of the mass of the satellite - design objectives is therefore to support the largest possible payload with the smallest possible platform.
Major Spacecraft Sub-Systems:
Payload Block Diagram:
One satellite in the GSO can "see" about one third of the Earth's surface so can provide telephone, data and TV links over the whole of that area.
However, coverage is impossible to the polar regions (beyond about ±80º latitudes). (An aircraft at 30,000 feet can still communicate at latitude 85º)
MSS satellite antennas commonly have maximum gain (about 21 dBi) at the edge of coverage, and lower gain (18 dBi) at beam centre.
Three geostationary satellites at 120º spacing provide full Earth coverage (except for the polar regions).
|Satellite Beam Options|
A choice of smaller beam means higher gain and EIRP but at the cost of a reduced coverage and possibly connectivity.
Space Transportation System (space shuttle) can deliver to LEO/Parking orbit only. Also the payload must have additional PAM/PKM to achieve GTO. Total STS payload bay launch capability is very large, but must take account of the support cradle and also the PAM stage.
All prices are quoted for a dedicated launch. Estimated prices are shown underlined:
*Long March have quoted a price of $30 million, but in order to protect the interests of western launch vehicles, the US has used the COCOM rules to prevent Long Match from launching any vehicle commercially for less than $58 million. This is likely to apply also to the Soviet Proton and other soviet vehicles.
|Target orbit (e.g. GSO) cannot, usually, be achieved in a single
trajectory. Launch vehicles are "multi-staged" so propulsion is a series of distinct
Launch Flight Plan
|The launch flight plan is different for
Expendable and Reusable Launch Vehicles.
The flight plan will be comprised of clear stages seperated by motor "burns"
The objective is usually to achieve target orbital position with the minimum fuel used, but this takes the maximum time to achieve.
An operator can trade off fuel for speed. However, you cant refual a satellite in orbit (yet!).
Reusable Launch Vehicle
Elliptical transfer orbit - "Hohmann Transfer Ellipse" - is minimum energy path between two circular orbits.
Expendable Launch Vehicle
|Maintaining The Satellite In Orbit|
Attitude Control and Stabilisation
Stationkeeping and Stabilisation require two elements:
|Sources of Positional Data|
Inertial Space (gyroscopes and accelerometers)
Spinning Earth Sensor (Earth Horizon Telescope)
Reaction Wheels (e.g. Olympus)
Rubber diaphragm or capillary action used to ensure fuel is expelled from output pipe. Heaters are used to increate specific impulse.
Commonly used for both apogee motor and stationkeeping. Tanks commonly oversized and fuel added until satellite has specified wet mass.
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