Lecture 2, Space Segment

Presentation / Lecture 2, Space Segment

Date Submitted: 06 June 2001

Written by RPC Telecommunications Ltd.. Website: http://www.rpctelecom.com

  Rate This Article

This is the second in the series of general satcom tutorial lectures submitted by RPC Telecommunications.

Comment On This Article

 Printable Version
-Section 1
Orbit Theory: Kepler's Laws.
-Section 2
Orbit Theory: Formulae.
-Section 3
The Real World.
-Section 4
Types & Implications of Orbit.
-Section 5
Satellite Hardware.
-Section 6
Launch Vehicles.
-Section 7
Flight Plan & Maintaining Satellite In Orbit.
-Section 8
Stationkeeping & Stabilisation.

Kepler And Satellite Ephemeris Formulae

Semi-Major Axis of satellite orbit:

Smaj = ((T/(2*π))2*Mu)(1/3)

Notes:

T is the orbit period (seconds)
Mu is the product of the gravitational constant and the earth's mass.
Mu=G*Me=6.67*10-11 * 5.974 * 1024
Mu=3.986*1014 metres3/second

Semi-Minor Axis of satellite orbit:

Smin = (Smaj2-C2)0.5 metres

Notes:

C = E*Smaj=Smaj-Pe=Ap-Smaj
E is the eccentricity of the orbit
Pe-R is the perigee (or the minimum) satellite altitude in metres.
Ap-R is the apogee (or the maximum) satellite altitude in metres.
R is the radius of the Earth (6378 * 103 metres)

Mean Anomaly

M = Ecc - E*sin(Ecc)

Notes:

M is the mean anomaly and the mean angle between perigee and satellite, as seen from the centre of the earth at any date and time.
M=ω*time + Anom
Anom is the reference value of the mean anomaly (M) at a specified date and time.
ω is the mean angular velocity of the satellite as seen from the centre of the earth = 2*π/T
Ecc is the eccentric anomaly which can be treated as a variable defined by the equation above.

Virtual Anomoly

V = tan-1(tan(Ecc/2)*((1+E)/(1-E)))

Notes:

V is the virtual anomoly or the true angle between perigee and satellite, as seen from the centre of the earth, at any date and time.

Radius of Satellite Orbit

Radius = Smaj*(1-E2)/(1-E*cos(V)) metres

Notes:

Radius of the satellite orbit is the distance between the satellite and the centre of the earth at any date and time.
Satellite altitude = Radius - R

Satellite Linear Speed

Speed = (Mu*(2/Radius - 1/Smaj))0.5

Notes:

Speed of the satellite is the linear speed of the satellite at any date and time.

Sub-satellite Latitude

Q = sin-1(sin(I)*sin(V+Per))

Notes:

Per is the argument of the perigee and the angle between the perigee and the ascending node i.e. the angle between perigee and the equator (satellite heading north) within the satellite's orbital plane.
I is the satellite orbit inclination angle and is the maximum angle between the equatorial plane and the plane of the satellite orbit.

Sub-satellite Longitude

U = cos-1(cos(V+Per)/cos(Q)) + Asc

Notes:

Asc is the ascending node and the angle in the earth's equatorial plane between Aries and the satellite's orbital plane (satellite heading north).

Geometric Formulae
Great circle angle between an earth station and a satellite

Ngc = cos-1(sin(P)*sin(I)+cos(P)*cos(I)*cos(Us-Ue))

Notes:

P is the earth station latitude
Ue is the earth station longitude
I(t) is the satellite (instantaneous) inclination
Us(t) is the (instantaneous) longitude of the sub-satellite point

Earth station elevation angle to the satellite

Ee = tan-1((cos(Ngc)-R/(H+R))/sin(Ngc))

Notes:

R is the radius of the earth (6378*103m)
H(t) is the (instantaneous) altitude of the satellite (geostationary satellite: 35786*103m)
Ngc is the great circle angle

Earth station azimuth angle to the satellite

Ae = tan-1(sin(Usat-Ues)/(sin(P)*cos(Us-Ue)-tan(I)*cos(P))

Notes:

It is necessary to investigate the signs of the numerator and denominator to find the quadrant in which the azimuth angle is.

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))

Notes:

Suffixes 1 and 2 denote angles to the first and second satellites.

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)))

Notes:

Z=(R2*cos2(P)+(R+H)2-2*R*cos(P)*(R+H)*cos(Us-Ue)0.5)
Z=(S2-R2*sin2(P))0.5
I (satellite inclination angle) is zero.

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))

Notes:

I (satellite inclination angle) is zero.

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))

Notes:

Esb is the satellite elevation angle in the direction of the boresight.
Ese is the satellite elevation angle in the direction of the earth station.
Asb is the satellite azimuth angle in the direction of the boresight.
Ase is the satellite azimuth angle in the direction of the earth station.

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

Notes:

A is the major axis of the ellipse
B is the minor axis of the ellipse
Q=tan-1((Ese-Esb)/(Ase-Asb)) - O
O is the orientation of the ellipse defined as the counter clockwise angle from a line parallel to the equatorial plane to the major axis of the ellipse.

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)))

Notes:

cos(Ngc)=cos(Us-Ue)*cos(P)
cos(Ngc)=cos(sin-1((R+H)/R*sin(Nso))-Nso)
Ngco is the great circle angle between an earth station and a geostationary satellite (can be found by letting I=0 in our first equation in this section).
Nso=cos-1(cos(Es)*cos(As))
Nso is the angle between the sub-satellite point and the earth station as determined from the geostationary satellite (can be found by letting Esb=0 in our equation for Ns above).

Earth station longitude expressed in terms of satellite azimuth and elevation angles

Us-Ue = sin-1(sin(As)*tan(P)/tan(Es))

Next: Section 3 - The Real World.

 
 

Home | Satellite School | Presentations | Calculators | Software | Forum | Submit | Books | Contact Us

 

Copyright 2018 Satcom Online (About This Site)