AT737
Satellite Orbits and Navigation 1
AT737 Satellite Orbits and Navigation 2
Newton’s Laws
1. Every body will continue in its state of rest or of uniform motion in a straight line except insofar as it is compelled to change that state by an impressed force.
2. The rate of change of momentum is proportional to the impressed force and takes place in the line in which the force acts.
3. Action and reaction are equal and opposite.
AT737 Satellite Orbits and Navigation 3
Newton's Second Law is the familiar
where F is force, m is mass, a is acceleration, v is velocity, and t is time.
dt
dvmmaF
Newton’s Laws (continued)
AT737 Satellite Orbits and Navigation 4
Newton’s Law of Universal Gravitation
The force of attraction between two point masses m1 and m2 separated by a distance r is
where G is the Newtonian (or universal) gravitation constant (6.67259 x 10-11 N m2 kg-
2).
221
r
mGmF
AT737 Satellite Orbits and Navigation 5
Circular Orbit Example
CentripetalForce 2
2
r
mGm
r
mv e GravitationalForce
-23 14 s m10 3.986005eGm
32
2 4r
GmT
e
Period v
rT
2 The NOAA satellites
orbit at about 850 km above the surface (r = 7228 km) and therefore have a period of about 102 minutes.
AT737 Satellite Orbits and Navigation 6
Kepler’s Laws1. All planets travel in elliptical paths with the sun
at one focus.
2. The radius vector from the sun to a planet sweeps out equal areas in equal times.
3. The ratio of the square of the period of revolution of a planet to the cube of its semimajor axis is the same for all planets revolving around the sun.
The same laws apply if we substitute satellite for planet and earth for sun, but the proportionality constant is different.
AT737 Satellite Orbits and Navigation 7
Ellipse Geometry
a = semimajor axis = eccentricity (0-1) = true anomalyr = radius
cos1
)1( 2
a
r Equation of an Ellipse
AT737 Satellite Orbits and Navigation 8
Kepler’s Equation
3
2
sin)(
a
Gm
Tn
eettnM
e
p
M = Mean anomalyn = mean motion constanttp = time of perigeal passagee = eccentric anomaly = eccentricity
Angles M, e, and θ are zero at perigee.
cos1
coscos
cos1
coscos
e
e
e
NOTE: All angles in radians.
AT737 Satellite Orbits and Navigation 9
Right Ascension & Declination
= declination = right ascension
Need a coordinate system to orient orbital plane in
space
AT737 Satellite Orbits and Navigation 10
Orientation Angles
i = inclination angle = argument of perigee = right ascension of ascending node
i < 90° progradei > 90° retrograde
AT737 Satellite Orbits and Navigation 11
Classical Orbital Elements
Element Symbol
Semimajor axis* a
Eccentricity
Inclination i
Argument of perigee o
Right ascension of ascending node
o
Mean anomaly** Mo
Epoch time to*Two-line elements give orbits per day instead of a**ESA uses true anomaly instead of mean anomaly
AT737 Satellite Orbits and Navigation 12
Sources of Orbital Elements
NOAA Satellite Information System
http://noaasis.noaa.gov/NOAASIS/ml/quicklook.html(current TBUS and TLEs for GOES and NOAA satellites)
T.S. Kelso’s CelesTrak sitehttp://celestrak.com(TLEs for a lot of satellites—still in business in spite of Space Track)
New Government Sitehttp://www.space-track.orgEstablished by Public Law 108-136, Section 913
AT737 Satellite Orbits and Navigation 13
Keplerian Orbits
Viewed from space, Keplerian orbits are constant and simple.
Viewed from a point rotating with the earth, Keplerian orbits are complex.
AT737 Satellite Orbits and Navigation 14
Orbit Perturbing Forces
Force Source
Nonspherical gravitational fieldNonspherical, nonhomogeneous Earth
Gravitational attraction of other bodies
Sun, moon, planets
Radiation pressure Solar radiation
Particle flux Solar wind
Lift and drag Residual atmosphere
Electromagnetic forces
Interaction of electrical currents in the satellite with Earth’s magnetic field
AT737 Satellite Orbits and Navigation 15
Perturbation Equations
2
2
2 sin312
11
r
rJ
r
GmU eee
i
a
rJnn
dt
dM ee 22/322
2 sin2
311
2
31
i
a
rJn
dt
d ee cos12
3 222
2
i
a
rJn
dt
d ee 2222
2 sin2
521
2
3
U is the gravitational potential energy
(g = -U)
ree = equatorial radius of Earth = 6,378,137 m
J2 = 1.08263 x 10-3
a, , and i are unperturbed
Anomalistic mean motion constant
AT737 Satellite Orbits and Navigation 16
…and more equations
nT
2
Anomalistic period—the time from perigee to moving perigee
The reciprocal of this is what you get in two-line elements in place of the semimajor axis
dtd
nT
2~ Synodic or nodal period—
the time from ascending node to ascending node
AT737 Satellite Orbits and Navigation 17
Where is that satellite?A step-by-step calculation guide
1. Find the orbital elements of the satellite you are interested in.
2. Update the variable elements (M, , & ) to the time (t ) that you are interested in:
M = Mo + (t – to)(dM/dt), etc.
3. Use Kepler’s equation to calculate the true anomaly ().
4. Use ellipse equation to calculate r, the distance of the satellite from the center of the earth.
AT737 Satellite Orbits and Navigation 18
Calculations continued…
5. Calculate the argument of latitude: +
(measures angular distance from equator).
6. Calculate latitude: = sin-1(sin sin i)
7. Calculate the right ascension of the satellite at time t:
cos
cossintan 1 i
sat
= right ascension of ascending node as calculated in step 2
AT737 Satellite Orbits and Navigation 19
Calculations completed
8. Calculate the right ascension of Greenwich (the prime meridian) at time t:
Greenwich = 99.965° + 360.985645 twhere t is the time in days (and fraction) between time t and 0000 UTC 1 January 2000.
9. Calculate the longitude of the satellite: = sat - Greenwich
Homework
Top Related