APPLIED PHYSICS – GRAVITATION
ARTIFICIAL SATELLITES*
EMILIA DANA SELEŢCHI
PhD student - University of Bucharest, Department of Atomic and Nuclear Physics, Măgurele,
P.O. Box, MG-11, RO-077125, Bucharest, ROMANIA, e-mail: [email protected]
Received December 21, 2004
Presenting “modelling” as a modern method of scientific investigation, by
combining different types of pattern: icons (images) or ideals (logical-mathematical),
this paper presents an overview on satellite system. By using “the algorithm” as a way
of studying objects and phenomena or as a method of solving practical and theoretical
problems, we intend to describe the optimal transfer problem of a spacecraft between
two given orbits and we have chosen to discuss the conceptual basis founded on
relativity in the Global Positioning System.
Key words: geophysics, orbits, satellites, communication.
1. ORBIT DESCRIPTIONS
Experience with satellite communications has demonstrated that satellite
systems can satisfy many military requirements. They are reliable, survivable,
secure, and a cost effective method of telecommunications. Satellite links are
unaffected by the propagation variations that interfere with hf radio. They are also
free from the high attenuation of wire or cable facilities and are capable of
spanning long distances.
Orbits generally are described according to the physical shape and the angle
of the inclination. The shape of an orbit is established by the initial launch
parameters and the later deployment techniques used.
Angle of inclination is the angle between the equatorial plane of the Earth
and the orbital plane of the satellite. An orbit with an angle of inclination of 90 0 or
near 90 0 is called a “polar orbit”. The satellite can survey the whole of the Earth’s
surface, including the poles in a few days. An equatorial orbit has an inclination
angle of 00 .
*
Paper presented at the 5th International Balkan Workshop on Applied Physics, 5–7 July 2004,
Constanţa, Romania.
Rom. Journ. Phys., Vol. 51, Nos. 1–2, P. 121–130, Bucharest, 2006
122
Emilia Dana Seleţchi
2
Low Earth Orbit satellites are placed into a circular inclined orbit at an
altitude of roughly 700 to 1600 km above the Earth’s surface. The advantages of
this type of orbit are very low costs per satellite, modest demands on antenna,
receiver and transmitter performance and very low propagation latency times. The
disadvantages of this model are the limited footprint on the Earth’s surface, so
these orbits require many satellites to provide global coverage.
A Geosynchronous Orbit is a circular, low inclination orbit about Earth
having a period of 23 hours, 56 minutes, 4.09 seconds. The altitude is about 35 680
km. These orbits allow the satellite to observe almost a full hemisphere of the Earth
and are therefore ideal for TV and Radio Broadcasting and international
communications. These satellites have poor resolution and geosynchronous orbits
require more energy and rocket fuel.
Table 1
Typical satellite orbits
Types of satellite orbits
Altitude
LEO
(Low Earth Orbit)
MEO
(Medium Earth Orbit)
GEO (Geosynchronous Earth Orbit)
Below 2 000 km
(1250 miles)
10 000 km
(6250 miles)
35 680 km
(22 300 miles)
The orbital period of the
satellite
Of 90 min to 2 hours
≈
6 hours
24 hours
The orbits of satellites about a central massive body can be described as
either circular or elliptical. A circular orbit is a special type of elliptical orbit. A
satellite orbiting about the Earth in circular motion is moving with a constant speed
and remains at the same height above the Earth’s surface. In this particular case
there is no component of force in the direction of motion. Since kinetic energy
depends on the speed of an object, the amount of kinetic energy will be constant
throughout the satellite’s motion. And since potential energy depends on the height
of an object, the amount of potential energy will be constant throughout the
satellite’s motion. So if the kinetic energy and the potential energy remain constant
the total mechanical energy of system remains constant.
1.1 CIRCULAR ORBITS
Consider a satellite with mass m orbiting a planet (the Earth).
R = the radius of orbit for the satellite (distance from center of planet) where:
R = R0 + h
R0 = 6,37 ⋅ 106 m , M = 5,98 ⋅ 1024 Kg , G = 6,673 ⋅ 10−11
(1)
Nm 2
Kg 2
3
123
Artificial satellites
h = the height above the Earth’s surface, M = the Earth’ mass, G = universal
gravitational constant, g 0 = acceleration of gravity at the Earth’ surface.
The orbital speed of the satellite is given by the following equation:
GM
g0
= R0
R0 + h
R0 + h
v0 =
(2)
The first cosmic speed and the period of the satellite are given by:
v1 = R0
g0
= g 0 R0 ≈ 7,9 Km / s ,
R0
T1 = 2π
R0
≈ 1h 25 min
g0
(3), (4)
The escape velocity:
v2 = 2 g0 R0 = 2 ⋅ v1 ≅ 11,2 km s
(5)
The rotation period of a satellite:
T = 2π
( R0 + h )
g0 R02
3
= 2π
( R0 + h )
GM
3
(6)
The radius of orbit for the satellite:
R=
3
g0 R02T 2 3 GMT 2
=
4π 2
4π 2
(7)
2π g0 R02
T
(8)
The orbital speed of the satellite:
v=
3
1.2 ELLIPTICAL ORBITS
A satellite orbiting the Earth in elliptical motion will experience a component
of force in the same or the opposite direction as its motion. The speed of a satellite
in elliptical motion is constantly changing, increasing as it moves closer to the
Earth and decreasing as it moves further from the Earth. The speed of the satellite
is greatest at perigee and least at apogee, so the satellite moves from A to B thus it
loses kinetic energy and gains potential energy. As the satellite moves from B back
to A, it gains speed and loses height subsequently there is a gain in kinetic energy
and a loss of potential energy. Yet throughout the entire elliptical trajectory, the
total mechanical remains constant.
124
Emilia Dana Seleţchi
4
If a satellite is moving on an elliptical trajectory, (the total mechanical energy E < 0
and the eccentricity e <1, we can write:
(1 − e )
2
x12 + x22 + 2 pex1 = p 2
⇒
2
pe 

 x1 +

x22
1 − e2 

+
=1
2
2
 p 
 p 

2 


2
1− e 
 1− e 
(9)
r
v
y
α
A
B
Fig. 2 – The elliptic orbit of the satellite.
a
b=
c=
p
1 − e2
(10)
p
(11)
1 − e2
pe
= ea = a 2 − b 2
1 − e2
(12)
p
1+ e
(13)
r1 = a − c = (1 − e)a =
5
125
Artificial satellites
r2 = a + c = (1 + e ) a =
p
1− e
(14)
p = the parameter of the ellipse, e = the eccentricity
The equation of the ellipse can be written as:
r (ϕ ) =
Considering e = 0 ⇒ x12 + x22 = r 2
p
1 + e cos ϕ
(15)
we obtain the equation of the circle.
ϕ = 0 ⇒ r1 =
p
1+ e
(16)
ϕ = π ⇒ r2 =
p
1− e
(17)
e=
r2 − r1
r1 + r2
(18)
p=
2r1r2
r1 + r2
(19)
The area A of an ellipse may be found by making the change of coordinates
b
x ' ≡   x and y ' ≡ y from the elliptical region R to the new region R ' . The
a
equation of the ellipse becomes:
2
1 a '
y '2
x
+
=1


a 2  b  b2
(20)
⇒ x '2 + y '2 = b 2 , so R ' is a circle o radius b.
−1
−1
∂x  ∂x ' 
a
b
=
 =  =
a
b
∂x '  ∂x 
 
The Jacobian is:
∂ ( x, y )
(
'
∂ x,y
'
)
∂x
∂x '
=
∂x
∂y '
∂y
a
∂x '
= b
∂y
0
∂y '
0
1
=
(21)
a
b
(22)
126
Emilia Dana Seleţchi
A=
∂ ( x, y )
∫∫ dx dy = ∫∫ ∂ ( x y ) dx dy
'
R
'
'
'
=
R
a
b
6
∫∫ dx dy
'
'
R
=
( )
a
π b 2 = π ab
b
(23)
The equation for momentum conservation is given by:
m v1 r1 = m v2 r2
(24)
The equation for energy conservation has the form:
−G M m 1
−G M m 1
+ m v12 =
+ m v22
2
2
r1
r2
(25)
v1 =
2GMr2
=
r1 ( r1 + r2 )
2 µ r2
r1 ( r1 + r2 )
(26)
v2 =
2GMr1
=
r2 ( r1 + r2 )
2 µ r1
r2 ( r1 + r2 )
(27)
where v1 is the speed at the perigee, v2 is the speed at the apogee and µ represents
the gravitational parameter of the central body.
µ ≡G M
(28)
The speed for a satellite on an elliptical motion is then given by [5]:
v0 ≥ R0
2 g0 R0h
( R0 + h ) ( R0 + h )
2
sin 2 α − R02 

(29)
2. A STUDY OF THE OPTIMAL TRANSFER PROBLEM OF A SPACECRAFT
BETWEEN TWO GIVEN ORBITS
The main goal of this chapter is to find the minimum cost trajectory, in terms
of fuel consumed, to transfer a spacecraft from parking orbit around a planet to a
higher orbit around the planet. The first step involved in this maneuver is the
application of an impulse in order to inserting the spacecraft into an elliptical orbit
with r1 = r0 and r2 = rf .
7
127
Artificial satellites
LEO
GTO
r1
GEO
r2
Fig. 2 – Hohmann Transfer Orbit.
The velocity in the initial circular orbit (LEO orbit) is given by:
vc1 =
GM
=
r0
µ
(30)
r1
r1 is the radius of the initial circular obit
r2 is the radius of the final circular orbit
The velocity in the Hohmann transfer orbit at r1 is given by:
v1 =
2 1 
2 µ r2
= µ − 
r1 ( r1 + r2 )
 r1 at 
The semi-major axis of the transfer orbit is given by: at =
(31)
r1 + r2
2
(32)
The standard Hohmann transfer has a cost of:
∆v
H1
=
2 µ r2
−
r1 ( r1 + r2 )
µ
2 1 
= µ −  −
r1
 r1 at 
µ
r1
The initial velocity increment is applied to get on geostationary orbit (GTO) at
perigee.
The velocity in the transfer orbit at final orbit height is given by:
(33)
128
Emilia Dana Seleţchi
8
2 1
2 µ r1
= µ − 
r2 ( r1 + r2 )
 r2 at 
v2 =
(34)
The last steep is to circularize the orbit. The velocity in the final circular orbit
(GEO orbit) is then given by:
vC 2 =
µ
(35)
r2
The final velocity increment is given by:
∆v
H2
=
µ
r2
2 1
− µ − 
r2
 r2 at 
2 µ r1
=
r2 ( r1 + r2 )
−
µ
(36)
The average angular frequency of a body in elliptical orbit is defined by:
n=
2π
T
(37)
The mean motion appears in Kepler’s third law as:
n2 a3 = G M = µ
(38)
The orbital period of the Hohmann transfer orbit is given by:
T=
2π
a3
= 2π
= 2π
µ
n
( r1 + r2 )
3
8µ
(39)
The time taken to transfer has the form:
t=
1
T =π
2
( r1 + r2 )
3
8µ
(40)
3. INVESTIGATION OF SIMPLIFIED MODELS FOR ORBIT DETERMINATION
One of the most important objectives of this paper is to investigate simplified
models to determine in real time the orbit of an artificial satellite, using single
frequency GPS (Global Positioning System) measurements. The space applications
would be limited to LEO satellites. A GPS receiver measures pseudoranges and
pseudorange rates to the satellites.
The orbit determination problem can be formulated by using one of the best
methods, the Kalman Filter [3].
9
129
Artificial satellites
The model for a GPS pseudorange measurement is given by:
y c ( xk , t k ) = ρ k + β k
Where ρ =
( xGPS − x )
2
+ ( yGPS − y ) + ( zGPS − z )
2
(41)
2
is the geometric range, x, y,
z are the positional states of the user satellite at reception time, xGPS , yGPS and
zGPS are the positional states of the GPS satellite at transmission time (corrected
for light time delay), and βk are the errors steaming from ionosphere path delay,
GPS satellite and receiver clock offsets and other errors.
The equation of the pseudorange in L1 frequency is given by;
Pk1 = ρ k + I k + c [ dtGPS (tk ) − dtU (tk ) ] − ε k
(42)
where Pk1 is the pseudorange in L1, ρ k is the geometric range, I k is the
ionospheric delay, c is the vacuum speed of light, dtGPS (tk ) is the GPS satellite
clock offset, dtU (tk ) is the receiver clock offset, tk is the observation instant in
GPS time and ε k is a remnant error supposed random gaussian.
The information for the GPS satellite clocks is transmitted via the broadcast
navigation message in the form of three polynomial coefficients a f 0 , a f 1 , a f 2
with a reference time toc . The clock correction of the GPS satellite for the epoch
tGPS is given by:
∆tsv = a f 0 + a f 1 (t sv − toc ) + a f 2 (t sv − toc ) + ∆t R
2
(43)
t sv = tGPS − ∆t sv , ∆t R is a small relativistic clock correction caused by the orbital
eccentricity e, the semimajor axis of the orbit a, and ϕ .
Since GPS has the potential to enhance military operations other than weapon
delivery and will also be available to civilian users, several applications for GPS
have been developed. The DGPS (Differential GPS) applications include:
instrument approach, all weather helicopter operations, narrow channel maritime
operations, reference station for testing/calibration of navigation equipment and
surveying for mapping and positioning.
CONCLUSIONS
A Hohmann transfer orbit is the most efficient intermediate orbital path to
transfer a spacecraft from one circular orbit to another using a very low amount of
energy.
130
Emilia Dana Seleţchi
10
By using measurements of pseudorange between an object and each of
several earth orbiting GPS satellites, the object can be very accurately located in
space. A Kalman filter characterizes the noise sources in order to minimize their
effect on the desired receiver outputs. The major sources of range error for
nondifferential GPS are: selective availability errors, ionospheric delay,
tropospheric delay, ephemeredes error and satellite clock error. DGPS was
developed to increase the positioning accuracy and it also enhances GPS integrity
by compensating for anomalies in the satellite ranging signals and navigation data
message.
REFERENCES
1. P. W. Binning, Satellite Orbit Determination Using GPS Pseudoranges Under SA. Advances in
The Astronautical Sciences, Vol. 95, part I, 1997, pp. 183–193.
2. R.A. Broucke; A.F.B.A. Prado, Orbital Planar Maneuvers Using Two and Three-Four (Through
Infinity) Impulses. Journal of Guidance, Control and Dynamics, Vol. 19, No. 2, pp. 274–282.
3. Ana Paula Marins Chiaradia; Helio Koiti Kuga; A.F.B.A. Prado, Investigation of Simplified Models
for Orbit Determination Using single Frequency GPS Measurements, Instituto Nacional de
Pesquisas Espacias – Brazil;.
4. A. Leick – GPS satellite Surveying, second edition, John Wiley & Sons, INC., 1995.
5. Ioan Merches, Lucian Burlacu, Mecanica analitica si a Mediilor Deformabile – Editura Didactica
si Pedagogica, Bucuresti 1983, pp. 55–67.
6. G. Seeber – Satellite Geodesy: Foundations, Methods and Aplications, Walter de Gruyter, BerlinNew York, 1993.
7. S.C. Wu, T.P. Yunck, C.L. Thornton, Reduced Dynamic technique for precise orbit determination
of low Earth Satellites, J. Guidance, Control and Dynamics, Vol. 14, 1991, pp. 24–30