ENG 491CU1
Fall 2011
Erik Kroeker
Outline
 Kepler’s Laws
 Newton’s Laws and the n-Body Problem
 Two-Body Problem
 Defining Captured Orbits
 Keplerian Elements
 TLE
 Changing Orbits
 Unique Orbits
 Relevance to IlliniSat
Kepler’s Laws
 The orbit of every planet is
an ellipse with the Sun at
one of the two foci.
 A line joining a planet and
the Sun sweeps out equal
areas during equal
intervals of time.
 The square of the orbital
period of a planet is
directly proportional to
the cube of the semimajor axis of its orbit.
Newton’s Laws
 First Law
 Every body remains in a state of rest or uniform motion
(constant velocity) unless it is acted upon by an external
unbalanced force.
 Second Law
 A body of mass m subject to a force F undergoes an
acceleration a that has the same direction as the force and a
magnitude that is directly proportional to the force and
inversely proportional to the mass. (i.e., F = ma.)
 Third Law:
 The mutual forces of action and reaction between two bodies
are equal, opposite and collinear.
Newton’s Laws of Gravitation
 Every point mass attracts every single other point mass
by a force pointing along the line intersecting both
points. The force is directly proportional to the
product of the two masses and inversely proportional
to the square of the distance between the point
masses:
m1m2
F G 2
r
G = 6.67 x 10^-11 N m^2 / kg
The n-Body Problem
 Given only the present positions and velocities of a group of celestial
bodies, predict their motions for all future time and deduce them for all
past time.
m j q j  G
k j
m j mk (qk  q j )
qk  q j
3
, j  1,..., n
 Historical Aside: King Oscar II Prize
 Prize was to go to the first to prove: Given a system of arbitrarily many
mass points that attract each according to Newton's law, under the
assumption that no two points ever collide, try to find a representation
of the coordinates of each point as a series in a variable that is some
known function of time and for all of whose values the series converges
uniformly.
 The prize was finally awarded to Poincaré, even though he did not solve
the original problem (and his solution contained a serious error).
Two-Body Problem
F12 ( x1 , x2 )  m1 x1
F21 ( x1 , x2 )  m2 x2
 By adding them, we find that the center of mass does not move:
m1 x1  m2 x2  (m1  m2 ) R  F12  F21  0
m1 x1  m2 x2
R
m1  m2
R0
Two-Body Problem
 By subtracting them and dividing each force by its
respective mass, we find that the displacement vector acts
as:
 F12 F21   1
1 
r  x1  x2  



 F12
 m1 m2   m1 m2 
or
 r  F12 ( x1 , x2 )  F (r )
m1m2


 1
1  m1  m2



m
m
 1
2 
1
Two-Body Problem
 By adding the motion of the center of mass of the system,
and the displacement of the vector, we can achieve an
analytic solution for the motion of the bodies.
m2
x1 (t )  R (t ) 
r (t )
m1  m2
m1
x2 (t )  R (t ) 
r (t )
m1  m2
Making Simple, Simpler
 Let us first assume that we have a large and a small body,
such that the force applied to one body is negligibly small
(Sun and Earth, Earth and Satellite).
 In this case, we can consider one body to remain fixed, and
the other body moves about it.
Mm
Mm
r 
r G 2
M m
r
GM
r  2 , m  M
r
Conic Sections and Orbits
 The solution to the simplified two-body problem are
from the set of mathematical solutions referred to as
conic sections.
 In analytic geometry, a conic may be defined as a plane
algebraic curve of degree 2.
Defining Orbits
Defines shape of ellipse
 Semi-Major Axis (a):
distance from the
center of the ellipse to
the farthest point.
 Eccentricity (e):
measure of how much
the orbit deviates from
a circle.
e = 0 (circular orbit)
0<e<1 (elliptic orbit)
e>1 (hyperbolic orbit)
Defining Orbits
Defines orientation of
orbital plane w.r.t a
reference plane
 Inclination(i): angle of
tilt between the orbital
plane and the reference
plane
 Longitude of
Ascending Nodes (Ω):
the angle at which the
satellite rises above the
reference plane.
 Argument of Periapsis
(ω): angle from the line
of nodes to the position
of periapsis.
Defining Orbits
Defines position on ellipse
 True Anomaly (f): angle
from perigee to current
location.
Reference Frames and
Nomenclature
 Sun – Ecliptic Plane, First Point of Ares
 Earth – Equatorial Plane, First Point of Ares
 Galaxy – Galactic Plane, Center of Milky Way through Sun
 Currently the First Point of Ares is in the constellation of
Pisces, as the point moves with time. Therefore, the orbit
must be defined from an epoch.
 Perisapsis and Apoapsis
 Earth: Perigee and Apogee
 Sun: Perihelion and Apohelion
 Galaxy: Perigalacticon and Apogalacticon
TLEs
 A Two-Line Element set (TLE) is a set of orbital
elements that describe the orbit of an earth satellite.
Example for ISS:
ISS (ZARYA)
1 25544U 98067A 08264.51782528 -.00002182 00000-0 -11606-4 0 2927
2 25544 51.6416 247.4627 0006703 130.5360 325.0288 15.72125391563537
 Historical Aside: 80 Characters Long
 The TLE was originally design for 80 character wide
punch cards.
Field
Columns
Content
Example
1
01-01
Line number
1
2
03-07
Satellite number
25544
3
08-08
Classification
(U=Unclassified)
U
4
10-11
International Designator
(Last two digits of launch
year)
98
5
12-14
International Designator
(Launch number of the
year)
067
6
15-17
International Designator
(Piece of the launch)
A
7
19-20
Epoch Year (Last two digits
of year)
08
8
21-32
Epoch (Day of the year and
fractional portion of the
day)
264.51782528
9
34-43
First Time Derivative of the
Mean Motion divided by
two
-.00002182
10
45-52
Second Time Derivative of
Mean Motion divided by
six (decimal point
assumed)
00000-0
11
54-61
BSTAR drag term (decimal
point assumed)
-11606-4
12
63-63
The number 0 (Originally
this should have been
"Ephemeris type")
0
13
65-68
Element number
292
14
69-69
Checksum (Modulo 10)
7
Field
Columns
Content
Example
1
01-01
Line number
2
2
03-07
Satellite number
25544
3
09-16
Inclination [Degrees]
51.6416
4
18-25
Right Ascension of the
Ascending Node [Degrees]
247.4627
5
27-33
Eccentricity (decimal point
assumed)
0006703
6
35-42
Argument of Perigee
[Degrees]
130.5360
7
44-51
Mean Anomaly [Degrees]
325.0288
8
53-63
Mean Motion [Revs per
day]
15.72125391
9
64-68
Revolution number at
epoch [Revs]
56353
10
69-69
Checksum (Modulo 10)
7
Overview
 Types of Orbits
 Circular vs. Elliptical
 Stationary (or Near Stationary Orbits)
 Sun-Synchronous
 Repeating Ground Trace
 Changing Orbits
 Non-Keplerian Orbits
Types of Orbits
 Circular – constant distance from the Earth.
 Elliptical – variable distance from the Earth.
Types of Orbits
Types of Orbits
 Geostationary – holds at one point over the equator.
 Geosynchronous – returns to the same point over the
Earth everyday at the same time.
Types of Orbits
Types of Orbits
 Molniya – Dwells over high inclination locations for
extended periods of time.
Types of Orbits
 Sun-synchronous – the orbital plane is always in the
same orientation relative to the sun.
 Terminator or Dusk/Dawn – rides along the terminator
between night and day. Always illuminated by Sun.
 Noon/Midnight – Always looking at the Earth at noon or
midnight local time.
Types of Orbits
Types of Orbits
 Repeating Ground Trace – The ground trace aliases
with itself continuously.
Changing Orbits
 In order to change the orbit, you must change this
energy of the orbit. This can be achieved by:
 Drag forces removing energy from the body.
 Artificially adding energy to the system.


From internal sources (e.g. second stage, thrusters)
From external sources (e.g. solar sails)
Changing Orbits
 Why Change Orbits?
 Inclination Changes
 Rendezvous
 Altering a Ground Path
 Escaping Orbits, Orbit Transfers
Changing Orbits
Some Rules of Thumb for Orbit Changing
 To change inclination, it is easiest to change at the
equatorial crossing.
 To add energy, it is most efficient to add energy at
periapsis (highest velocity). This is known as the
Oberth effect.
 To change from one circular orbit to another, two
maneuvers are required.
 One to change the orbit to an ellipse, and the second to
turn it back into a circle.
Others Types of Orbits
 Non-Keplerian Orbits
 By continuously varying the orbital energy it is possible
to maintain unique orbits, impossible to achieve with a
normal orbit.
 One example is a statite, which in theory would use solar
radiation pressure to stay fixed above one of the poles.
Relevance to IlliniSat
 Altitude is approximately 200-900 km.
 Orbital period is 88-103 min
 Maximum eclipse period is 35 min
 Revolutions per day is 14-16
 Orbit lifetime predicted as 14 years
 Maximum time in view of UIUC is roughly 13 minutes