Download Chapter 1 Two-Body Orbital Mechanics 1.1

Chapter 1
Two-Body Orbital Mechanics
1.1 - Birth of Astrodynamics: Kepler’s Laws
Since the time of Aristotle the motion of planets was thought as a combination of
smaller circles moving on larger ones. The official birth of modern Astrodynamics
is in 1609, when Johann Kepler published his first two laws of planetary motion.
The third followed in 1919.
First Law – The orbit of each planet is an ellipse with the sun at a focus.
Second Law – The line joining the planet to the sun sweeps out equal
areas in equal times.
Third Law – The square of the period of a planet is proportional to the
cube of its mean distance from the sun.
Kepler’s results were made possible by the accurate observations of the
astronomer Tycho Brahe, but Kepler was unable to provide any explanation of
the planetary motion.
2.2 - Newton’s Laws of Motion
In Book I of his Principia (1687) Newton introduces the three laws of motion
First Law – Every body continues in its state of rest or of uniform motion in
a straight line unless it is compelled to change that state by forces
impressed upon it.
Second Law – The rate of change of momentum is proportional to the
force impressed and is in the same direction as that force.
Third Law – To every action there is always opposed an equal reaction.
The first law permits to identify an inertial system. The second law can be
expressed mathematically as


F  ma


where F is the resultant of the forces acting on the constant mass m and a is
the acceleration of the mass measured in an inertial reference frame. A different
equation applies to a variable-mass system. The third law has just permitted to
deal with a dynamical problem by using an equilibrium equation. Its scope is

however wider: for instance, the presence of the action F on m implies an action

 F on another portion of an ampler system.
1.3 - Newton’s Law of Universal Gravitation
In the same book Newton enunciated the law of Universal Gravitation by stating
that two bodies attract one another with a force proportional to the product of
their masses and inversely proportional to the square of the distance between
them, that is,
F G
Mm
r2
where the Universal Gravitational Constant G  6.673 1011m3kg-1s-2 . The law is
easily extended from point masses to bodies with a spherical symmetry.
1.4 - The N-Body Problem
The analysis of a space mission studies the motion of a spacecraft, which is the
Nth body in a system that comprises other N-1 celestial bodies. The knowledge of
the position of all the N bodies is however necessary, as there is mutual
attraction between each pair of bodies: the so called N-Body Problem arises.
An inertial reference frame is required to apply Newton’s second law of
dynamics to each jth body. The problem is described by N second-order nonlinear vector differential equations



R j  Fgj  Foj m j
j  1,2,..., N



where Fgj is the sum of the actions on mj of all the other N-1 bodies, which are
assumed to have point masses, according to Newton’s law of gravitation. The

additional term Fgj takes into account that

usually the bodies are not spherically symmetric

the spacecraft may eject mass to obtain thrust

other forces (atmospheric drag, solar pressure, electromagnetic forces,
etc.) act on the spacecraft
and represents the resultant force on mj due to these perturbations.
The interest is in the motion relative to specific celestial bodies, for
instance, the 1st. The corresponding equation can be subtracted from all the
others, and the relevant equations are in fact N-1.
The spacecraft mass is negligible and cannot perturb the motion of the
celestial bodies, which is regularly computed by astronomers and provided to the
Astrodynamics community in the form of ephemeris. The remaining vector
second-order differential equation, describing the spacecraft motion relative to
the first body, is transformed in a system of six scalar first-order differential
equations, which is solved numerically. An analytical solution can be only
obtained if one assumes the presence of just one celestial body, and neglects all
the other actions on the spacecraft than the point-mass gravitational attraction.
1.5. Equation of Motion in the Two-Body Problem
Consider a system composed of two bodies of masses M and m; the bodies are
spherically symmetric and no forces other than gravitation are present. Assume


an inertial frame; vectors R and  describe the positions of masses M and m,
respectively. The position of m relative to M is
  
  
  
r   R

r    R

r    R
where all the derivatives are taken in a non-rotating frame.
The law of Universal Gravitation is used when the second law of dynamics
is separately used to describe the motion of each mass

Mm 
m   G 3 r 
r
Mm 

MR G 3 r
r
By subtracting the second equation from the first (after an obvious simplification
concerning masses) one obtains

M m 
GM 
r  G
r  3 r
3
r
r
as in practical systems M >>m (Earth and moon with M  81.3 m are the most
balanced pair of bodies in the solar system).
The Equation of Relative Motion is usually written in the very simple form

 
r   3 r
r
after introducing the gravitational parameter  = GM; typical numerical values are
  1.372 1011 km 3 s -2
(sun)
  3.986 10 5 km 3 s -2
(earth)
  4.903 10 3 km 3 s -2
(moon)
It is worthwhile to remark that

the relative position r is measured in a non-rotating frame which is not

rigorously inertial

the relative motion is practically independent of the mass of the secondary
body; a unit mass will be assumed in the following.
1.6 - Potential Energy
The mechanical work done against the force of gravity to move the secondary
body from position 1 to position 2
 2 
 
L12   3 r  ds   2 dr     Eg 2  Eg1
r
r
r2 r1
1
1
2
 
does not depend on the actual trajectory and one deduces that the gravitational
field is conservative. The work can be expressed as the difference between the
values at point 1 and 2 of the potential energy, which contains an arbitrary
constant
Eg  

r
C

Eg  

r
The constant C is conventionally assumed to be zero in Astrodynamics. The
maximum value of the potential energy is zero, when the spacecraft is at infinite
distance from the central body.
1.7 - Constants of the Motion
After the equation of motion is written in the form
  
r  3 r  0
r

its scalar product with r provides, by using Eq. A.5 in Appendix,
         

r  r  3 r  r  v  v  3 r  r  v v  2 r  0
r
r
r

d v2  
  0
dt  2 r 
During the motion the specific mechanical energy, which the sum of kinetic and
potential energy, is constant
v2 
E
  const
2
r
This result is the obvious consequence of the conservative nature of the
gravitational force, which in the two-body problem is the only action on the
spacecraft.

The vector product of r by the equation of motion gives (using Eq. A.2)
 
r  r  0

 
d  
d  
r  r  r  v   0
dt
dt
Therefore, the angular momentum
  
h  r  v  const
which is normal to the orbital plane, is a constant vector and the spacecraft
motion remains in the same plane. In other words, being the unique action radial,
no torque acts on the satellite and the angular momentum is constant.

Finally, the cross product of the equation of motion with h


 
     d      
r  h  3 r  h 
r h  3 r r r  0
r
dt
r
using Eq. A.6 becomes



 r r r  d   
d         
d  
r


r  h  3 r  r r  r  r r 
v  h    2  v  h     0
 r r  dt 
dt
r
dt
r





 


The vector
  
 v h r
e
  const

r
is another constant of the spacecraft motion in the two-body problem.
1.8 - Trajectory Equation
The dot product
       
  r v  h r  r r v  h
h2
r e 


r 
r

r


provides
r
h2 
1  e cos


where is the angle between the vectors r and e . The radius attains its

minimum value for  =0: the constant vector e is directed from the central body
to the periapsis.
A conic is the locus of a point that moves so that the ratio of its absolute
distance r from a given point (focus) to its absolute distance d from a given line
(directrix) is a positive constant e (eccentricity)
r
p
e
d
d  r cos

r
p
1  e cos
which is the equation of a conic section, written in polar coordinates with the
origin at a focus. By comparison with the trajectory equation, one deduces that

in the two-body problem the spacecraft moves along a conic section that
has the main body in its focus; Kepler’s second law is demonstrated and
extended from ellipses to any type of conics

the semi-latus rectum p of the trajectory is related to the angular
momentum of the spacecraft by p  h 2 


the eccentricity of the conic section is the magnitude of e , which is named
eccentricity vector.
1.9 – The Conic Sections
The name of the curves called conic sections derives from the fact that they can
be obtained as the intersections of a plane with a right circular cone. If the plane
cuts across a half-cone, the section is an ellipse; one obtains a circle, if the plane
is normal to the axis of the cone, and a parabola in the limit case of a plane
parallel to a generatrix of the cone. The two branches of a hyperbola are
obtained when the plane cuts both the half-cones. Degenerate conic sections (a
point, one or two straight lines) arise if the plane passes through the vertex of the
cone.
1.9.1 – Ellipse
The canonical equation of the ellipse in Cartesian coordinates is
x2 y 2

1
a2 b2
(a  0)
where a is supposed to be a positive quantity. The foci are located at F( c ,0) ,
with
c  a2  b 2
e
c
1
a
The distance of a point of the ellipse from the prime focus (c > 0) is found by
considering
r 2  ( x  c )2  y 2  x 2  2cx  c 2  b 2 
b2 2 c 2 2
2
x  2 x  2cx  a 2  a  ex 
2
a
a
r  a  ex
and, in a similar way, it is found the distance from the second focus
r   a  ex
The sum of the last two equations provides
r  r   2a
which proves that an ellipse can be drawn using two pins and a loop of thread.
The periapsis and apoapsis radii, corresponding to x   a , are
rP  a (1  e)
rA  a (1  e)
and can be combined to obtain
a
rA  rP
2
e
rA  rP
rA  rP
The abscissa of a generic point of the ellipse x  c  r cos is used to obtain the
polar equation
r  a  ex  a  e 2a  e r cos

r
a (1  e 2 )
1  e cos
The length of the semi-latus rectum is therefore p  a (1  e 2 ) .
1.9.2 – Hyperbola
The canonical equation of the hyperbola in Cartesian coordinates
x2 y 2

1
a2 b2
(a  0)
shows that the curve does not intersect the y-axis. The arbitrary choice a  0
allows obtaining the same polar equation as for the ellipse. The foci are located
at F( c ,0) , with
c   a2  b 2
e
c
1
a
By posing
x   cos 
y   sin 
the equation is rewritten in polar coordinates with the origin in the center of the
xy-frame
 2 cos2 
a2

 2 sin2 
b2
1
b 2 cos2   a 2 sin2  

a 2b 2
2
0
Points of the hyperbola exist only if tan   b a ; the limit condition
tan2  
b2
a2
1
b2

 1  e2
2
2
cos  a

provides the direction of the asymptotes
cos  
1
e
The prime focus is the left one and only the left branch of the hyperbola is
significant in Astrodynamics, as gravity is an attractive force. The distance of a
point of the hyperbola from the prime focus (c < 0) is found by considering
r 2  ( x  c )2  y 2  x 2  2cx  c 2  b 2 
r  a  ex

b2 2 c 2 2
2
x  2 x  2cx  a 2  a  ex 
2
a
a
rP  a (1  e)
and, by using the abscissa of a generic point of the hyperbola ( x  c  r cos )
r  a  ex  a  e 2a  e r cos

r
a (1  e 2 )
1  e cos

p  a (1  e 2 )
i.e., the same expressions as for the ellipse, but a < 0 and e > 1.
1.9.3 – Parabola
The parabola may be derived from the hyperbola in the limit case of unit
eccentricity.
cos   1

0
The asymptotes are parallel and their intersection is at the infinite ( c   , a   ).
The semi-latus rectum is however finite and the polar equation of the parabola is
r
p
1  cos

rP 
p
2
1.10 - Relating Energy and Semi-major Axis
The constant values of the angular momentum and total energy can be evaluated
at any point of the trajectory; in particular at the periapsis, where the spacecraft
velocity is orthogonal to the radius vector
v P2 
h2

E
  2
2 rP 2rP rP
h  rP v P
where
h 2   a (1  e 2 )
rP  a(1  e)
One obtains
 a (1  e 2 )
1 

  
E    2

2
2a
 2a (1  e) a (1  e) 
that is, a very simple relationship between the specific mechanical energy to a
geometrical parameter of the conics, the semi-major axis.
The last relationship between geometrical and energetic parameters of the
conics is obtained by recasting the above equations
e  1
h2
h2
 1 2E 2
a

The degenerate conics have zero angular momentum and therefore unit
eccentricity.
Appendix – Vector Operations
The results of some vector operations are given in this appendix.
A.1.
A.2.
A.3.
A.4.
 
a  a  a2
 
aa  0
 
 
a  b  b  a
     
a b c  ab c
This equality is found by considering that either side represents the volume of
the parallelepiped which has the vectors as the corners.
A.5.
 
a  a  a a

 
d  
a  a   d a 2
dt
dt
This result of vector analysis is very important in orbital mechanics and rocket


propulsion: assigned a vector variation da , its component parallel to a
increases the vector magnitude, whereas the perpendicular component just

rotates a , keeping its magnitude constant.
        
a  b  c  a  c  b  b  c a
A.6.


 