1. Celestial Mechanics II
Orbital energy and angular momentum
Elliptic, parabolic and hyperbolic orbits
Position in the orbit versus time

2. Orbital Energy KINETIC POTENTIAL per unit mass
The orbital energy – the sum of
kinetic and potential energies
per unit mass – is a constant in
the two-body problem (integral)

3. The Vis-viva Law

4. Vis-viva law, continued
This is a form of the
energy integral, where
the velocity is given as
a function of distance
and orbital semi-major
axis.
VERY USEFUL!

5. Geometry versus Energy
E<0  0  e < 1: Ellipse
E=0  e = 1: Parabola
E>0  e > 1: Hyperbola
Orbital shape determined
by orbital energy!

6. Special velocities Circular orbit (r  a): Circular velocity
Parabolic orbit (a  ):
Escape velocity
Note: The parabolic orbit is a borderline case that
separates entirely different motions: periodic (elliptic)
and unperiodic (hyperbolic). The slightest change of
velocity around ve may have an enormous effect (chaos)

7. Geometry vs Angular Momentum
The semilatus rectum p is positive for all kinds of orbits
(note that a is negative for hyperbolic orbits). It is a
well-defined geometric entity in all cases.
In terms of the perihelion distance q, we have:
ellipse
parabola
hyperbola

8. Orbital position versus time: The choice of units
Gravitational constant:
SI units ([m],[kg],[s]) G = m3kg-1s-2
Gaussian units ([AU],[M],[days])
k = AU3/2M-1/2days-1
Kepler III:
k2 replaces G
m1 = 1; m2 = 1/ (Earth+Moon)
P = (sidereal year); a = 1

9. Orbital position versus time: Why we need it
Ephemeris calculation
– ephemeris = table of positions in the sky (and, possibly, velocities) at given times, preparing for observations
Starting of orbit integration
– this means integrating the equation of motion, which involves positions and velocities

10. Orbital Elements
These are geometric parameters that fully describe an orbit and an object’s position and velocity in it
Since the position and velocity vectors have three components each, there has to be six orbital elements
Here we will only be concerned with three of them (for the case of an ellipse, the semi-major axis, the eccentricity, and the time of perihelion passage)

11. “Anomalies” These are angles used to describe the location of an
object within its orbit
 = true anomaly
E = eccentric anomaly
M = mean anomaly
M increases linearly with time and can thus easily be
calculated, but how can we obtain the others?

12. Introducing the eccentric anomaly
Express this in terms of
E and dE/dt

13. Differential equation for E

14. Integration
Mean motion: n
Mean anomaly: M

15. Kepler’s Equation
Finding (position, velocity) for a given orbit at a given time:
Solve for E with given M
Finding the orbit from (position, velocity) at a given time:
Solve for M with given E
(much easier)

16. Position vs time in elliptic orbit
• Calculate mean motion n from a (via P)
• Calculate mean anomaly M
• Solve Kepler’s equation to obtain the eccentric anomaly E
• Use E to evaluate position and velocity components

17. Solving Kepler’s equation
Iterative substitution:
(simple but with slow convergence)
Newton-Raphson method:
(more complicated but with
quicker convergence)

18. Handling difficult cases, 1
Solving Kepler’s equation for nearly parabolic orbits is
particularly difficult, especially near perihelion, since
E varies extremely quickly with M
Trick 1: the starting value of E
Under normal circumstances, we can take E0 = M
But for e  1, we develop sin E in a Taylor series:
and hence:

19. Handling difficult cases, 2
Trick 2: higher-order Newton-Raphson
Taylor series development of M:
With
and
we get:

20. Two useful formulae  

21. Parabolic orbits a and e=1, but p=a(1-e2)=h2/ remains finite
Kepler III loses its meaning
No center, no eccentric anomaly
No mean anomaly or mean motion, but we will find substitutes

22. Parabolic eccentric anomaly N
Introduce:
From the angular momentum equation:
we get:
But:
Thus:
and:
N plays a role similar to E in Kepler’s equation

23. Mean motion, mean anomaly
Introduce the parabolic mean motion:
and the parabolic mean anomaly:
We get:
Cubic equation for N(M), called Barker’s Equation

24. Solving Barker’s equation
Numerically, it can be done by simple Newton-Raphson
using N0 = M
Analytically, there is a method involving auxiliary variables:

25. Position vs time in parabolic orbit
• Calculate parabolic mean motion n from q
• Calculate parabolic mean anomaly M
• Solve Barker’s equation to obtain the parabolic eccentric anomaly N = tan(/2)
• Use N to evaluate position and velocity components

26. The Hyperbola
Semi-major axis
negative for hyperbolas!

27. Hyperbolic deflection
Impact parameter: B (minimum distance along the straight line)
Velocity at infinity: V
Angle of deflection:  =  - 2
Gravitational focusing: q < B

28. Hyperbolic eccentric anomaly H

29. Differential equation for H

30. Hyperbolic Kepler Equation
Hyperbolic mean motion:
Hyperbolic mean anomaly:

31. Solving the hyperbolic Kepler equation
The methods are the same as for the elliptic, usual Kepler equation
The same tricks can be applied for nearly parabolic orbits
The derivatives of trigonometric functions are replaced by those of hyperbolic functions

32. Position vs time in hyperbolic orbit
• Calculate hyperbolic mean motion n from |a|
• Calculate hyperbolic mean anomaly M
• Solve the hyperbolic Kepler equation to obtain the eccentric anomaly H
• Use H to evaluate position and velocity components

33. Examples q=2.08 AU Ellipse: a=2.48, e=0.16 Hyperbola:
a= AU, e=2.5
 positions 253 days after perihelion
Ellipse: v=22.3 km/s
Parabola: v=29.3 km/s
Hyperbola: v=38.7 km/s
Velocity at perihelion —