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

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

2 Orbital Energy KINETIC per unit mass POTENTIAL 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): v c 2 = µ r Circular velocity Parabolic orbit (a ): v e 2 = 2µ r Escape velocity v e = 2 " v c Note: The parabolic orbit is a borderline case that separates entirely different motions: periodic (elliptic) and unperiodic (hyperbolic). The slightest change of velocity around v e may have an enormous effect (chaos)

7 Geometry vs Angular Momentum h = µp 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: p < 2q p = 2q p > 2q ellipse parabola hyperbola

8 Orbital position versus time: The choice of units Gravitational constant: SI units ([m],[kg],[s]) G = m 3 kg -1 s -2 Gaussian units ([AU],[M ],[days]) k = AU 3/2 M -1/2 days -1 Kepler III: k 2 replaces G m 1 = 1; m 2 = 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 semimajor 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 ν = 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 E 0 = M But for e 1, we develop sin E in a Taylor series: M = E " esin E # E " sin E = 1 6 E 3 +K and hence: E 0 = ( 6M) 1/ 3

19 Handling difficult cases, 2 Trick 2: higher-order Newton-Raphson Taylor series development of M: " M = M 0 + dm % $ ' + # de & 0(E 1 " $ 2# With M 0 = M E 0 d 2 M de 2 % ' & 0 ((E) ( ) and "E 0 = M # M 0 we get: ( dm /de) 0 " $ # d 3 M de 3 % ' & 0 ((E) 3 +K "E = $ & % dm de ' ) + 1 $ & ( 0 2% d 2 M de 2 M # M 0 ' ) 0 + ( 0"E 1 $ & 6 % d 3 M de 3 ' ) ( 0 ("E 0 ) 2 +K

20 Two useful formulae

21 Parabolic orbits a and e=1, but p=a(1-e 2 )=h 2 /µ remains finite p = 2q; h = 2µq 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: N = tan " 2 From the angular momentum equation: r 2 " = h we get: ( 1+ N 2 ) 2 " = h q = 2µ 2 q 3 But: dn = ( 1+ N 2 ) d" 2 Thus: ( 1+ N 2 )dn = µ 2q 3 dt and: N plays a role similar to E in Kepler s equation

23 Mean motion, mean anomaly Introduce the parabolic mean motion: n = µ 2q 3 and the parabolic mean anomaly: M = n(t " T) We get: M = N N 3 Cubic equation for N(M), called Barker s Equation

24 Solving Barker s equation Numerically, it can be done by simple Newton-Raphson using N 0 = 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(ν/2) Use N to evaluate position and velocity components

26 The Hyperbola q = a(1" e) p = a(1" e 2 ) Semi-major axis negative for hyperbolas! cos" # = $ 1 e

27 Hyperbolic deflection Impact parameter: B (minimum distance along the straight line) Velocity at infinity: V Angle of deflection: ψ = π - 2φ Gravitational focusing: q < B " tan # 2 = µ V $ h = µ V $ 2 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 ο positions 253 days after perihelion Velocity at perihelion Ellipse: v=22.3 km/s Parabola: v=29.3 km/s Hyperbola: v=38.7 km/s