Parameters in the Orbital Plane : Expansion of the True Anomaly In Celestial Mechanics

 

Motion and angular variations from it in the orbital plane are a key aspect of celestial mechanics. The diagram below shows assorted "orbital elements" for a planet of mass m2 (the Sun is m1) and this can be used as a basis for orbital energy analysis and also to predict future positions. 

Energy constants in celestial mechanics are very useful for quickly coming to terms with specific properties of an orbit such as shown in the accompanying sketch- designating a generic orbit in x-y-z space. In the diagram, w   is the argument of the perihelion, W is the longitude of the ascending node, f is the true anomaly and i is the inclination of the orbit. The critical or key parameter here is h, the angular momentum vector for the orbiting system. It may be useful here to refer to the diagram (b) in the figure used for angular momentum vector (z) in an atomic system:

Getting specific, assuming r and r' are r (radius vector) and d r/dt, respectively, the magnitude h, of the angular momentum vector is:

h = r x r’ =

(y z’ - z y’)

(z x’ - x z’) = (c1 c2 c3)

(x y’ - y z’)

so (r x r’) = (c1/ h, c2/ h, c3/h)

and inserting variables one finds:

c1/ h = sin W sin (i)

c2/ h = - cos W sin (i)

c3/h = cos(i)

Now since the inclination of Earth's orbit to the ecliptic  (i) is known (23.5 deg) and therefore cos(i) can be determined, then sin(i) can be as well. Also h can be determined, since:

 h = c3 / cos(i) = GMm a (1 – e ²) ^{1 /2} 

where all the constants are known (a = semi-major axis of orbit, e = eccentricity of orbit)

The energy (vis viva) equation is: ½V ² - m  /r = c, and the c's - energy integration constants- are  found as shown below)  Also, h can be determined, since: h = c3 / cos(i) .  

Also h =  [c1 ² +  c2 ²   +  c3 ²]  ^½   (We also know  W =  11.26 deg.)

To fix ideas the longitude of the ascending node, W  is one of the key orbital elements (difference between mean anomaly  M and argument of the perihelion,  w )  where M can be obtained from a table based on observations, and w can be obtained using a Fourier expansion of the mean anomaly, M, e.g.:

 w  = M + (2e – e³ / 4) sin M + 5 e²/4 sin 2M + ... etc.

Once W is known, and c1, c2 and c3 are computed, the astronomer is ready to compute the position of a planet, say Jupiter, forty or so years in the future.  The actual calculations are formidable but can be found in a good celestial mechanics text, or using the orbital element formulae can be worked out using a computer.

We end with the basis to compute the true anomaly f. We start with a series expansion of  f in terms of E:

tan f/2  =  Ö {(1 + e) / (1 - e)}   tan E/2

And then get an expansion in terms of complex logs. i.e.:  

log _e Z = if,   log _e W = iE  

Such that:

Ö {(1 + e) / (1 - e)}    =  (1  + b )/ (1 -   b)

Note:  exp (ix) = cos x + i sin x

cos (x) =  [exp  (ix) +  exp (-ix)]/ 2

sin (x) =  [exp  (ix) -  exp (-ix)] / 2

Then:   tan f/ 2  = 

1/i [exp  (if/2) -  exp (-if/2)/ exp  (if/2) +  exp (-if/2)]

= 

Ö {(1 + e) / (1 - e)} 1/i [exp (iE/2) -  exp (-iE/2)/ exp  (iE/2) +

  exp (-iE/2)]

Where:   i f/ 2  =  ½   log Z

Then:  

exp(if/2) =  exp (log Z ^1/2 )=   Z ^1/2 / exp  (iE/2) =  W ^1/2

In complex format:

Ö {(1 + e) / (1 - e)} 1/i [exp (iE/2) -  exp (-iE/2)/ exp  (iE/2) +

  exp (-iE/2)]

Þ Ö Z  -  1/ Ö Z /  Ö Z  +  1/ Ö Z  =   (Z - 1)/ ( Z + 1)

=  (1  + b )/ (1 -   b)   (W - 1)/ ( W + 1)

Cross multiplying:

(Z - 1) (1 -   b)( W + 1)  =  (Z + 1)  (1  + b ) (W - 1)

So that:    Z =  (W -  b)/  ( 1 -  Wb)

The final form for the expansion is then:

Z =  W ( 1 -  W ^-1 b) ( 1 -  Wb) ^-1

Taking the logs of both sides:

log Z = log W + log (1 - b W ^-1 ) + log (1 - b W) ^-1

Whence:  if  = iE

Suggested Problems:

1) The Earth's aphelion distance is 1.01671 AU and its perihelion distance is 0.98329 AU. Given its eccentricity e = 0.016, then use this information and any other (from previous posts) to find:

a) the velocities at aphelion and perihelion

b) the energy constants C(A) and C(P) at each of these points, and h, the 'specific relative angular momentum'.

c) Hence or otherwise, use the vis viva equation to confirm the results you obtained in (a)

2)Explain the significance of the expression:

Z =  (W -  b)/  ( 1 -  Wb)

3)What  can you deduce regarding the relationship between: f,  E  and  w ?  What common factor unites them?