Orbit Visualization and Ephemeris Generation:
One of the first steps in orbit determination
is ephemeris generation. An ephemeris is a
table of values that gives the position of an
object in the sky as a function of time. With
a known orbit we want to tabulate the future
positions and velocities of the orbiting
object.
The big picture of this assignment is that
given the classical orbital elements
(a,e,v,i,) we wish to display the orbit of
this asteroid using Newtonian gravitation
and a central force model and calculate the
position and velocity of that object at that
time.
Wikipedia 2008
Recall that previously you placed the earth, sun and your asteroid in a solar
system. You drew the orbit based
on an entered eccentricity, semimajor axis and inclination. You
drew arrows that connected the
sun to the earth (Something we
obtain from JPL from the Horizon
system), the earth to the asteroid
(What we measure from the CCD)
and the sun asteroid vector
(Determined as the sum of the
earth sun and earth asteroid
vector.) The output of your
program should have looked something this:
Now we would like to implement the other orbital elements into our model. If
we can model these elements completely, we can run our model forward and
backward in time and find the position of the asteroid at any time. This is the
process of ephemeris generation.
1. Make any necessary modifications to your program from Monday so that
you can display the orbit of your asteroid given any (e,a and i)
2. To implement the inclination rotation, you are rotating about the x axis.
You probably implemented the inclination by assuming that the initial
momentum was all in one direction, say the z direction and then
multiplying the y and z components of the
0
0  0  
0 
1
momentum by -sine(i) and cosine(i)

  

 0 cos(i )  sin( i )  0     sin( i )v 
respectively. A more elegant way to do
 0 sin( i ) cos(i )  v   cos(i )v 
this is to apply a rotation matrix.

  

3. Vpython provides a simple way to apply these rotation matrices using
the “rotate” command.
newvector =
rotate(oldvector, angle=pi/4., axis=axis, origin=pos)
The rotate function applies a transformation to the specified object
oldvector.). The transformation is a rotation of angle radians,
counterclockwise around the line defined by origin and origin+axis. By default,
rotations are around the object's own position and axis. Modify you program
so that it uses the “rotate” command to implement the inclination.
HINT: You might want to display a set of X, Y and Z axis centered on the Sun
to help you visualize the rotations.
Make sure you understand how the rotate command works before
proceeding!
4. Now you need to implement the longitude of the ascending node().
Based on the diagram at the very top, you should see that this is a
second rotation of the orbit, but this time about the y axis. Use the
vector rotate command to implement the longitude of the ascending
node. (Hint: you will need to rotate both the momentum vector and the
position vector)
5. Draw the line of nodes. This is a straight line from the ascending node
to the descending node. Also draw a line to indicate the origin of the
system. Call Prof Mason or a TA over when you are done so that
they can see you work.
Origin Line
Line of Nodes
6. Now implement the argument of periapsis (). This is yet another
rotation. However this one is a bit trickier. By inspection of the
diagram at the top, it is clear that we need to rotate the ellipse about an
axis that is perpendicular to the plane defined by the orbital plane of the
ellipse. Use another rotation of the position vector and momentum
vector about an axis perpendicular to the plane of the ellipse to
implement the argument of periapsis. Hint: you might want to think
about the definition of the cross product. If you get stuck, talk to your
neighbor or ask for help. Keep checking your progress against the
Wikipedia picture at the vey top and the picture above.
If you use the following values you should produce a picture that pretty
closely matches the Wikipedia picture and the picture below:
e = 0.4,a = 3e11, i = .80, = 4.72, = 1.57
Call Prof Mason or a TA over when you have finished your work.
7. The last element is the “starting point” of the
asteroid along its orbit. We can’t simply do
another rotation, because the asteroid needs to
propagate forward along the ellipse, not on a
circle! This will require some thinking!!!!
The simplest thing to start with is the true anomaly.
The true anomaly is the angle between the direction of
perihelion and the current position of the asteroid as
seen from the sun.
If we know the true anomaly, we can write the rectangular coordinates of the
ellipse as
 x' 
 x   r cos 
  sin 
  

 '
GM 

e  cos 
 y    r sin   and taking the derivative  y  
2 
a (1  e ) 

z  0 
 z' 
0


  

 
Where:
a(1  e 2 )
r
1  e cos v
 x' 
 0 
 '
GM 

e  1
Note that in the case that  = 0, the expression reduces to  y  
2 
a (1  e ) 

 z' 
 0 
 
or
y'
GM
(e  1)
a (1  e 2 )
Which simplifies to:
GM (e  1)
y' 
a (e  1)
This is equivalent to the perigee velocity we derived in class.
Implement these conditions as the initial conditions for the position
and velocity of your asteroid. Your asteroid should now start off
rotated on the ellipse.
Mean Anomaly:
Unfortunately, True Anomaly is not typically reported in the description of
orbits! What is typically reported is Published Mean Anomaly. I didn’t make
this stuff up so please bear with me for a bit! The first thing we have to do is
get the Mean anomaly(M) from the published Mean anomaly(Mp). In order to
do this we also need to know the ephemeris time. This is simply a time when
astronomers agree to start their results. The MPC recomputes this every year
and most recently, ephemeris time (TE) is June 18,2009. Since time is linear,
we just calculate the Mean Anomaly (M) by how much time has elapsed since
the ephemeris time.
GM
 t – TE   Mp
a3
Where t is the current Julian Date and Mp is the published Mean Anomaly
NOTE: Mp is usually published in DEGREES. However the units in this
equation assume that you are working in radians.
M 
Once you have the Mean Anomaly, you can calculate the eccentric anomaly by
inverting the Kepler Equation using Newton’s Method: M = E – e sin E. Recall
that we discussed this in class on Monday.
Exercise: Write a function to take a value for M and a tolerance and return
the value for E. Here is some sample data to help:
e = 0.5 M = 45o E = 72.29o
Note: You can also check you function by plugging E back into M = E – e sin E
Once we have the Eccentric Anomaly, we are in the homestretch. Dr Ran
derived the rectangular coordinates of the initial position and velocity of the
asteroid as a function of a, e and E as:
GM   sin( E ) 
 x' 
 x   a cos E  ae 
  



a 3  1  e2 cos E 
2
'
and
y

y

a
1

e
sin
E

  

  1  e cos E 
'


z 

z 

0
0
  

 


Where this position is in ecliptic coordinates with the x axis coinciding with the
perihelion.
Now change your program to accept Published Mean Anomaly and Ephemeris
Time instead of True anomaly.
Be sure to get Prof Mason or a TA to check you off at this point.
Congratulations! The hard part is done. We can generate the position and
velocity vectors in ecliptic coordinates of an asteroid at a given (a,e,Mp,i,)
and Te.
These are just the position and velocity vectors your program generates after
applying the Mean Anomaly and the three rotations.
8. First we need to convert the coordinates into ecliptic(The earth velocity
vector given below has a non zero z component!) and then equatorial
coordinates.
Convert r to equatorial by rotating the
coordinates by the tilt of earth relative
to the ecliptic :
 xeq   1
0
0  x1 
  
 
 yeq    0 cos( )  sin(  )  y1 
 z   0 sin(  ) cos( )  z 
 1 
 eq  
This is a rotation around the x axis. Is that still
appropriate in our coordinate system? Where  =
23.439281o
9. Now determine the new earth
asteroid vector (also known as the
range vector or ) by adding the
earth-sun vector to the new r
vector. To do this accurately, we need to put the earth in the right
place relative to the sun! Modify your program so that it places the
earth in the right place in the sky relative to date and time by
incorporating your program from the homework.
10. Now solve for your RA and DEC using ρ̂ Be sure to worry about angle
ambiguity!
You ephemeris project is to implement all of these ideas in a program that takes a set of orbital
elements plus a JD and generates the RA and DEC of the object at that time.
Wikipedia 2008