This assignment is DUE on Friday, May 8.
The following table gives the right ascensions and declinations of an asteroid, at 10-day intervals.
| Date | Right Ascension | Declination |
| 1975 Dec 24.0 | 4 h 6.3 m | +18° 50' |
| 1976 Jan 13.0 | 3 h 56.2 m | +19° 41' |
| 1976 Feb 2.0 | 3 h 57.6 m | +20° 52' |
Using either Laplace's or Gauss' method (your choice), calculate the position and velocity vectors of the comet, relative to the Sun for the middle date. You will need to look up the rectangular coordinates of the Sun in the ephemeris. The equator and equinox of the observations are 1950, so be sure to use the correct table for the position of the Sun when you get to that point in the calculation!
HINTS:
Use a spreadsheet for this! You'll save yourself a lot of time and grief.
Be sure to apply checks. For example, make sure that whenever you have a vector that's supposed to be a unit vector, you check that the sum of the squares of its components equals one. That way you can avoid blunders.
Let t0 be the middle date, and let t-1 and t1 be the first and last dates, respectively. Convert time from ephemeris days to dimensionless time tau by using tauj=k( tk- t0), where k is the Gaussian constant of gravity.
The three dates are deliberately equally spaced. This will allow you to use the simple interpolation formulas if you are using Laplace's method.
Compute the unit vectors to the object for the three dates. If using Laplace's method, compute the first and second derivatives with respect to tau with the interpolation formulas. Compute the constants in the equation for the triangle (remember that the cosine of an angle is equal to the dot product between the two vectors pointing along the sides of the angle, divided by the product of their lengths). Perform the dot and cross products required and solve iteratively for rho and r at the middle time. Compute the required position and velocity vectors.
If using Gauss' method, you'll need to compute T-1= t1- t0, etc. as given in the notes. Compute the coefficients of the f and g functions that you will need. Compute the equation rho=A+B/ r3, and the equation for the triangle (as for Laplace's method). Solve iteratively for rho and r; then compute the position and velocity vectors as shown in the notes.
EXTRA CREDIT
For extra credit, you may calculate the classical elements of the orbit from the position and velocity vectors of the asteroid that you obtained in the main problem. This will count an extra 50%, so I hope that you will do this.
For extra credit, you may calculate the position and velocity vectors using both Laplace's and Gauss' method. This will also count an extra 50%.