1. Celestial Mechanics III
Time and reference frames
Orbital elements
Calculation of ephemerides
Orbit determination

2. 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

3. Celestial mechanics units
The usage of Gaussian units is typical of celestial
mechanics applications
In spaceflight applications, SI units are also
necessary to use
In Oort Cloud dynamics, one often uses the year
as unit of time. The gravitational constant is then
= 42 to very good approximation

4. History of time keeping
Before 1960: Earth’s rotation provided the basic clock, measured by astronomical observations of the sidereal day = 23h56m4.1s (Earth’s spin period).
One mean solar day = 24h = 86400s.
Universal Time (UT) = Greenwich mean time (GMT).
Locally observed UT = UT0 (the results differ due to polar motion).
Corrected UT = UT1 (but the rate still varies because the Earth is not a perfect rotator). Lunisolar tides lead to Length Of Day (LOD) variations.

5. LOD variations Note: the LOD differs from 86400 s by about 2 ms!
From Time Service Dept., US Naval Obs.
Note: the LOD differs from s by about 2 ms!

6. History of time keeping, ctd
From 1960: Solar system orbital motions provided the basic clock, measured by astronomical observations of orbiting objects.
Ephemeris time ET = uniform time, as required by the Newtonian equations of motion.
The ephemeris second was defined as: 1/31,556, of the tropical year at epoch 1900
UT1 was used temporarily, until ET was available and the correction T=ET-UT1 was published.

7. History of time keeping, ctd
In 1967: the frequency of a hyperfine transition in the 133Ce isotope defines the SI second to be the duration of 9,192,631,770 periods of the radiation corresponding to this transition.
This is equivalent to one ephemeris second.
International Atomic Time (TAI) is based on the SI second and maintained by a large number of clocks operating at standards laboratories.
Zero point: UT1 – TAI  0 on Jan. 1, 1958
Since 1984: Terrestrial Time (TT) is used in astrometry and ephemeris calculations, thus replacing ET.
Zero point: TT – TAI = s (ET–UT1 in 1958).

8. History of time keeping, ctd
Currently, on average, one mean solar day = UT seconds  TT seconds. In about 500 days, the difference between TT and UT increases by 1s.
Coordinated Universal Time (UTC) increases at the TAI rate like TT, but by introducing leap seconds, it is kept close to UT1. It defines civil time and provides the connection between astronomical and atomic times.
The first leap second was introduced in 1972 with a starting value of TAI–UTC = 10 s. The latest one (nr. 24) was introduced on 31 Dec

9. Julian dates
A continuous count of days and fractions since noon UT on Jan 1, 4713 BCE (Before Christian Era) on the Julian calendar; the day numbers are now approaching 2.5 million
Very useful for ephemeris calculations
Converters between Julian dates and calendar dates can be found at:
A practical formula, valid for the 20th and 21st centuries, is:

10. Fundamental reference frames
Ecliptic frame, basically heliocentric, couples to the orbits of objects
Equatorial frame, basically geocentric, couples to the astrometric observations
Each one is defined by a fundamental plane, which cuts the celestial sphere along a great circle.
The points of intersection are the equinoxes – the vernal equinox is used as reference direction.

11. Reference frames, ctd
The angle between the two planes (“obliquity of the ecliptic”) was =2326’20” on 1 Jan
Due to lunisolar precession of the Earth’s spin axis, the vernal equinox drifts by ~50” per year.
Planetary precession of the ecliptic plane causes smaller effects in both  and the equinox.
Since we need fixed reference frames, we use a standard epoch (currently “J2000.0”) to define the equator and equinox.

12. Angular coordinates
Use (xeq,yeq,zeq) and (xec,yec,zec) as Cartesian geocentric coordinates. Let the xeq and xec axes point toward the vernal equinox, and let the zeq and zec axes point toward the respective poles.
In spherical geocentric coordinates we instead use:
 = geocentric distance
 = right ascension
 = declination
 = ecliptic longitude
 = ecliptic latitude
(index ‘2000’ means they refer to the standard equator and equinox of 2000)

13. Coordinate transformation

14. Orientation of orbit w.r.t. ecliptic

15. Inclination
i is the angle from the pole of the ecliptic to the pole of the orbital plane (0 < i < )
Prograde orbits: i < /2
Retrograde orbits: i > /2

16. Longitude of the ascending node
Ascending node vector
•  is measured along the ecliptic, counterclockwise as seen from the North pole, from the vernal equinox to the ascending node

17. Argument of perihelion
 is counted along the orbital plane, counterclockwise as seen from its North pole, from the ascending node to the perihelion direction

18. Orbital elements Semi-major axis a, or perihelion distance q = a(1–e)
We have identified six orbital elements, which can be grouped as follows:
Semi-major axis a, or perihelion distance q = a(1–e)
Eccentricity e
Time of perihelion passage T, or mean anomaly at a given epoch M0
Inclination i
Longitude of the ascending node 
Argument of perihelion , or longitude of perihelion  = +
Conversion from orbital position to ecliptic frame
Orbital position at given time

19. Conversion matrix
This transforms vectors from the orbital frame to the ecliptic frame, e.g., the position and velocity vectors:

20. Ephemeris calculation
Calculate mean motion n=2π/P, P=2πa3/2/k
Calculate mean anomaly M=n(t-T)
Solve Kepler’s equation to obtain eccentric anomaly E
Use E to calculate the position (and velocity) vector(s) X (and dX/dt) in the orbital frame
Use {i,ω,} and “Homer’s” transformation matrix to obtain position (and velocity) in the heliocentric ecliptic frame

21. Ephemeris calculation, ctd
Find the heliocentric ecliptic coordinates of the Earth
Calculate the geocentric position vector of the object
Calculate the distance  of the object
Correct for planetary aberration (light time correction)
If the light time is  = /c, repeat the calculation of the object for a time t–

22. Ephemeris calculation, ctd
Convert to equatorial coordinates
Find the right ascension and declination of the object
Repeat for a set of regularly spaced dates

23. Orbit determination
Essential in order to identify and keep track of moving objects like Near-Earth asteroids or comets
Clearing house: IAU Minor Planet Center
Search programs, discovery statistics for numbered minor planets as of March 10, 2009:
LINEAR ,780
Spacewatch 22,036
NEAT ,670
LONEOS ,000

24. Way of procedure
Preliminary orbit determination according to one of several methods (calculate orbital elements from few observed positions)
Orbit improvement (reduction of uncertainties in orbital elements by using many observed positions)
Linkage of several oppositions of an asteroid or apparitions of a comet (including chance identifications)

25. The Method of Gauss Assume 3 observations available: (j,j,tj); j=1,3
Thus 3 geocentric equatorial unit vectors are known:
We can also find the Sun’s geocentric positions at the 3 times of observation
We can write 3 heliocentric position vectors of the object using unknown geocentric distances:
(at time tj)

26. The Method of Gauss, ctd Idea: find the three  values;
then use the three r values to derive the heliocentric position and velocity at the middle observation
Transform from position and velocity to orbital elements
Since motion is planar:
Coefficients given by:

27. The Method of Gauss, ctd
We obtain a vector equation, equivalent to 3 scalar equations, from which j can be solved, if c1 and c3 are known:
The method will be to work with successive approximations and iterate until convergence
Start with an initial guess for c1 and c3

28. Estimating the c parameters
Half the vector product is the area of the triangle; A is the area overswept by r
by Kepler II:
Similarly:
yj are the sector-to-triangle ratios (close to 1)

29. Transforming the vector equation
Transform from the equatorial to a new coordinate system, which we denote ‘C’:
Multiplying the vector equation for j by the transformation matrix, in the new system we have:

30. Solving for the ’s
Due to our choice of axes in the C system, the scalar equations become separable in the three ’s
The scalar products i, i,i are known quantities

31. Solving for the ’s, ctd

32. Algorithm Use {j} to calculate Δj unit direction vectors
Use these to calculate {2, 2, 2, 3, 3} and ReqC
Transform Δj and R,j vectors to C system
We need to know c1 and c3! Assume y2/ y1= y2/ y3=1
Calculate the geocentric distances!
Calculate the approximate heliocentric position vectors of the object!
But now we can estimate overswept areas, i.e., more realistic values of c1 and c3 can be calculated. Iterate!

33. Numerical example

34. Orbit Improvement : Orbital elements : Algorithm (e.g., two-body)
: Ephemerides
Apply an algorithm to the orbital elements to obtain an ephemeris
Compute the Jacobian of the algorithm:
Further observations show discrepancies with the ephemeris

35. Orbit Improvement, ctd
Identify the Jacobian with the ratio of finite differences:
With many observations, we get a system of many equations involving known residuals . Use a statistical method like the least-squares method to find the  that minimizes the residuals.