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

Orbital position versus time: The choice of units Gravitational constant: SI units ([m],[kg],[s]) G = 6.67259 10-11 m3kg-1s-2 Gaussian units ([AU],[M],[days]) k = 0.01720209895 AU3/2M-1/2days-1 Kepler III: k2 replaces G m1 = 1; m2 = 1/354710 (Earth+Moon) P = 365.2563835 (sidereal year); a = 1

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

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.

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 86400 s by about 2 ms!

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: 1/31,556,925.9747 of the tropical year at epoch 1900 UT1 was used temporarily, until ET was available and the correction T=ET-UT1 was published.

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 = 32.184 s (ET–UT1 in 1958).

History of time keeping, ctd Currently, on average, one mean solar day = 86400 UT seconds  86400.002 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. 2008.

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: http://aa.usno.navy.mil/data/docs/JulianDate.html

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.

Reference frames, ctd The angle between the two planes (“obliquity of the ecliptic”) was =2326’20” on 1 Jan. 2000. 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.

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)

Coordinate transformation

Orientation of orbit w.r.t. ecliptic

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

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

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

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

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

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

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–

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

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 104,780 Spacewatch 22,036 NEAT 19,670 LONEOS 13,000

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)

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)

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:

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

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)

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:

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

Solving for the ’s, ctd

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!

Numerical example

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

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.