Celestial Mechanics

Celestial Mechanics I : 

Celestial Mechanics I

Today’s Lecture : 

Today’s Lecture First orbit determination according to the method of Gauss (calculate orbital elements from few observed positions) Orbit improvement (reduction of uncertainties in orbital elements by using many observed positions)

Why is this important? : 

Why is this important? Planned observations requires known orbits Scientific value of orbit statistics Potentially Hazardous Asteroids (PHA)

Search Programmes : 

Search Programmes Discovery statistic, numbered asteroids LINEAR (81641) NEAT (11464) Spacewatch (11007) LONEOS (9365) UDAS (105). Number 68 worldwide Minor Planet Center (MPC) International Astronomical Union (IAU)

Gauss’ Method : 

Gauss’ Method Position- and velocity vectors at one point in orbit  {a,e,i,ω,,T} Gauss’ method assumes three sets of observations available How do we find the geocentric distances?

Derivation of Gauss’ Method : 

Derivation of Gauss’ Method Take advantage of orbital motion in a plane!

Derivation of Gauss’ Method : 

Derivation of Gauss’ Method Take advantage of Kepler II

Derivation of Gauss’ Method : 

Derivation of Gauss’ Method

Derivation of Gauss’ Method : 

Derivation of Gauss’ Method

Algorithm : 

Algorithm Use {αj,δ j} to calculate Δj unit direction vectors Use Δj 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!

Steffensen’s Method : 

Steffensen’s Method Check if y*=y(1)-2 y(2)+y(3) is close to zero. If so, y= y(1). If not, update y(1)

Numerical example : 

Numerical example

Orbit Improvement : 

Orbit Improvement : Orbital elements : Method (e.g., two-body) : Ephemerides

Orbit Improvement : 

Orbit Improvement