Position of Earth on Celestial Sphere at Input Universal Time
by R C Chakraborty, July 12, 2015, Pages 68 – 163.
(This is Sec. 4, pp 68 – 163, of Orbital Mechanics – Model & Simulation Software (OM-MSS), Sec 1 to 10, pp 1 – 402.)
Earth is a sphere, the third planet from the Sun and the fifth largest of the eight planets in the Solar System.
Planets order from the Sun : Mercury, Venus, Earth, Mars, Jupiter, Saturn, Uranus, Neptune.
Earth Rotates on its axis passing through the North and South Poles. The rotation is counterclockwise looking down at North Pole. This rotation results daytime in area facing Sun and night time in area facing away from Sun. Since we are on Earth, we do not sense its rotation, but experience by observing the relative motion of the Sun (like from a moving vehicle we see the surroundings move).
The time for Earth to make a complete rotation is approximately 24 hours (exactly 23.9344699 hours or 23 hours, 56 minutes, 4.0916 seconds). The earth’s orbit around the sun is not a circle, it is slightly elliptical. Therefore, distance between earth and sun varies throughout the year.
To Compute the Position of Earth on Celestial Sphere at any instant, we first need to Compute Position of Sun on celestial sphere and then at same instant Compute Position of Earth on celestial sphere. For the Position of Sun on celestial sphere, much has been computed / illustrated in previous section (Ref. https://myreaders.wordpress.com/2015/07/11/position-of-sun-on-celestial-sphere-at-input-universal-time/).
The Position of Earth on celestial sphere is characterized by computing around 120 orbital parameters. The number is large, because some parameters are computed using more than one model equation, that require different inputs. This helps in validation of results and understanding the different input considerations.
The Orbital Parameters that Characterize the Position of Earth on Celestial Sphere, are put into following groups :
1. GST Greenwich sidereal time and GHA Greenwich hour angle in 0 to 360 deg, at input UT time YY MM DD HH.
2. Earth Log in 0 to 360 deg and Lat in +ve or -ve in 0 to 90 deg pointing to Sun Ecliptic Log (Lsun) at time input UT.
3. LST Local sidereal time using GST over three longitudes, Greenwich log, Sun mean log (Lmean), & Sun epliptic log (Lsun) .
4. ST0 sidereal time over Greenwich longitude = 0.0, at time input Year JAN day 1 hr 00.
5. ST sidereal time, at time input UT, over three log, Greenwich log, Sun mean log (Lmean), and Sun epliptic log (Lsun).
6. H hour angle in 0 to 360 deg using ST over five longitudes, Greenwich, Lmean, Lsun, Earth Sub Sun point SS, Earth Observation point EP, at time input UT.
7. Delta E is Equation of Time in seconds, using p_julian_day, n_sun, w_sun at time input UT.
8. GST Greenwich sidereal time, and GHA Greenwich hour angle 0 to 360 deg at time when earth is at perihelion.
9. ST sidereal time & MST mean sidereal time at different instances, using Earth mean motion rev per day and Julian century days from YY 2000_JAN_1_hr_1200.
10. Earth orbit radius, sub sun point on Earth surface & related parameters, using SMA, e_sun, T_sun, w_sun etc.
11. Earth center(EC) to Sun center(SC) Range Vector [rp, rq, r] in PQW frame (perifocal coordinate system).
12. Transform_1 Earth position EC to SC Range Vector [rp, rq] in PQW frame To Range Vector [rI, rJ, rK] in IJK frame (inertial system cord).
13. Transform_2 Earth point EP (lat, log, hgt) To EC to SC Range Vector [RI, RJ, RK, R] in IJK frame.
14. Transform_3 Earth position EC to SC Range Vectors [rI, rJ, rK] & [RI, RJ, RK] To EP to SC Range Vector [rvI, rvJ, rvK] in IJK frame.
15. Transform_4 Earth point EP to SC Range Vector [rvI, rvJ, rvK] in IJK frame To EP to SC Range Vector [rvS, rvE, rvZ] in SEZ frame.
16. Elevation (EL) and Azimuth (AZ) angle of Sun at Earth Observation point EP.
17. Distance in km from Earth observation point (EP) to Sub Sun point (SS) and Earth Velocity meter per sec in orbit at time input UT.
18. Earth State Position Vector [X, Y, Z] in km at time input UT.
19. Earth State Velocity Vector [Vx, Vy, Vz] in meter per sec at time input UT.
20. Earth Orbit Normal Vector [Wx, Wy, Wz] in km and angles Delta, i, RA at time input UT; Normal is line perpendicular to orbit plane.
21. Transform Earth State Vectors To Earth position Keplerian elements.
22. Transform Earth position Keplerian elements To Earth State Vectors .
The values of all these parameters are Computed are at Standard Epoch JD2000 and when Earth is at Perihelion, Aphelion, Equinoxes, and Solstices. The time at perihelion, aphelion, equinoxes, and solstices, were computed earlier for the input year in section 2. (Ref. https://myreaders.wordpress.com/2015/07/11/positional-astronomy-earth-orbit-around-sun/).
For complete post (Page 68 – 163) Move on to Website URL :