Orbital  Mechanics  :  Model   &  Simulation  

Motion   of  Earth,   Sun,  Moon   &   Satellites  Motion   in  Orbit.  

 Updated  Dec. 20,  2015

Orbital  Mechanics Model & Simulation Software (OM-MSS), Astronomical Time Standards and  Time  Conversions,  Positional  Astronomy - Earth Orbit  Around  Sun, Position of Sun  on  Celestial Sphere,  Position  of  Earth on Celestial  Sphere,  Satellites  Orbit Elements - Ephemeris,  Keplerian elements, State vectors, Satellites Motion Around Earth  - Orbital  & Positional parameters, Satellite  Pass  for Earth  Stn - Prediction  of  Ground  Trace are  addressed.

 

Motion of Earth Sun Moon Satellites Motion in Orbit

Bookmark      

We look in to space  from  Earth,  observe Sun, Planets, Moon,  and Satellites in   Motion.

Earth, is 3rd  planet from Sun, takes  around  365.25 days to moves  around  Sun in an Elliptical orbit. The average  distance  from  the Earth  to  the Sun  is  called  one Astronomical Unit (AU);   1 AU = 149,597,870.7 km.

 

Mars,  is  4th planet from  Sun,  that  takes 686.971  Earth days  to  orbit around Sun. The  orbital path of Mars is highly eccentric. Mars &  Earth move along  their orbits,  and come near to one  another  approximately every  two years. On Apr. 08, 2014, the  near  or  close distance between  Mars and  Earth was  92.4  million km.

 

Moon moves  around  Earth in the  same  kind  of  orbit.  The Moon is the  Earth's only natural  Satellite. The average  distance  of  the Moon from the  Earth is 384,403 km.

 

A Satellite  is  an  artificial  object, intentionally  placed  into  orbit.  Thousands of Satellites are  launched  into orbit  around  Earth.

 

A  few Satellites called Space  Probes  have been placed into orbit  around  Moon, Mercury, Venus, Mars,  Jupiter,  Saturn, etc.

 

Understanding  the motion of Earth  around  Sun,  and the  motion  of  Moon  and Satellites around Earth  is  of  interest  to  many.

 

 

Monographs,  Articles   &   Invited   Talks

 

 

 

Introduction to Orbital  Mechanics Software OM-MSS

by R C  Chakraborty,  June 23, 2015,  Pages 1 – 5. 

 

Earth, Sun, Moon &  Satellites  Motion  in  Orbit -  Model &  Simulation  Software.

The  Software  is  written in 'C'  Language. The  Compiler  used  is  Dev C++ and  the Platform is

Windows 7, 64 bit Laptop.  The Source Code around 30,000 Lines,  is  Compiled.

The 'OM-MSS.EXE' File generated  is  of  Size 1.5 KB.  The Executable File,  < OM-MSS.EXE >, is RUN 

Step-by-Step for  a Set  of  Inputs.  

The  execution of OM- MSS  Illustrates its  Scope,  Capability, Accuracy,  and Usage.

The  OM-MSS  Software  includes  the following  :

(a) Astronomical   Time  Standards  and  Time   Conversions   Utilities :

     GMT -  Greenwich Mean Time,  LMT - Local Mean Time,  LST - Local Sidereal

     Time,  UT - Universal  Time,   ET - Ephemeris  Time, JD - Julian  Day,  Standard 

     Epoch J2000, Gregorian  Calendar  date  and more.

(b) Positional Astronomy  of  Earth,  Sun,  Moon, &  Satellites  Motion  in  Orbit,  includes  computations   of  :

1

Position of Sun  and Position of Earth  on  Celestial Sphere at Epoch ; 

2

Keplerian  elements :  Inclination,  RA of asc. Node,  Eccentricity, Arg. of Perigee,

Mean Anomaly,  Mean Motion;

3

Motion Irregularities : Mean,  Eccentric and  True  anomaly in deg;

4

Precise  Time  at  Earth Orbit  Points :  Perihelion, Aphelion,  Equinoxes, Solstices,

Semi-Major &  Minor-axis;

5

Astronomical years :  Anomalistic,  Tropical, and  Sidereal  Years;

6

Four Seasons :  Spring, Summer,  Autumn  and Winter start  time  and duration;

7

Position of Satellites around Earth : Keplerian elements and State Vectors 

at epoch, and  computing,  Sub-Sat point lat/log,  EL & AZ angles, Distances, Velocity,  and more;

8

Satellite  Pass, Ground Trace  for Earth  Stn using  NASA/NORAD  2-line bulletins;

 

(c) Customized Utilities and products  : On special  request either developed  or  configured  and generated.

 

   

 

 

Astronomical Time Standards  and Time Conversions

by R C  Chakraborty,  June 23, 2015,  Pages 6 – 32. 

 

First  look  into   few   preliminaries   and   then  time  conversion   utilities.

Time is a dimension in which the events can be ordered from the past through the present into the future.  Our clocks are set to run (approximately) on solar time (sun time).  For astronomical observations, we need to use sidereal time (star time).

 

Time   Standards   and   designations :

Solar Time,  Sidereal Time,  Equation of time,  Precession,  Nutation,  Hour Angle HA,   GMT,  GMST,  LMT,

LMST,  Universal Time UT, International Atomic Time TAI,   Ephemeris Time ET,  Gregorian calendar,

Julian Day JD.

 

The  Precise   time  conversion   utilities :

1.

Conversion of Universal Time To Julian Day;

2.

Conversion of Julian Day To Universal Time;

3.

Conversion of Fundamental Epoch To Julian day and Julian century;

4.

Add or Subtract time (days, hour, minute seconds) to or from input time;

5.

Julian day for start of any Year;

6.

Solar Time : Local Mean Solar Time (LMT) over observer's Longitude;

7.

Sidereal Time : Greenwich universal time at hour 0.0 (ST0) and GMST;

8.

Greenwich Sidereal Time (GST), Hour Angle (GHA) & Mean Sidereal Time (MST);

9.

Local Mean Sidereal Time (LMST) over observer's Longitude ;

10

Time Conversions : LMT to LST, LST to LMT,  LMT to LMST,  LMST to LMT .

 

    

 

 

Positional Astronomy  - Earth  Orbit Around Sun

by R C  Chakraborty,  June 23, 2015,  Pages 33 – 56. 

 

First  look into few Preliminaries about 'Positional Astronomy' and Prediction of Astronomical Events.

 

Preliminaries about Positional Astronomy : Kepler's Laws of Planetary Motion, Celestial  Coordinate      Systems, Celestial Orbit in astronomy,  Orbit Elements  or  Parameters, anomalies  or  irregularity  in  the motion   of  a planet. 

 

Prediction of Astronomical Events : Anomalies, Equinoxes, Solstices, Years  & Seasons  utilities :

1.

Precise  value for  Mean, Eccentric  & True anomalies;  

2.

Precise  time  for Earth  to  reach Perihelion &  Aphelion  points;

3.

Precise  time  for Earth  to  reach Vernal &  Autumnal  equinox points;

4.

Precise  time  for Earth  to  reach Summer &  Winter  solstice  points;

5.

Precise  time  for Earth  to  reach Semi-major &  Semi-major  axis  points;

6.

Duration of Anomalistic, Tropical &  Sidereal years;

7.

Start  time  & durations  of  seasons -  Spring, Summer,  Autumn, &  Winter.

 

   

 

 

Position   of  Sun  on   Celestial   Sphere  at   input   universal   time  (ut)

by R C  Chakraborty,  June 23, 2015,  Pages 57 – 67.

 

Sun  is  a star at the  center  of  our Solar  System.   Earth moves  in  an  elliptical  orbit around the  Sun. 

The  Position  of  Sun on Celestial  Sphere  is  represented by computing following  parameters  :

1.

Semi-major axis (SMA),

2.

Mean movement per day (n sun),

3.

Mean distance (As),

4.

Mean anomaly (m sun), 

5.

True anomaly (T sun), 

6.

Eccentric anomaly (E sun),

7.

Right ascension (Alpha),

8.

Declination (Delta),

9.

Mean longitude (Lmean), 

10.

Ecliptic longitude (Lsun),

11.

Nodal elongation (U sun),

12.

Argument of perigee (W sun),

13.

Obliquity of ecliptic (Epcylone),

14.

Mean dist (d_sun),

15.

Radial distance (Rs). 

 

 

The  values  of  all these  parameters  are Computed are  at  Standard  Epoch JD2000   and   when  Earth  is at Perihelion,  Aphelion, Equinoxes, and  Solstices.

 

    

 

 

Position   of  Earth  on   Celestial   Sphere  at   input   universal   time  (ut)

by R C  Chakraborty,  June 23, 2015,  Pages 68 – 163.

 

Earth  is  a sphere,  the third  planet  from  the  Sun and  the fifth  largest of   the eight  planets in the

Solar  System. Earth  Revolves  around  Sun in a  counter clock wise direction. The  complete  orbit 

(360 deg)  is  one Sidereal year, occurs every  365.256363  mean  Solar days. The  Position  of  Sun on

Celestial  Sphere  is  represented  by computing  following  parameters :

GST Greenwich sidereal time and GHA Greenwich hour angle in deg at input UT .

2.

Earth Log and Lat in deg, pointing to Sun Ecliptic Log (Lsun) at input UT .

3.

LST  over  Greenwich log, Sun  Mean  log (Lmean), &  Sun Epliptic log (Lsun)

4.

ST0  over  Greenwich log  at  input Year JAN day 1 hr 00 .

5.

ST   over  Greenwich log,  Sun mean log (Lmean), & Sun Epliptic log (Lsun) .

6.

H  hour angle in deg over Greenwich log, Lmean, Lsun,  Sub Sun point SS, Earth  Observation point EP at input UT .

7.

Delta  E Equation of Time in sec, using  p_julian_day, n_sun, w_sun at input UT .

8.

GST and GHA at time when Earth is at Perihelion .

9.

ST & MST using Earth mean motion rev per day and Julian century days from Std  Epoch J2000 .

10.

Earth orbit radius &  sub sun point 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)

12.

Transform  EC to SC Range Vector[rp, rq] in PQW frame To Range Vector[rI, rJ, rK] in IJK frame (inertial system cord).

13.

Transform  Earth point EP (lat, log, hgt) To EC to SC Range Vector[RI, RJ, RK, R] in IJK frame.

14.

Transform  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  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 EP to Sub Sun point SS and  Earth Velocity in orbit at input UT.

18.

Earth State Position Vector [X, Y, Z] in km at input UT.

19.

Earth State Velocity Vector [Vx, Vy, Vz] in meter per sec at input UT.

20.

Earth Orbit Normal Vector  [Wx, Wy, Wz] in km and angles Delta, i, RA at input UT;

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 at Standard Epoch JD2000 and  when  Earth is at

Perihelion,  Aphelion, Equinoxes, and  Solstices.

 

   

 

 

Satellites   Orbit   Elements  - Ephemeris,   Keplerian elements,  State vectors  

by R C  Chakraborty,  June 23, 2015,  Pages 164 – 192.

 

A  satellite is an object that moves  around  a larger object. 

 

Satellite  Ephemeris is  Expressed  either   by  'Keplerian  elements'   or  by 'State Vectors', that uniquely identify a  specific  orbit.

 

The Keplerian elements are  encoded as text in different  formats.  The  most  common  format  is

NASA/NORAD 'Two-Line  Elements' (TLE).  For  all satellites,  you can  download  the NASA/NORAD 'two-line elements' (TLE)  from  CelesTrak Web site,  URL http://celestrak.com/NORAD/elements/ .  

 

State Vectors represents, Position (X, Y, Z) and Velocity (Vx, Vy, Vz) of a Satellite orbital trajectory in time.

 

   

 

 

Satellites   Motion  Around   Earth -  Orbital   &   Positional  Parameters

by R C  Chakraborty,  June 23, 2015,  Pages 193 – 267.

 

The  Satellites  Orbit around Earth, Counterclockwise,  in  the same way  as  Earth orbits around Sun.

The  Satellite motion around Earth  is  represented  by computing  following parameters :

1.

UT Year and  Days  decimal of year : Convert into UT & Julian day.

2.

Sat orbit Semi major axis in km, Ignoring and Considering earth oblatenes.

3.

Sat Mean motion in rev per day, Ignoring and Considering earth oblateness.

4.

Sat Mean motion in rev per day, Ignoring and Considering earth oblateness.

5.

Sat Rate of change of Right ascension and Argument of perigee in deg per day at  time_t.

6.

Sat Mean,  Eccentric & True anomaly in deg at time_t considering earth oblateness.

7.

Sat position vector[rp, rq] from Earth Center(EC) to Sat in PQW frame (perifocal coordinate system)

8.

Sat Position Range Vector from Earth Center(EC) to Satellite(SAT) - Range Vector[rI rJ rK r] components in km in IJK frame.

9.

GST and  GHA deg, at input at time_t.

10.

Sat Orbit point direction :  Right ascension(Alpha) and Declination(Delta) in deg.

11.

Sat Log & Lat in deg at time_t; (ie  sub-sat point log & lat on earth    surface ).

12.

Sat height in km from EC to Sat and from Earth surface to Sat at time_t.

13.

Distance of  sub-sat point To Earth stn (ES) in km over Earth surface at time_t.

14.

Local sidereal time(LST) and Local mean time(LMT) over Sub-Sat point Log on  earth.

15.

Local sidereal time(LST) and Local mean time(LMT) Over Earth Stn (ES) or Earth point(EP) Log.

16.

Earth stn Position Vector from Earth Center(EC) to Earth Stn(ES) : Range  Vector[RI, RJ, RK, R] components in IJK frame.

17.

Sat Position Range Vector from Earth Stn(ES) to SAT :  Range Vector[rvI, rvJ,  rvK, rv] components in km in IJK frame

18.

Sat Position Range Vector from Earth Stn(ES) to SAT :  Range Vector[rvS, rvE, rvZ, rv] components in km in SEZ frame

19.

Elevation(EL) and Azimuth(AZ) angle of Satellite at Earth Observation point EP or ES.

20.

Sat Velocity meter per sec in orbit.

21.

Sat Velocity Vector [vX, vY, vZ] in meter per sec in orbit in frame XYZ.

22.

Sat Pitch and Roll angles.

23.

Sat State Vectors -  Position [ X, Y, Z ] in km  and velocity [ Vx, Vy, Vz ] in meter per sec at time_t.

24.

Satellite Direction (right ascension alpha & declination delta in deg)

25.

Sat Angular momentum km sqr per sec :  Hx Hy Hz H from state vector pos and  vel.

26.

Sat Orbit normal Vector : Wx Wy Wz W Delta Alpha from  r_sat_pos in IJK frame, i, RA.

27.

Sat Position Keplerian elements computed using State Vector, at time input UT.

28.

Sat position State Vectors, computed using Keplerian elements at time input UT.

The  values  of  all these  parameters  are computed using  NASA/NORAD  'two-line elements' (TLE) of the Satellite  from CelesTrak  Web site.   The Satellites considered for  computation are  LANDSAT 8, SPOT 6, CARTOSAT 2B, ISS (ZARYA), GSAT-14 ,  and Moon.

 

    

 

 

Satellite  Pass   for   Earth   Stn   -   Prediction  of   Ground  Trace  

by R C  Chakraborty,  June 23, 2015,  Pages 268 – 390.

Satellites,  look like slow-moving  Stars, are  most  visible when they are  in  Sunlight  while the

viewer is in darkness.

A  typical Satellite  in  low Earth  orbit (LEO) circles  the Earth  about 16 times  each  day.

The  Orbital Velocity of a  LEO satellite  is  about 7500 meters/sec.

The  Orbital Velocity of a  Geo-stationary  satellite is about  3007  meters/sec.

The  Moon, the  only  natural satellite  of  earth has  orbital velocity about  1003  meters/sec.

 

Satellite  Pass  for Earth  Stn, is Computed   for following  six satellites : 

LANDSAT 8,   SPOT 6,   CARTOSAT-2B,  ISS (ZARYA),  GSAT-14,  and  MOON.

The  'Satellites Pass'  goes through  a Time_Step  of   2 minutes (120 sec). 

For  Moon  the Time_Step  is  of  1 hr (3600 sec).

The  input is respective Satellite's  NASA/NORAD  'Two-Line Elements'(TLE).

The  Output  is  predictions  of   instantaneous   ground trace  coordinates, look angles at each

Time_Step  on   computer  screen,  in  a Table  form  where  respective   columns   indicate  :

1.

Orbit no

2.

Node Ascending or Descending

3 -6

Input time GMT D H M S

7.

True Anomaly

8.

Sat  Height  from  earth surface

9.

Sat  at  Perigee,  or  Equator,  or  Apogee

10.

Sat  Velocity

11-12

Latitude &  Longitude at sub-satellite  point on earth  surface,

13.

Sat  Slant Range  from  earth stn,

14.

Distance of sub-satellite  point from earth  stn,

15-16

Sat  Pitch &  Roll  angles,

17-18

Sat  Elevation &  Azimuh  angles  at  earth stn,

19

Access to Sat  through On Board  Computer  or  Direct  Line  Of  Sight based  on  elevaion  angle  at  ES,

23-26

Local  Mean  Time  at  earth stn D H M S

27-28

Sun  Elevation &  Azimuh  angles  at  earth stn,

29

Distance of sub-Sun  point on earth  surface from earth  stn,

30

Line number

 

   

 

 

Orbital  Mechanics -  Model &  Simulation  Software (Om-Mss)

by R C  Chakraborty,  June 23, 2015,  Page 1 – 402. 

 

Presented  the Earth, Sun,  Moon  & Satellites Motion in Orbit - Model  & Simulation Software with   Examples,  Problems  and Software  Driven   Solutions.

The  Software  is  written in 'C'  Language. The  Compiler  used  is  Dev C++ and  Platform is a  Windows 7, 64 bit  Laptop. The  Source  Code around 30,000 Lines,  is  Compiled. The 'OM-MSS.EXE'   File  generated is of Size 1.5 KB.   The Executable File,  < OM-MSS.EXE >, is RUN  Step-by-Step  for a  Set of  Inputs.

 

The  execution of OM- MSS  illustrates its  Scope,  Capability, Accuracy,  and Usage. The  Results  seen   on Computer Screen are  put in a  File, that in effect becomes  'A Monograph of Orbital  Mechanics with  Examples,  Problems  and Software  Driven  Solutions',  Page 1 – 402,  which  includes  the following : 

1

Introduction to OM-MSS Software

2

Astronomical Time Standards  and Time Conversions.

3

Positional Astronomy  - Earth  Orbit Around Sun, Anomalies  & Astronomical Events   (equinoxes, solstices, years  & seasons).

4

Position of Sun  on  Celestial Sphere at input  universal time (ut).

5

Position of Earth  on  Celestial Sphere at input  universal time (ut).

6

Satellites Orbit  Elements  - Ephemeris, Keplerian elements,  State vectors

7

Satellites Motion Around Earth  - Orbital  & Positional parameters at epoch.

8

Satellite  Pass  for Earth  stn -  Prediction  of  Ground  Trace coordinates, Look angles,  Universal/Local  time  & more.

 

   

 

 

 

 

  ----------------------------------------------------------------------------------------------------------------------------

 Artificial Intelligence

 Soft Computing

Image Proc Comp Vision

Orbital Mechanics

Remote Sensing

Science  Technology

Projects Academic

Education Junior Level

  ----------------------------------------------------------------------------------------------------------------------------

                 Sitemap GIF 20x20       Diigo Feed    

 

 

                 Sitemap GIF 20x20       Diigo Feed            

[Home] [Tech. & Sci.] [Remote Sensing] [Communication] [Computers] [Electronics] [Orbital Mechanics]