Differences between revisions 2 and 5 (spanning 3 versions)

Near Circular Orbits

MORE LATER add pointers to orbit discussions

$ r = (1-e^2) a

Periods of M orbits

M orbits describe the number of minutes an orbit takes travel once around the earth and return to the same position overhead. For server sky, these are integer fractions of a 1440 minute synodic day; this makes it easier to calculate the sky position given the orbital parameters and the time of day. the M288 orbit returns to the same apparent position 5 times per day, or 365.256...*5 times per year. That position moves around the earth 365.256...+1 times per year. So the total number of orbits per year, relative to the stars, is 365.256...*6+1 orbits per year. That is divided into the year length in seconds to yield the

$ sidereal orbit time = ( 365.256... * 86400 ) / ( 365.256... * 6 + 1 ) = 86400 / ( 6 + 1/365.256... ) =

1 year = 365.256363004 days of 86,400 seconds, or 31558149.7635456 seconds

symbol		LEO 300Km	M288	M360	GEO	Moon	Earth	units
\mu	gravitation param.	3.98600448E+14	3.98600448E+14	3.98600448E+14	3.98600448E+14	3.98600448E+14	1.32712440E+20	m³/s²
	relative to	earth	earth	earth	earth	earth	Sun
a_g	gravity	8.938095E+00	2.437062E+00	1.911369E+00	2.242078E-01	2.697573E-03	5.930053E-03	seconds
T	sidereal period	5431.009959	14393.432269	17270.543331	86164.099662	2360591.577436	31558149.763546	seconds
T_s	synodic period	5431.944772	14400.000000	17820.000000	86400.000000	2551442.900000	31558149.763546	seconds
T/2\pi	Sidereal / 2pi	864.372081	2290.785853	2748.692283	13713.440926	375698.721205	5022635.529703	seconds
\omega	Angular velocity	1.15690919E-03	4.36531419E-04	3.63809367E-04	7.29211585E-05	2.66170722E-06	1.99098659E-07	rad/sec
	orbits/year	5810.73318	2192.53822	1827.28185	366.25640	13.36879	1.00000
R	semimajor axis	6678000.00	12788970.60	14440980.32	42164169.86	384399000.00	149598261000.00	meters
R_a	apogee radius	6678000.00	12838970.60	14490980.32	42164169.86	405696000.00	152098232000.00	meters
R_p	perigee radius	6678000.00	12738970.60	14440980.32	42164169.86	363104000.00	147098290000.00	meters
e	eccentricity	0.000000	0.001951	0.001728	0.000000	0.055401	0.016711
V_0	mean velocity	7725.84	5582.79	5253.76	3074.66	1023.16	29784.81	m/s
V_a	apogee velocity	7725.84	5571.90	5244.68	3074.66	966.47	29287.07	m/s
V_p	perigee velocity	7725.84	5593.68	5262.84	3074.66	1079.84	30282.55	m/s
C_3	orb. specific energy	-59688596.59	-31167516.16	-27602035.26	-9453534.82	-1036944.55	-887125553.00	J/kg

-  ⇤ ← Revision 2 as of 2011-04-01 01:30:32 → 
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+   ← Revision 5 as of 2011-04-01 02:17:56 → ⇥
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  Editor: KeithLofstrom
  Comment:
-Deletions are marked like this.
+Additions are marked like this.
 Line 15:
-M orbits describe the number of minutes an orbit takes travel once around the earth and return to the same position overhead.  For server sky, these are integer fractions of a 1440 minute synodic day; this makes it easier to calculate the sky position given the orbital parameters and the time of day.  the M288 orbit returns to the same position 5 times per day, for example, and makes six synodic orbits in one synodic day.  But not exactly.  Because the earth orbits around the sun, the position of the sun makes a complete turn around the earth in one year, 365.256363004 days of 86,400 seconds, or 31558149.7635456 seconds
+M orbits describe the number of minutes an orbit takes travel once around the earth and return to the same position overhead.  For server sky, these are integer fractions of a 1440 minute synodic day; this makes it easier to calculate the sky position given the orbital parameters and the time of day.  the M288 orbit returns to the same apparent position 5 times per day, or 365.256...*5 times per year.  That position moves around the earth 365.256...+1 times per year.  So the total number of orbits per year, relative to the stars, is 365.256...*6+1 orbits per year.  That is divided into the year length in seconds to yield the
 Line 17:
-||   symbol   ||                         || LEO 300Km      || M288           || M360           || GEO            || Moon           || Earth           ||   units   ||
|| $ \mu $    || gravitational parameter || 3.98600448E+14 || 3.98600448E+14 || 3.98600448E+14 || 3.98600448E+14 || 3.98600448E+14 || 1.32712440E+20  || m^3^/s^2^ ||
||            ||           relative to   || earth          || earth          || earth          || earth          || earth          || Sun             ||           ||
|| $ a_g $    || gravity                 || 8.938095E+00   || 2.437062E+00   || 1.911369E+00   || 2.242078E-01   || 2.697573E-03   || 5.930053E-03    || seconds   ||
|| $ T $      || sidereal period         || 5431.009959    || 14393.432015   || 17270.542964   || 86164.090540   || 2360584.685000 || 31558149.763546 || seconds   ||
|| $ T_s $    || synodic period          || 5445.879572    || 14400.000000   || 17820.000000   || 86400.000000   || 2551442.900000 || 31558149.763546 || seconds   ||
|| $ T/2\pi $ || Sidereal / 2pi          || 864.372081     ||  2290.785853   ||  2748.692283   || 13713.440926   ||  375698.721205 ||  5022635.529703 || seconds   ||
|| $ \omega $ || Angular velocity        || 1.15690919E-03 || 4.36531419E-04 || 3.63809367E-04 || 7.29211585E-05 || 2.66170722E-06 ||  1.99098659E-07 || rad/sec   ||
||            || orbits/year             || 5810.73318     ||   2192.53822   ||     1827.28185 ||      366.25640 ||      13.36879  ||       1.00000   ||           ||
|| $ R $      || semimajor axis          || 6678000.00     ||  12788970.60   || 14440980.32    || 42164169.86    || 384399000.00   || 149598261000.00 ||  meters   ||
|| $ R_a $    || apogee radius           || 6678000.00     ||  varies        ||   varies       || 42164169.86    || 405696000.00   || 152098232000.00 ||  meters   ||
|| $ R_p $    || perigee radius          || 6678000.00     ||  varies        ||   varies       || 42164169.86    || 363104000.00   || 147098290000.00 ||  meters   ||
|| $ e $      || eccentricity            || 0.000000       ||  varies        ||   varies       || 0.000000       ||  0.055401      ||      0.016711   ||           ||
|| $ V_0 $    || mean velocity           ||  7725.84       ||  5582.79       ||   5253.76      ||  3074.66       ||   1023.16      ||      29784.81   ||  m/s      ||
|| $ V_a $    || apogee velocity         ||  7725.84       ||  varies        ||   varies       ||  3074.66       ||   966.47       ||      29287.07   ||  m/s      ||
|| $ V_p $    || perigee velocity        ||  7725.84       ||  varies        ||   varies       ||  3074.66       ||   1079.84      ||      30282.55   ||  m/s      ||
|| $ C_3 $    || orbit specific energy   || -59688596.59   || -31167516.16   || -27602035.26   || -9453534.82    ||   -1036944.55  || -887125553.00   ||  J/kg     ||
+$ sidereal orbit time = ( 365.256... * 86400 ) / ( 365.256... * 6 + 1 ) = 86400 / ( 6 + 1/365.256... ) = 


1 year = 365.256363004 days of 86,400 seconds, or 31558149.7635456 seconds

||   symbol   ||                      || LEO 300Km      || M288           || M360           || GEO            || Moon           || Earth           ||   units   ||
|| $ \mu $    || gravitation param.   || 3.98600448E+14 || 3.98600448E+14 || 3.98600448E+14 || 3.98600448E+14 || 3.98600448E+14 || 1.32712440E+20  || m^3^/s^2^ ||
||            ||         relative to  || earth          || earth          || earth          || earth          || earth          || Sun             ||           ||
|| $ a_g $    || gravity              || 8.938095E+00   || 2.437062E+00   || 1.911369E+00   || 2.242078E-01   || 2.697573E-03   || 5.930053E-03    || seconds   ||
|| $ T $      || sidereal period      || 5431.009959    || 14393.432269   || 17270.543331   || 86164.099662   || 2360591.577436 || 31558149.763546 || seconds   ||
|| $ T_s $    || synodic period       || 5431.944772    || 14400.000000   || 17820.000000   || 86400.000000   || 2551442.900000 || 31558149.763546 || seconds   ||
|| $ T/2\pi $ || Sidereal / 2pi       || 864.372081     ||  2290.785853   ||  2748.692283   || 13713.440926   ||  375698.721205 ||  5022635.529703 || seconds   ||
|| $ \omega $ || Angular velocity     || 1.15690919E-03 || 4.36531419E-04 || 3.63809367E-04 || 7.29211585E-05 || 2.66170722E-06 ||  1.99098659E-07 || rad/sec   ||
||            || orbits/year          || 5810.73318     ||   2192.53822   ||     1827.28185 ||      366.25640 ||      13.36879  ||       1.00000   ||           ||
|| $ R $      || semimajor axis       || 6678000.00     ||  12788970.60   ||  14440980.32   ||  42164169.86   || 384399000.00   || 149598261000.00 ||  meters   ||
|| $ R_a $    || apogee radius        || 6678000.00     ||  12838970.60   ||  14490980.32   ||  42164169.86   || 405696000.00   || 152098232000.00 ||  meters   ||
|| $ R_p $    || perigee radius       || 6678000.00     ||  12738970.60   ||  14440980.32   ||  42164169.86   || 363104000.00   || 147098290000.00 ||  meters   ||
|| $ e $      || eccentricity         || 0.000000       ||  0.001951      ||  0.001728      ||  0.000000      ||  0.055401      ||      0.016711   ||           ||
|| $ V_0 $    || mean velocity        ||  7725.84       ||  5582.79       ||  5253.76       ||  3074.66       ||   1023.16      ||      29784.81   ||  m/s      ||
|| $ V_a $    || apogee velocity      ||  7725.84       ||  5571.90       ||  5244.68       ||  3074.66       ||    966.47      ||      29287.07   ||  m/s      ||
|| $ V_p $    || perigee velocity     ||  7725.84       ||  5593.68       ||  5262.84       ||  3074.66       ||   1079.84      ||      30282.55   ||  m/s      ||
|| $ C_3 $    || orb. specific energy || -59688596.59   || -31167516.16   || -27602035.26   || -9453534.82    ||   -1036944.55  || -887125553.00   ||  J/kg     ||

Diff for "NearCircularOrbits"

Near Circular Orbits

Periods of M orbits