Differences between revisions 5 and 7 (spanning 2 versions)

Near Circular Orbits

MORE LATER add pointers to orbit discussions

General Elliptical Orbits

In the orbital plane, neglecting the J_2 spherical oblateness parameter :

gravitational parameter	\mu = a^3 \omega^2 = G M
semimajor axis	a = \sqrt[3]{ \mu / \omega^2 }
angular velocity	\omega = \sqrt{ \mu / a^3 }
period	2 \pi / \omega = 2 \pi \sqrt{ a^3/ \mu }
velocity	v_0 \sqrt{ \mu / a (1 - e^2) }
eccentricity	e = ( r_a - r_p ) / ( r_a + r_p )
true anomaly, orbit angle from focus	\theta
eccentric anomaly, ellipse center angle	E =( e+\cos(\theta) )/( 1+e\cos(\theta))
mean anomaly,	M = E - e \sin( E )
time from perigee	t = M / \omega
radius	r = (1-e^2) a / ( 1 + e \cos( \theta ) )	r_p = ( 1 - e ) a	r_a = ( 1 + e ) a
perpendicular velocity	v_{\perp} = v_0 ( 1 + e \cos( \theta ) )	v_p = ( 1 + e ) v_0	v_a = ( 1 - e ) v_0
radial velocity	v_r = e v_0 \sin( \theta )	periapsis ( general )	apoapsis ( general )
total velocity ( tangent to orbit )	v = \sqrt{ \mu \left( 2/r - 1/a \right)}	perigee (earth)	apogee (earth )
orbit energy parameter	C_3 = \mu / a	perihelion ( sun )	apohelion ( sun )

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... ) = 14393.43227 seconds

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

Note: The following table uses the classic formula \omega^2 a^3 = \mu from the books, and does not take into account the oblate spheroid shape of the Earth, which adds centripedal force for low orbits and thus should increase \omega and | C_3 | . So please check these numbers.

The eccentricities for the M orbits assume orbits mapped onto a 50km minor radius toroid.

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
a	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 5 as of 2011-04-01 02:17:56 → 
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  Editor: KeithLofstrom
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+   ← Revision 7 as of 2011-04-01 05:22:12 → ⇥
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  Editor: KeithLofstrom
  Comment:
-Deletions are marked like this.
+Additions are marked like this.
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+----
MORE LATER add pointers to orbit discussions
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== General Elliptical Orbits ==
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-MORE LATER add pointers to orbit discussions
+In the orbital plane, neglecting the $ J_2 $ spherical oblateness parameter :

|| gravitational parameter                 || $ \mu = a^3 \omega^2 = G M                 $ ||
|| semimajor axis                          || $ a   = \sqrt[3]{ \mu / \omega^2 }         $ ||
|| angular velocity                        || $ \omega = \sqrt{ \mu / a^3 }              $ ||
|| period                                  || $ 2 \pi / \omega = 2 \pi \sqrt{ a^3/ \mu } $ ||
|| velocity                                || $ v_0  \sqrt{ \mu / a (1 - e^2) }          $ ||
|| eccentricity                            || $ e = ( r_a - r_p ) / ( r_a + r_p )        $ ||
|| true anomaly, orbit angle from focus    || $ \theta                                   $ ||
|| eccentric anomaly, ellipse center angle || $ E =( e+\cos(\theta) )/( 1+e\cos(\theta)) $ ||
|| mean anomaly,                           || $ M = E - e \sin( E )                      $ ||
|| time from perigee                       || $ t = M / \omega                           $ ||
|| radius                                  || $ r = (1-e^2) a / ( 1 + e \cos( \theta ) ) $ || $ r_p = ( 1 - e ) a   $ || $ r_a = ( 1 + e ) a   $ ||
|| perpendicular velocity                  || $ v_{\perp} = v_0 ( 1 + e \cos( \theta ) ) $ || $ v_p = ( 1 + e ) v_0 $ || $ v_a = ( 1 - e ) v_0 $ ||
|| radial velocity                         || $ v_r = e v_0 \sin( \theta )               $ ||  periapsis ( general )  ||   apoapsis ( general )  ||
|| total velocity ( tangent to orbit )     || $ v = \sqrt{ \mu \left( 2/r - 1/a \right)} $ ||  perigee (earth)        ||   apogee (earth )       ||
|| orbit energy parameter                  || $ C_3 = \mu / a                            $ ||  perihelion ( sun )     ||   apohelion ( sun )     ||

----
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-$ r = (1-e^2) a

-----
+----
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+Line 33:
 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:
+Line 35:
-$ sidereal orbit time = ( 365.256... * 86400 ) / ( 365.256... * 6 + 1 ) = 86400 / ( 6 + 1/365.256... ) =
+Sidereal orbit time = ( 365.256... * 86400 ) / ( 365.256... * 6 + 1 ) = 86400 / ( 6 + 1/365.256... ) = 14393.43227 seconds
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+Line 37:
+----
1 year = 365.256363004 days of 86,400 seconds, or 31558149.7635456 seconds
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+Line 40:
-year = 365.256363004 days of 86,400 seconds, or 31558149.7635456 seconds
+----
Note:  The following table uses the classic formula $ \omega^2 a^3 = \mu $ from the books, and does not take into account the oblate spheroid shape of the Earth, which adds centripedal force for low orbits and thus should increase $ \omega $ and $ | C_3 | $.  So please check these numbers.

The eccentricities for the M orbits assume orbits mapped onto a 50km minor radius toroid.
-Line 31:
+Line 54:
-|| $ R $      || semimajor axis       || 6678000.00     ||  12788970.60   ||  14440980.32   ||  42164169.86   || 384399000.00   || 149598261000.00 ||  meters   ||
+|| $ a $      || semimajor axis       || 6678000.00     ||  12788970.60   ||  14440980.32   ||  42164169.86   || 384399000.00   || 149598261000.00 ||  meters   ||

Diff for "NearCircularOrbits"

Near Circular Orbits

General Elliptical Orbits

Periods of M orbits