Differences between revisions 23 and 25 (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	\large \mu = a^3 \omega^2 = G M
semimajor axis	\large a = \sqrt[3]{ \mu / \omega^2 }
angular velocity	\large \omega = \sqrt{ \mu / a^3 }
sidereal period	\large T = 2\pi / \omega = 2\pi \sqrt{ a^3/\mu }
eccentricity	\large e = ( r_a - r_p ) / ( r_a + r_p )
velocity	\large v_0 = \sqrt{ \mu / ( a (1 - e^2) ) }
true anomaly orbit angle from focus	\Large \theta
eccentric anomaly ellipse center angle	\large E = \Large { { e+\cos(\theta) } \over { 1+e\cos(\theta)}}		periapsis	apoapsis
mean anomaly,	\large M = E - e \sin( E )	earth	perigee	apogee
time from perigee	\large t = M / \omega	sun	perihelion	apohelion
radius	\large r=a\Large{{1-e^2}\over{1+e\cos(\theta)}}	\huge\rightarrow	\large r_p =( 1-e )a	\large r_a =( 1+e )a
perpendicular velocity	\large v_{\perp}= v_0 ( 1+e \cos( \theta ))	\huge\rightarrow	\large v_p =(1+e)v_0	\large v_a =(1-e)v_0
radial velocity	\large v_r = e v_0 \sin( \theta )
total velocity tangent to orbit	\Large v=\LARGE \sqrt{{{2\mu}\over{r}}-{{\mu}\over{a}}}
orbit energy parameter	\large C_3 = \mu / a

Fictional Forces in Orbit

In the rotating frame of a circular orbit, counterclockwise viewed from above the orbital plane, the directions are

direction	unit vector	description
x	\vec i	Tangential to (along the line of) the orbit, in the orbital plane, pointing clockwise or westward
y	\vec j	Radially outwards from the center of rotation, in the orbital plane
z	\vec k	Perpendicular to the orbital plane, northwards

The rotation is expressed as \vec \Omega = \omega \hat k , where \omega is the angular velocity of the orbit . The radial vector \vec r is composed of \vec r = x \vec i + y \vec j + z \vec k .

Coriolis acceleration: \Large \ddot{\vec r}_{Coriolis} = -2 \vec \Omega \times \dot { \vec r }

Centrifugal acceleration: \Large \ddot{\vec r}_{Centrifugal} = - \vec \Omega \times \vec \Omega \times \vec r

FROM HERE DOWN, WORK IN PROGRESS, NOT VERIFIED:

Centripedal (gravity) acceleration: \Large \ddot{\vec r}_{Centripedal} = \omega^2 ( 2 y \vec k - x \vec i - z \vec j )

In scalar equations:

\LARGE \ddot x = -2 \omega \dot y - \omega^2 x

\LARGE \ddot y = 2 \omega \dot x + 3 \omega^2 y

\LARGE \ddot z = -\omega^2 z

Locus of Elliptical Orbit Position in Rotating Frame

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					1.3271244E+20	m³/s²
	relative to	earth	earth	earth	earth	earth	Sun
a_g	gravity	8.938095	2.437062	1.911369E	0.02242078	0.002697573	0.005930053	s
T	sidereal period	5431.00996	14393.43227	17270.54333	86164.09966	2360591.5774	31558149.7635	s
T_s	synodic period	5431.944772	14400	17820	86400	2551442.9	31558149.7635	s
T/2\pi	Sidereal / 2pi	864.37208	2290.78585	2748.69228	13713.440926	375698.721205	5022635.5297	s
\omega	Angular velocity	1.1569092E-03	4.3653142E-04	3.6380937E-04	7.2921159E-05	2.6617072E-06	1.9909866E-07	rad/s
	orbits/year	5810.7332	2192.5382	1827.2819	366.25640	13.36879	1.00000
a	semimajor axis	6678000	12788970.60	14440980.32	42164169.86	384399000	149598261000	m
R_a	apogee radius	6678000	12838970.60	14490980.32	42164169.86	405696000	152098232000	m
R_p	perigee radius	6678000	12738970.60	14440980.32	42164169.86	363104000	147098290000	m
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	-59688597	-31167516	-27602035	-9453535	-1036945	-887125553	J/kg

-  ⇤ ← Revision 23 as of 2011-04-01 22:34:23 → 
  Size: 7503
  Editor: KeithLofstrom
  Comment:
+   ← Revision 25 as of 2011-04-02 00:08:01 → ⇥
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  Editor: KeithLofstrom
  Comment:
-Deletions are marked like this.
+Additions are marked like this.
 Line 38:
-Coriolis acceleration:              $ \Large \vec a_{Coriolis}    = -2 \vec \Omega \times \dot { \vec r } $
+Coriolis acceleration:              $ \Large \ddot{\vec r}_{Coriolis}    = -2 \vec \Omega \times \dot { \vec r } $
 Line 40:
-Centrifugal acceleration:           $ \Large \vec a_{Centrifugal} = -  \vec \Omega \times \vec \Omega \times \vec r $
+Centrifugal acceleration:           $ \Large \ddot{\vec r}_{Centrifugal} = -  \vec \Omega \times \vec \Omega \times \vec r $
 Line 42:
-Centripedal (gravity) acceleration: $ \Large \vec a_{Centripedal} =  \omega^2 ( 2 y \vec k - x \vec i - z \vec j ) $
-Line 44:
+Line 43:
-In scalar equations:
+FROM HERE DOWN, WORK IN PROGRESS, NOT VERIFIED:

Centripedal (gravity) acceleration: $ \Large \ddot{\vec r}_{Centripedal} =  \omega^2 ( 2 y \vec k - x \vec i - z \vec j ) $

In scalar equations:
-Line 75:
+Line 80:
-|| $ 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  || 2360591.57743 || 31558149.7635 || seconds   ||
|| $ T_s $    || synodic period       || 5431.944772   || 14400.000000  || 17820.000000  || 86400.000000  || 2551442.90000 || 31558149.7635 || seconds   ||
|| $ T/2\pi $ || Sidereal / 2pi       || 864.372081    ||  2290.785853  ||  2748.692283  || 13713.440926  || 375698.721205 ||  5022635.5297 || seconds   ||
|| $ \omega $ || Angular velocity     || 1.1569092E-03 || 4.3653142E-04 || 3.6380937E-04 || 7.2921159E-05 || 2.6617072E-06 || 1.9909866E-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  ||  meters   ||
|| $ R_a $    || apogee radius        || 6678000.00    ||  12838970.60  ||  14490980.32  ||  42164169.86  || 405696000.00  || 152098232000  ||  meters   ||
|| $ R_p $    || perigee radius       || 6678000.00    ||  12738970.60  ||  14440980.32  ||  42164169.86  || 363104000.00  || 147098290000  ||  meters   ||
+|| $ a_g $    || gravity              || 8.938095  || 2.437062  || 1.911369E  || 0.02242078  || 0.002697573  || 0.005930053  || s   ||
|| $ T $      || sidereal period      || 5431.00996   || 14393.43227  || 17270.54333  || 86164.09966  || 2360591.5774 || 31558149.7635 || s   ||
|| $ T_s $    || synodic period       || 5431.944772   || 14400  || 17820  || 86400  || 2551442.9 || 31558149.7635 || s   ||
|| $ T/2\pi $ || Sidereal / 2pi       || 864.37208    ||  2290.78585  ||  2748.69228  || 13713.440926  || 375698.721205 ||  5022635.5297 || s   ||
|| $ \omega $ || Angular velocity     || 1.1569092E-03 || 4.3653142E-04 || 3.6380937E-04 || 7.2921159E-05 || 2.6617072E-06 || 1.9909866E-07 || rad/s   ||
||            || orbits/year          || 5810.7332    ||   2192.5382  ||    1827.2819 ||     366.25640 ||     13.36879  ||       1.00000 ||           ||
|| $ a $      || semimajor axis       || 6678000 ||  12788970.60  ||  14440980.32  ||  42164169.86  || 384399000  || 149598261000  ||  m   ||
|| $ R_a $    || apogee radius        || 6678000 ||  12838970.60  ||  14490980.32  ||  42164169.86  || 405696000  || 152098232000  ||  m   ||
|| $ R_p $    || perigee radius       || 6678000 ||  12738970.60  ||  14440980.32  ||  42164169.86  || 363104000  || 147098290000  ||  m  ||
-Line 88:
+Line 93:
-|| $ C_3 $    || orb. specific energy || -59688596.59  || -31167516.16  || -27602035.26  || -9453534.82   ||  -1036944.55  || -887125553    ||  J/kg     ||
+|| $ C_3 $    || orb. specific energy || -59688597  || -31167516 || -27602035  || -9453535   ||  -1036945  || -887125553    ||  J/kg     ||

Diff for "NearCircularOrbits"

Near Circular Orbits

General Elliptical Orbits

Fictional Forces in Orbit

Locus of Elliptical Orbit Position in Rotating Frame

Periods of M orbits