Differences between revisions 6 and 22 (spanning 16 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 \vec a_{Coriolis} = -2 \vec \Omega \times \dot { \vec r }

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

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

In scalar equations:

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

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

\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	8.938095E+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
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	J/kg

-  ⇤ ← Revision 6 as of 2011-04-01 03:58:54 → 
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  Editor: KeithLofstrom
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+   ← Revision 22 as of 2011-04-01 22:33:21 → ⇥
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  Editor: KeithLofstrom
  Comment:
-Deletions are marked like this.
+Additions are marked like this.
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-|| 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 = \omega a = \sqrt{ \mu / a }        $ ||
|| 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 )     ||
+|| gravitational parameter                 || $ \large \mu = a^3 \omega^2 = G M                 $ ||<|7-3>{{attachment:Anomaly.png|Anomaly|width=400}}||
|| 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<<BR>>orbit angle from focus|| $ \Large \theta                                   $ ||
|| eccentric anomaly<<BR>>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<<BR>>tangent to orbit    || $ \Large v=\LARGE \sqrt{{{2\mu}\over{r}}-{{\mu}\over{a}}}$||
|| orbit energy parameter                  || $ \large C_3 = \mu / a                            $||
 Line 27:
+=== 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 \vec a_{Coriolis}    = -2 \vec \Omega \times \dot { \vec r } $

Centrifugal acceleration:           $ \Large \vec a_{Centrifugal} = -  \vec \Omega \times \vec \Omega \times \vec r $

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

In scalar equations:   

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

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

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

----
=== Locus of Elliptical Orbit Position in Rotating Frame ===
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+Line 58:
-== Periods of M orbits ==
+=== Periods of M orbits ===
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+Line 72:
-||   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   ||           ||
|| $ 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     ||
+||   symbol   ||                      || LEO 300Km     || M288          || M360          || GEO           || Moon          || Earth         ||   units   ||
|| $ \mu $    || gravitation param.   ||<:-5>                 3.98600448E+14                                               || 1.3271244E+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  || 2360591.57743 || 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   ||
|| $ 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    ||  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