Differences between revisions 4 and 28 (spanning 24 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.

		LEO 300Km	M288	M360	GEO	Moon	Earth	units
\mu	gravitation param.	3.98600448e14					1.3271244e20	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	14393.4323	17270.5433	86164.100	2360591.577	31558149.76	s
T_s	synodic period	5431.944772	14400	17820	86400	2551442.9	31558149.76	s
T/2\pi	Sidereal / 2pi	864.37208	2290.78585	2748.69228	13713.44093	375698.7212	5022635.5297	s
\omega	Angular velocity	1.15690e-03	4.36531e-04	3.63809e-04	7.29212e-05	2.66171e-06	1.99099e-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	1.4959826e11	m
R_a	apogee radius	6678000	12838970.60	14490980.32	42164169.86	405696000	1.5209823e11	m
R_p	perigee radius	6678000	12738970.60	14440980.32	42164169.86	363104000	1.4709829e11	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 4 as of 2011-04-01 01:40:58 → 
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+   ← Revision 28 as of 2011-04-02 00:18:55 → ⇥
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  Editor: KeithLofstrom
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-Deletions are marked like this.
+Additions are marked like this.
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+----
MORE LATER add pointers to orbit discussions
-----
== 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                 || $ \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                            $||

----
=== 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 ===
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+----
=== Periods of M orbits ===
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+Line 65:
+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 10:
+Line 67:
-$ r = (1-e^2) a
+Sidereal orbit time = ( 365.256... * 86400 ) / ( 365.256... * 6 + 1 ) = 86400 / ( 6 + 1/365.256... ) = 14393.43227 seconds
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+Line 69:
------
== Periods of M orbits ==
+----
1 year = 365.256363004 days of 86,400 seconds, or 31558149.7635456 seconds
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+Line 72:
-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
+----
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.
-Line 17:
+Line 75:
-||   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    || 14393.432015   || 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     ||  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     ||
+The eccentricities for the M orbits assume orbits mapped onto a 50km minor radius toroid.
-Line 35:
+Line 77:
+||            ||                      || LEO 300Km     || M288          || M360          || GEO           || Moon          || Earth         ||   units   ||
|| $ \mu $    || gravitation param.   ||<:-5>                 3.98600448e14                                               || 1.3271244e20 || m^3^/s^2^ ||
||            ||<)>      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   || 14393.4323  || 17270.5433  || 86164.100  || 2360591.577 || 31558149.76 || s   ||
|| $ T_s $    || synodic period       || 5431.944772   || 14400  || 17820  || 86400  || 2551442.9 || 31558149.76 || s   ||
|| $ T/2\pi $ || Sidereal / 2pi       || 864.37208    ||  2290.78585  ||  2748.69228  || 13713.44093  || 375698.7212 ||  5022635.5297 || s   ||
|| $ \omega $ || Angular velocity     || 1.15690e-03 || 4.36531e-04 || 3.63809e-04 || 7.29212e-05 || 2.66171e-06 || 1.99099e-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  || 1.4959826e11  ||  m   ||
|| $ R_a $    || apogee radius        || 6678000 ||  12838970.60  ||  14490980.32  ||  42164169.86  || 405696000  || 1.5209823e11  ||  m   ||
|| $ R_p $    || perigee radius       || 6678000 ||  12738970.60  ||  14440980.32  ||  42164169.86  || 363104000  || 1.4709829e11  ||  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     ||

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