Size: 7419
Comment:
|
Size: 13254
Comment:
|
Deletions are marked like this. | Additions are marked like this. |
Line 3: | Line 3: |
---- MORE LATER add pointers to orbit discussions |
|
Line 10: | Line 8: |
|| gravitational parameter || $ \large \mu = a^3 \omega^2 = G M $ ||<|7-3>{{attachment:Anomaly.png|Anomaly|width=400}}|| | ||<-2:> '''circular orbits''' ||<|10-3>{{attachment:Anomaly.png|Drawing from Wikipedia|width=400}}|| || gravitational parameter || $ \large \mu = a^3 \omega^2 = G M $ || |
Line 14: | Line 13: |
|| $J_2$ Speedup factor || $ \large \omega'/\omega = 1+1.5 |J_2| (R_{eq}/a)^2$ || ||<-2:> '''elliptical orbits''' || |
|
Line 17: | Line 18: |
|| 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 $||<)> sun || perihelion || apohelion || |
|| eccentric anomaly<<BR>>ellipse center angle||$\large E=\arccos\Large\left({{e+\cos(\theta)}\over{1+e\cos(\theta)}}\right)$|| || periapsis || apoapsis || || mean anomaly || $ \large M = E - e \sin( E ) $||<)> earth || perigee || apogee || || time from perigee || $ \large t = M / \omega $||<)> sun || perihelion || apohelion || |
Line 22: | Line 23: |
|| 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 $|| |
|| radial velocity || $ \large v_r = e v_0 \sin( \theta )$||<:>$\huge\rightarrow$||$ 0 $||$ 0 $|| || total velocity<<BR>>tangent to orbit|| $\Large v=\LARGE \sqrt{{{2\mu}\over{r}}-{{\mu}\over{a}}}$||<:>$\huge\rightarrow$||$\large v=v_p,~ r=r_p$||$ \large v=v_a,~ r=r_a$|| || orbit energy parameter || $ \large C_3 = \mu / a = v_p v_a $||<-3:> $\large \mu = a v_p v_a ~ ~ ~ ~ ~ ~ ~ ~ ~ 2 a = r_p + r_a $|| || angular momentum || $ \large L =\Large \sqrt{ { 2 \mu ~ r_p ~ r_a } \over { r_p + r_a } } $ || || $ \large v_p = \Large \sqrt{ { 2 \mu ~ r_a } \over { r_p ( r_p + r_a ) } } $ || $ \large v_a = \Large \sqrt{ { 2 \mu ~ r_p } \over { r_a ( r_p + r_a ) } } $ || ---- === Changing perigee by changing apogee velocity === $ \large v_a^2 = 2 \mu ( 1/r_a - 1/(r_a + r_p) ) $ $ \LARGE { { \partial v_a } \over { \partial r_p } } = { \mu \over { v_a ( r_a + r_p )^2 } } $ for nearly circular orbits $ \LARGE { { \partial v } \over { \partial r } } \approx { v \over { 4 r } } $ |
Line 29: | Line 40: |
In the rotating frame of a circular orbit, counterclockwise viewed from above the orbital plane, the directions are | In the rotating frame of a circular orbit, counterclockwise viewed from above the orbital plane, the directions are |
Line 35: | Line 46: |
The rotation is expressed as $ \vec \Omega = \omega \hat k $, where $ \omega $ is the angular velocity of the orbit . | The rotation is expressed as $ \vec \Omega = \omega \vec k $, where $ \omega $ is the angular velocity of the orbit . |
Line 38: | Line 49: |
Coriolis acceleration: $ \Large \ddot{\vec r}_{Coriolis} = -2 \vec \Omega \times \dot { \vec r } $ | Coriolis acceleration: $ \large \ddot{\vec r}_{Coriolis} = -2 \vec \Omega \times \dot { \vec r } $ |
Line 40: | Line 51: |
Centrifugal acceleration: $ \Large \ddot{\vec r}_{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 44: | Line 54: |
. note - these are probably the Hill-Clohessy-Wiltshire equations | |
Line 45: | Line 56: |
Centripedal (gravity) acceleration: $ \Large \ddot{\vec r}_{Centripedal} = \omega^2 ( 2 y \vec k - x \vec i - z \vec j ) $ | Centripedal (gravity) acceleration: $ \large \ddot{\vec r}_{Centripedal} = \omega^2 ( 2 y \vec k - x \vec i - z \vec j ) $ |
Line 47: | Line 58: |
In scalar equations: | Scalar equations: |
Line 49: | Line 60: |
$ \large \ddot x = 2 \omega \dot y $ | |
Line 51: | Line 62: |
$ \LARGE \ddot x = -2 \omega \dot y - \omega^2 x $ | $ \large \ddot y = -2 \omega \dot x + 3 \omega^2 y $ |
Line 53: | Line 64: |
$ \LARGE \ddot y = 2 \omega \dot x + 3 \omega^2 y $ $ \LARGE \ddot z = -\omega^2 z $ |
$ \large \ddot z = -\omega^2 z $ |
Line 60: | Line 69: |
For small eccentricities e, the deviation of the position of an object in a nearly circular eccentric orbit deviates from a ideal circular orbit as an ellipse, with a width of 4e and a height of 2e. This is difficult to compute analytically, but easy to show numerically. Here is a plot the normalized loci of the x,y position in the rotating frame, for various values of eccentricity: | |
Line 61: | Line 71: |
{{attachment:locus01.png}} [[attachment:locus01.pl | perl data generator]] [[attachment:locus01.dat | locus data]] [[attachment:locus01.gp | gnuplot command file]] If the whole orbit rotates east, clockwise when viewed from the north, then the object follows the locus above clockwise, with $ \theta = 0 $ at the bottom of the ellipsoid. The figure is very close to an ellipse for small e, but not exactly so. The program computes the radial distance rr, and the radial position xr and yr as a function of $\theta$. The program then computes the eccentric anomaly E, and the mean anomaly M. It then rotates the position vector back along the orbit, to approximately vertical, then subtracts the unit vector representing the elliptical orbit. So, it is approximately like watching an object in an elliptical orbit from a position in a circular orbit with the same semimajor axis. The deviation from an ellipse is quite small, until the eccentricity gets larger than 0.01 or so. Server sky orbits will have eccentricities of less than 0.002 . In the rotating frame, these eccentric loci describe the path of an object subject to fictitious forces. This provides an additional check on our equations for fictitious forces. ---- ==== The math ==== $\large \dot{ \theta } = v_{\perp} / r = v_0 ( 1+e \cos( \theta )) / ( a (1-e^2) / {1+e\cos(\theta)} $ $\large \dot{ \theta } = ( \sqrt{ \mu / ( a (1 - e^2) ) } / a ( 1-e^2 ) ) ( 1 + e\cos( \theta ))^2 $ $\large \dot{ \theta } = ( \sqrt{ \mu / ( a^3 (1 - e^2)^3 ) } ( 1 + 2 e \cos( \theta ) + e^2 \cos( \theta )^2 ) $ for small perturbations, $ e^2 \approx 0 $, so $\large \dot{ \theta } \approx ( \sqrt{ \mu / a^3 } ( 1 + 2 e \cos( \theta ) ) $ In the rotating frame, $ \dot{ x } \approx a ( \dot{ \theta } - \omega ) $ where $ \omega \equiv \sqrt{ \mu / a^3 } $, so $\large \dot{ x } \approx a \sqrt{ \mu / a^3 } ( ( 1 + 2 e \cos( \theta ) - 1 ) $ $\large \dot{ x } \approx 2 e a \omega ( \cos( \theta ) ) $ Approximating $ \theta \approx \omega t $, we can integrate to get: $\Large x \approx -2 e a \sin( \theta ) $ ---- $\large \dot{ y } = v_r = e v_0 \sin( \theta ) \approx e a \omega \sin( \omega t ) $ Integrating: $\Large y \approx -e a \cos( \theta ) $ For small $ e $, these equations in $ y $ and $ x $ describe the locus calculated above. ---- ==== Testing the Equations for Acceleration ==== $ \large \ddot x ~ ? = ~ \large 2 \omega \dot y $ $ \large \ddot x = { { d^2 } \over { dt^2 } } ( -2 e a \sin( \theta ) ) = { d \over { dt } } ( -2 e a \omega \cos( \theta ) ) = \Large 2 e a \omega^2 \sin ( \theta ) $ $ \large 2 \omega \dot y = 2 \omega { d \over { dt } } -e a \cos( \theta ) = \Large 2 e a \omega^2 \sin ( \theta ) $ . . . equal! $ \large \ddot y ~ ? = ~ \large -2 \omega \dot x + 3 \omega^2 y $ $ \large \ddot y = { { d^2 } \over { dt^2 } } ( -e a \cos( \theta ) ) = { d \over { dt } } ( e a \omega \sin( \theta ) ) = \Large e a \omega^2 \cos( \theta ) $ $ \large -2 \omega \dot x + 3 \omega^2 y = -2 \omega { d \over { dt } }( -2 e a \sin( \theta ) + 3 \omega^2 ( - e a \cos( \theta ) ) ) $ $ \large ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ = 4 e a \omega^2 \cos( \theta ) - 3 e a \omega^2 \cos( \theta ) = \Large e a \omega^2 \cos( \theta ) $ . . . equal! The acceleration equations work! |
|
Line 67: | Line 140: |
Sidereal orbit time = ( 365.256... * 86400 ) / ( 365.256... * 6 + 1 ) = 86400 / ( 6 + 1/365.256... ) = 14393.43227 seconds | M288 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. J2 speedup fraction $ = - 3 J_2 ( a_E / a )^2 = 3 \times 1.082626683e-3 \times (6378107m)^2 / a^2 = 1.321245688e11m^2 / a^2 $ |
Line 70: | Line 147: |
1 year = 365.256363004 days of 86,400 seconds, or 31558149.7635456 seconds | === A Table of orbits === || || || LEO 300Km || M288 || M360 || GEO || Moon || Earth || units || || $ \mu $ || gravitation param. ||<:-5> 3.98600448e14 || 1.3271244e20 || m^3^/s^2^|| || ||<)> relative to ||<:-5> earth || Sun || || || $ J_2 $ || Oblate-ness ||<:-5> -1.082626683e-3 earth radius = 6378107m || -6e-7 || || ||$\omega'/\omega$||J2 speedup fraction|| 2.9627e-3|| 8.0782e-4|| 6.3356e-4 || 7.4318e-5 || 8.9417e-7 || 2e-17 || || || $ a_g $ || gravity || 8.938095 || 2.437062 || 1.9113693 || 0.02242078 || 0.002697573|| 0.005930053 || s || || $ T $ || sidereal period || 5431.010 || 14393.4323 || 17270.5433 || 86164.100 || 2360591.577|| 31558149.76 || s || || $ T_s $ || synodic period || 5431.945 || 14400 || 17820 || 86400 || 2551442.9 || 31558149.76 || s || || $ T/2\pi$|| Sidereal / 2pi || 864.3721 || 2290.78585 || 2748.69228 || 13713.44093|| 375698.7212|| 5022635.5297|| s || || $ \omega$|| Angular velocity || 1.1569e-03|| 4.36531e-04|| 3.63809e-04|| 7.29212e-05|| 2.66171e-06|| 1.99099e-07 || rad/s || || || orbits/year || 5810.733 || 2192.5382 || 1827.2819 || 366.25640 || 13.36879 || 1.00000 || || || $ a $ || semimajor axis || 6678000 || 12788971 || 14440980 || 42164170 || 384399000 || 1.4959826e11 || m || || $ R_a $ || apogee radius || 6678000 || 12838976 || 14490980 || 42164170 || 405696000 || 1.5209823e11 || m || || $ R_p $ || perigee radius || 6678000 || 12738976 || 14440980 || 42164170 || 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 || |
Line 72: | Line 168: |
---- 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. |
Note: This table uses the classic formula $ \omega^2 a^3 = \mu $, and does not take into account the oblate spheroid shape of the Earth, and many other perturbations. However, with the perturbations included, and with good data from ground stations and GPS to establish positions and velocities, we really can compute these numbers to this many decimal places. So while the numbers above are actually far less accurate, they represent the precision of the measurements we will someday compute. |
Line 77: | Line 172: |
|| symbol || || 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 || |
[[ Lageos | LAGEOS 1 and LAGEOS 2 ]] are two slightly lower satellites in inclined but highly circular orbits. REFS: . http://articles.adsabs.harvard.edu/full/1981AJ.....86..912G . http://en.wikipedia.org/wiki/Rotating_reference_frame . http://en.wikipedia.org/wiki/Kepler_orbit . http://en.wikipedia.org/wiki/Kepler%27s_laws_of_planetary_motion . http://en.wikipedia.org/wiki/Orbital_period . http://en.wikipedia.org/wiki/Mean_anomaly . http://en.wikipedia.org/wiki/Eccentric_anomaly ... where I got the picture, then modified it |
Near Circular Orbits
General Elliptical Orbits
In the orbital plane, neglecting the J_2 spherical oblateness parameter :
circular orbits |
||||
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 } |
|||
J_2 Speedup factor |
\large \omega'/\omega = 1+1.5 |J_2| (R_{eq}/a)^2 |
|||
elliptical orbits |
||||
eccentricity |
\large e = ( r_a - r_p ) / ( r_a + r_p ) |
|||
velocity |
\large v_0 = \sqrt{ \mu / ( a (1 - e^2) ) } |
|||
true anomaly |
\Large \theta |
|||
eccentric anomaly |
\large E=\arccos\Large\left({{e+\cos(\theta)}\over{1+e\cos(\theta)}}\right) |
|
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 ) |
\huge\rightarrow |
0 |
0 |
total velocity |
\Large v=\LARGE \sqrt{{{2\mu}\over{r}}-{{\mu}\over{a}}} |
\huge\rightarrow |
\large v=v_p,~ r=r_p |
\large v=v_a,~ r=r_a |
orbit energy parameter |
\large C_3 = \mu / a = v_p v_a |
\large \mu = a v_p v_a ~ ~ ~ ~ ~ ~ ~ ~ ~ 2 a = r_p + r_a |
||
angular momentum |
\large L =\Large \sqrt{ { 2 \mu ~ r_p ~ r_a } \over { r_p + r_a } } |
|
\large v_p = \Large \sqrt{ { 2 \mu ~ r_a } \over { r_p ( r_p + r_a ) } } |
\large v_a = \Large \sqrt{ { 2 \mu ~ r_p } \over { r_a ( r_p + r_a ) } } |
Changing perigee by changing apogee velocity
\large v_a^2 = 2 \mu ( 1/r_a - 1/(r_a + r_p) )
\LARGE { { \partial v_a } \over { \partial r_p } } = { \mu \over { v_a ( r_a + r_p )^2 } }
for nearly circular orbits \LARGE { { \partial v } \over { \partial r } } \approx { v \over { 4 r } }
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 \vec 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:
- note - these are probably the Hill-Clohessy-Wiltshire equations
Centripedal (gravity) acceleration: \large \ddot{\vec r}_{Centripedal} = \omega^2 ( 2 y \vec k - x \vec i - z \vec j )
Scalar equations:
\large \ddot x = 2 \omega \dot y
\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
For small eccentricities e, the deviation of the position of an object in a nearly circular eccentric orbit deviates from a ideal circular orbit as an ellipse, with a width of 4e and a height of 2e. This is difficult to compute analytically, but easy to show numerically. Here is a plot the normalized loci of the x,y position in the rotating frame, for various values of eccentricity:
perl data generator locus data gnuplot command file
If the whole orbit rotates east, clockwise when viewed from the north, then the object follows the locus above clockwise, with \theta = 0 at the bottom of the ellipsoid.
The figure is very close to an ellipse for small e, but not exactly so. The program computes the radial distance rr, and the radial position xr and yr as a function of \theta. The program then computes the eccentric anomaly E, and the mean anomaly M. It then rotates the position vector back along the orbit, to approximately vertical, then subtracts the unit vector representing the elliptical orbit. So, it is approximately like watching an object in an elliptical orbit from a position in a circular orbit with the same semimajor axis. The deviation from an ellipse is quite small, until the eccentricity gets larger than 0.01 or so. Server sky orbits will have eccentricities of less than 0.002 .
In the rotating frame, these eccentric loci describe the path of an object subject to fictitious forces. This provides an additional check on our equations for fictitious forces.
The math
\large \dot{ \theta } = v_{\perp} / r = v_0 ( 1+e \cos( \theta )) / ( a (1-e^2) / {1+e\cos(\theta)}
\large \dot{ \theta } = ( \sqrt{ \mu / ( a (1 - e^2) ) } / a ( 1-e^2 ) ) ( 1 + e\cos( \theta ))^2
\large \dot{ \theta } = ( \sqrt{ \mu / ( a^3 (1 - e^2)^3 ) } ( 1 + 2 e \cos( \theta ) + e^2 \cos( \theta )^2 )
for small perturbations, e^2 \approx 0 , so
\large \dot{ \theta } \approx ( \sqrt{ \mu / a^3 } ( 1 + 2 e \cos( \theta ) )
In the rotating frame, \dot{ x } \approx a ( \dot{ \theta } - \omega ) where \omega \equiv \sqrt{ \mu / a^3 } , so
\large \dot{ x } \approx a \sqrt{ \mu / a^3 } ( ( 1 + 2 e \cos( \theta ) - 1 )
\large \dot{ x } \approx 2 e a \omega ( \cos( \theta ) )
Approximating \theta \approx \omega t , we can integrate to get:
\Large x \approx -2 e a \sin( \theta )
\large \dot{ y } = v_r = e v_0 \sin( \theta ) \approx e a \omega \sin( \omega t ) Integrating:
\Large y \approx -e a \cos( \theta )
For small e , these equations in y and x describe the locus calculated above.
Testing the Equations for Acceleration
\large \ddot x ~ ? = ~ \large 2 \omega \dot y
\large \ddot x = { { d^2 } \over { dt^2 } } ( -2 e a \sin( \theta ) ) = { d \over { dt } } ( -2 e a \omega \cos( \theta ) ) = \Large 2 e a \omega^2 \sin ( \theta )
\large 2 \omega \dot y = 2 \omega { d \over { dt } } -e a \cos( \theta ) = \Large 2 e a \omega^2 \sin ( \theta ) . . . equal!
\large \ddot y ~ ? = ~ \large -2 \omega \dot x + 3 \omega^2 y
\large \ddot y = { { d^2 } \over { dt^2 } } ( -e a \cos( \theta ) ) = { d \over { dt } } ( e a \omega \sin( \theta ) ) = \Large e a \omega^2 \cos( \theta )
\large -2 \omega \dot x + 3 \omega^2 y = -2 \omega { d \over { dt } }( -2 e a \sin( \theta ) + 3 \omega^2 ( - e a \cos( \theta ) ) )
\large ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ = 4 e a \omega^2 \cos( \theta ) - 3 e a \omega^2 \cos( \theta ) = \Large e a \omega^2 \cos( \theta ) . . . equal!
The acceleration equations work!
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
M288 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.
J2 speedup fraction = - 3 J_2 ( a_E / a )^2 = 3 \times 1.082626683e-3 \times (6378107m)^2 / a^2 = 1.321245688e11m^2 / a^2
A Table of orbits
|
|
LEO 300Km |
M288 |
M360 |
GEO |
Moon |
Earth |
units |
\mu |
gravitation param. |
3.98600448e14 |
1.3271244e20 |
m3/s2 |
||||
|
relative to |
earth |
Sun |
|
||||
J_2 |
Oblate-ness |
-1.082626683e-3 earth radius = 6378107m |
-6e-7 |
|
||||
\omega'/\omega |
J2 speedup fraction |
2.9627e-3 |
8.0782e-4 |
6.3356e-4 |
7.4318e-5 |
8.9417e-7 |
2e-17 |
|
a_g |
gravity |
8.938095 |
2.437062 |
1.9113693 |
0.02242078 |
0.002697573 |
0.005930053 |
s |
T |
sidereal period |
5431.010 |
14393.4323 |
17270.5433 |
86164.100 |
2360591.577 |
31558149.76 |
s |
T_s |
synodic period |
5431.945 |
14400 |
17820 |
86400 |
2551442.9 |
31558149.76 |
s |
T/2\pi |
Sidereal / 2pi |
864.3721 |
2290.78585 |
2748.69228 |
13713.44093 |
375698.7212 |
5022635.5297 |
s |
\omega |
Angular velocity |
1.1569e-03 |
4.36531e-04 |
3.63809e-04 |
7.29212e-05 |
2.66171e-06 |
1.99099e-07 |
rad/s |
|
orbits/year |
5810.733 |
2192.5382 |
1827.2819 |
366.25640 |
13.36879 |
1.00000 |
|
a |
semimajor axis |
6678000 |
12788971 |
14440980 |
42164170 |
384399000 |
1.4959826e11 |
m |
R_a |
apogee radius |
6678000 |
12838976 |
14490980 |
42164170 |
405696000 |
1.5209823e11 |
m |
R_p |
perigee radius |
6678000 |
12738976 |
14440980 |
42164170 |
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 |
Note: This table uses the classic formula \omega^2 a^3 = \mu , and does not take into account the oblate spheroid shape of the Earth, and many other perturbations. However, with the perturbations included, and with good data from ground stations and GPS to establish positions and velocities, we really can compute these numbers to this many decimal places. So while the numbers above are actually far less accurate, they represent the precision of the measurements we will someday compute.
The eccentricities for the M orbits assume orbits mapped onto a 50km minor radius toroid.
LAGEOS 1 and LAGEOS 2 are two slightly lower satellites in inclined but highly circular orbits.
REFS:
http://en.wikipedia.org/wiki/Kepler%27s_laws_of_planetary_motion
http://en.wikipedia.org/wiki/Eccentric_anomaly ... where I got the picture, then modified it