Tuesday, June 03, 2014

Transient perhaps Hohmann Trivia

Object previously located directly above Earth's Equator (Geosynchronous Equatorial Orbit, GEO) GEO node will be compelled to locate well away from Earth's equatorial plane; although tether constraint will still enable object to remain geosynchronous.  Perhaps  Geosynchronous Sub-Orbit, GSO) node.
VOLUME 0: ELEVATIONAL
VOLUME I: ASTEROIDAL
VOLUME II: INTERPLANETARY
VOLUME III: INTERSTELLAR
zx 

Continue Semi-Orbits

Increase HohmannTransfers

SOSemimajor axisOrbit
Time		Angle
Diff		Cumm.
Angle		HTPerihelionAphelionTransfer TimeAngle
Diff		Cumm
Angle
(i)atδΔ(i)qQtδΔ
11.003 AU0.502Yr0.7°0.7°11.003 AU1.004 AU0.502Yr0.9°0.9°
21.003 AU0.502Yr0.7°1.4°21.004 AU1.005 AU0.503Yr1.2°2.0°
31.003 AU0.502Yr0.7°2.2°31.005 AU1.006 AU0.504Yr1.5°3.5°
41.003 AU0.502Yr0.7°2.9°41.006 AU1.007 AU0.505Yr1.8°5.3°
51.003 AU0.502Yr0.7°3.6°51.007 AU1.008 AU0.506Yr2.1°7.3°
61.003 AU0.502Yr0.7°4.3°61.008 AU1.009 AU0.507Yr2.4°9.7°
71.003 AU0.502Yr0.7°5.0°71.009 AU1.010 AU0.507Yr2.7°12.3°
81.003 AU0.502Yr0.7°5.8°81.010 AU1.011 AU0.508Yr2.9°15.3°
91.003 AU0.502Yr0.7°6.5°91.011 AU1.012 AU0.509Yr3.2°18.5°
101.003 AU0.502Yr0.7°7.2°101.012 AU1.014 AU0.510Yr3.5°22.0°
111.003 AU0.502Yr0.7°7.9°111.014 AU1.015 AU0.511Yr3.8°25.9°
121.003 AU0.502Yr0.7°8.6°121.015 AU1.016 AU0.511Yr4.1°30.0°
GivenAU + rL(2a)3/2
5.656		(t-.5)*360°ΣδiGiven1AU+i*rΔq + rΔ(q+Q)3/2
5.656		(t-.5)*360°Σδi
rL =384,000 km = 0.0026 AUrΔ =164,000 km = 0.0011AU
12 Semi-Orbits take 6.02 years for 8.6° lag.Estimate total of 43 yrs for 30° lag.12 Hohman Transfers can take 6.08 years for 30.0° lag.It depends on size of rΔ.
aaa
Mass of Sun and EarthSmith, Peter J. The Earth. New York: Macmillan Company, 1986.
MSol = 2 × 1030 kg
which dominates rest of Solar System is dominated by the Sun
For example, :
MSol = 343,000 * MEarth
Also, from circular motion, v = 2 π r /T, where is T is the "period" of circular motion.
Mean distance of Moon from Earth's center is 384,400 kms.

Moon/Earth/Sun Comparison
Bulk parametersMoon'>http://solarsystem.nasa.gov/planets/compchart.cfm?Object1=Moon">MoonEarth(kg)
7.35 x 1022
5.97 x 1024
2 x 1030
μ (=GM)(km3/s2)
xxx x 106
3.986 x 105
1.327 x 1011
Volume(km3)
2.20 x 1010
1.083 x 1012
1.412 x 1018
Equatorial radius(km)
1,738
6,378
est. 696,000

Polar radius(km)
1,736
6,356
working
Mean density(kg/m3)
3,350
5,515
1,408
Surface gravity(m/s2)
1.62
9.80
274.0
Escape velocitykm/s
2.38
11.2
617.7
Orbital velocitykm/s
1.023*
29.81**
n/a
* Lunar orbital speed for orbit around Earth.
**Terran orbital speed for orbit around Sun.

orbit around Earth
Orbital Dimensions
Perigee*
Semimajor axis
Apogee*
363,300 kms
384,400 kms
405,500 ms

Orbital Period
Revolution
Synodic
27.322 days
29.53 days
Orbital velocity 
Min.
Mean
Max.
0.964 km/sec
1.023 km/sec
1.076 km/sec
Inclination to ecliptic 
5.145o
Inclination to equator 
Min.
Mean
Max.
18.28o
23.47o
28.58o
Orbit eccentricity
0.055
Sidereal rotation period 
27 days 7.73 hrs
Obliquity to orbit 
6.68o
Recession rate from Earth
3.8 centimeters per year
* Apogee and perigee values are for an average orbit. Lunar orbit changes over the course of the year so the distance from the Moon to Earth ranges from roughly 357,000 km to 407,000 km.



Transits 13-24

Continue Semi-Orbits

STATIC HOHMAN TRANSFERS

SOSemi-Orbit
Time		Angle
Diff		Cumm.
Angle		HTAphelionPerihelionSemimajor axisTransfer TimeIncr. Ang. DiffCumm. Ang. Diff.
(i)tδΔ°
(i)
Q
q
a
t
δ
Δ
13 
0.502Yr
0.7°
9.4°
13 
1.016 AU
1.012 AU
1.014 AU
0.505Yrs
1.9°
32.0°
14 
0.502Yr
0.7°
10.1°
14 
1.016 AU
1.012 AU
1.014 AU
0.505Yrs
1.9°
33.9°
15 
0.502Yr
0.7°
10.8°
15 
1.016 AU
1.012 AU
1.014 AU
0.505Yrs
1.9°
35.8°
16 
0.502Yr
0.7°
11.5°
16 
1.016 AU
1.012 AU
1.014 AU
0.505Yrs
1.9°
37.7°
17 
0.502Yr
0.7°
12.2°
17 
1.016 AU
1.012 AU
1.014 AU
0.505Yrs
1.9°
39.7°
18 
0.502Yr
0.7°
13.0°
18 
1.016 AU
1.012 AU
1.014 AU
0.505Yrs
1.9°
41.6°
19 
0.502Yr
0.7°
13.7°
19 
1.016 AU
1.012 AU
1.014 AU
0.505Yrs
1.9°
43.5°
20 
0.502Yr
0.7°
14.4°
20 
1.016 AU
1.012 AU
1.014 AU
0.505Yrs
1.9°
45.4°
21 
0.502Yr
0.7°
15.1°
21 
1.016 AU
1.012 AU
1.014 AU
0.505Yrs
1.9°
47.4°
22 
0.502Yr
0.7°
15.9°
22 
1.016 AU
1.012 AU
1.014 AU
0.505Yrs
1.9°
49.3°
23 
0.502Yr
0.7°
16.6°
23 
1.016 AU
1.012 AU
1.014 AU
0.505Yrs
1.9°
51.2°
24 
0.502Yr
0.7°
17.3°
24 
1.016 AU
1.012 AU
1.014 AU
0.505Yrs
1.9°
53.2°
Given
(2a)3/2
5.656
(t-.5)*360°
year
Σδi
Given
1AU+i*rΔ
Q - rΔ
Q+q
2
(1+a)3/2
5.656
(t-.5)*360°
year
Σδi
a =1.0026 AU 
rΔ = 500,000km =0.0033AU 
24 Semi-Orbits take 12.05 years for 17.3° lag.
Estimate 43 yrs for 30° lag.
24 Hohman Transfers take 12.1 years for 53.2°lag.

Transits 25-32

Continue Semi-Orbits

DECREASING HOHMAN TRANSFERS

SOAngle
Diff		Cumm.
Angle		HTAphelionPerihelionSemimajor axisTransfer TimeIncr. Ang. DiffCumm. Ang. Diff.
(i)δΔ°

(i)
Q
q
a
t
δ
Δ
25 
0.7°
16.6°
25 
1.012 AU
1.011 AU
1.012 AU
0.504Yrs
1.6°
54.7°
26 
0.7°
17.3°
26 
1.011 AU
1.009 AU
1.010 AU
0.504Yrs
1.4°
56.1°
27 
0.7°
18.0°
27 
1.009 AU
1.008 AU
1.008 AU
0.503Yrs
1.2°
57.3°
28 
0.7°
18.7°
28 
1.008 AU
1.006 AU
1.007 AU
0.503Yrs
1.0°
58.3°
29 
0.7°
19.5°
29 
1.006 AU
1.005 AU
1.005 AU
0.502Yrs
0.8°
59.0°
30 
0.7°
20.2°
30 
1.005 AU
1.003 AU
1.004 AU
0.502Yrs
0.5°
59.6°
31 
0.7°
20.9°
31 
1.003 AU
1.001 AU
1.002 AU
0.501Yrs
0.3°
59.9°
32 
0.7°
21.6°
32 
1.001 AU
1.000 AU
1.001 AU
0.500Yrs
0.1°
60.0°
Given
(t-.5)*360°
year
Σδi
Given
1AU+i*rΔ
Q - rΔ
Q+q
2
(1+a)3/2
5.656
(t-.5)*360°
year
Σδi
t =0.502 yr per Semi-Orbit 
rΔ = 233,333km =0.0016 AU 
32 Semi-Orbits take 16.06 years for 21.6° lag.
32 Hohman Transfers take 16.2 years for 60.0°lag.
Static Circular Orbit
TimeAng. Disp
(T)α
1 yrs1.28°
2 yrs2.56°
3 yrs3.84°
……
10 yrs12.78°
20 yrs25.57°
30 yrs38.35°
40 yrs51.14°
44.4 yrs60.0°
GivenT*365.25*ω
ω =0.0035°/dy
xxxxxxx

posted by JimODell at 3:02 PM

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