Geosynchronous Orbits and Units

Sqrt[g[r, 6378140, 6.673 * 10^(-11), 5.9742 * 10^24] * r]

(r If[r<6378140, (6.673*10^-11 5.9742*10^24 r)/6378140^3, (6.673*10^-11 5.9742*10^24)/r^2])^(1/2)

vOrb[r_] := Sqrt[(6.673*10^-11 5.9742*10^24)/r^2 * r]

vOrb[6378140 + 200000]/1000 (* low orbit velocity (200km) is around 7.78 km/sec *)

7.78483

Plot[vOrb[alt + 6378140], {alt, 0, 2000000}, Frame→True, FrameLabel→ {"altitude above earth in meters", "orbital velocity in m/s"}]

[Graphics:../HTMLFiles/index_9.gif]

-Graphics -

Solve[vOrb[r] == 2 Pi r/(24 * 3600), r] (* calculate geosynchronous orbit radius *)

{{r→4.22431*10^7}}

4.22431 * 10^7 * Meter/EarthRadius (* 6.6 earth radii for radius of a geosynchronous orbit *)

6.62311

4.22431 * 10^7 - 6378140

3.5865*10^7

(3.5864 * 10^7)/6378140 (* 5.6 earth radii for height of a geosynchronous orbit ? *)

5.62296

mOrbRadius = 3.82 * 10^8 (* meters *)

3.82*10^8

(mOrbRadius * Meter)/EarthRadius(* moon ' s orbit is at almost 60 earth radii *)

59.8921

tOrb[alt_] := (2 Pi (alt + 6378140)/vOrb[alt + 6378140])

General :: spell1 : Possible spelling error: new symbol name \"tOrb\" is similar to existing symbol \"vOrb\". More…

Plot[tOrb[alt]/3600, {alt, 0, 40000000}, Frame→True] (* orbital time in hours versus altitude in meters *)

[Graphics:../HTMLFiles/index_26.gif]

-Graphics -

Compare, geosynchronous orbital velocity, moons orbital velocity around earth, our earth spin velocity, earths orbital velocity around sun

EarthRadius

6378140 Meter

N[Convert[(2 Pi EarthRadius)/(24 Hour), Meter/Second]]

(463.831 Meter)/Second

LightYear = 3 * 10^8 * 60 * 60 * 24 * 365. (* meters *)

9.4608*10^15

EarthMass

5.9742*10^24 Kilogram

In[3]:=

SolarMass = 1.991 * 10^30 * Kilogram

Out[3]=

1.991*10^30 Kilogram

In[4]:=

EarthOrbitalRadius = 1.496 * 10^11 * Meter

Out[4]=

1.496*10^11 Meter

In[5]:=

MoonOrbitalRadius = 3.84 * 10^8 * Meter

Out[5]=

3.84*10^8 Meter

In[6]:=

SunOrbitalRadius = 1.75 * 10^9 * EarthOrbitalRadius

Out[6]=

2.618*10^20 Meter

GravitationalConstant

(6.673*10^-11 Meter^2 Newton)/Kilogram^2

Convert[(GravitationalConstant * SolarMass)/EarthOrbitalRadius^(1/2), Meter/Second]

(29801. Meter)/Second

Convert[(GravitationalConstant * EarthMass)/MoonOrbitalRadius^(1/2), Meter/Second]

(1018.91 Meter)/Second

Convert[(2 * Pi * SunOrbitalRadius)/(238 * 10^6 * Year), Meter/Second]

(219162. Meter)/Second

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