Step 13 - The Velocity Vector

	3563.8925 km	5052.4587 km	6016.5229 km
r =	-346.24563 km	1056.2713 km	2392.5540 km
	6251.1122 km	5011.6366 km	3143.8252 km

Variable	Symbol	Value
Second Geocentric Range	r₂	7,194.4110 km
Gravitational Parameter	m	398600.44 km³/s²
Modulus of First N Row	N₁	1.4336385 x 10¹⁰ km³
Modulus of Second N Row	N₂	1,961,484.1 km²

	-7.0410564 x 10⁹ km³	-963,347.6 km²	14,100.605 km²
N =	1.1587796 x 10¹⁰ km³	1,585,426 km²	17,389.173 km²
	4.6560117 x 10⁹ km³	637,028.9 km²	-21,954.234 km²

We are nearly ready to determine the final state vector (position and velocity vectors) of the satellite. Several final vectors and terms are required. First, we need an intermediate vector that will be determined using the following equation:

Y_i = N₂₂r₃₂ - N₃₂r₂₂Y_j = N₃₂r₁₂ - N₁₂r₃₂Y_k = N₁₂r₂₂ - N₂₂r₁₂

Y_i = 7.2727036 x 10⁹ km³
Y_j = 8.0465103 x 10⁹ km³
Y_k = -9.0278558 x 10⁹ km³

Next, we need a handy value to "tidy up" the loose values determined in the previous step:

W = [ m / (N₁N₂) ]^1/2

W = 3.7649279 x 10^-6 km^-1·s^-1

Finally, the velocity vector can be found by using the following equation:

v_i = W [ Y_i / r₂ + N₁₃ ]
v_j = W [ Y_j / r₂ + N₂₃ ]
v_k = W [ Y_k / r₂ + N₃₃ ]

v_i = 3.8589872 km/s
v_j = 4.2763114 km/s
v_k = -4.8070493 km/s

Finally, the entire "state vector" satellite orbit elements are known and are stated in their entirety below. The epoch of this state vector corresponds to the second observation of the satellite.

Orbit Element	Symbol	Value
Epoch Time	t_epoch	23:05:28.22 UTC January 3, 2012
x Position	x	5052.4587 km
y Position	y	1056.2713 km
z Position	z	5011.6366 km
x Velocity	ẋ	3.8589872 km/s
y Velocity	ẏ	4.2763114 km/s
z Velocity	ż	-4.8070493 km/s

BACK TO STEP #12

PROCEED TO STEP #14

CASTOR HOME	SITE MAP	*GAUSS' METHOD*

Step 13: Velocity Vector Was Last Modified On September 23, 2013