# A satellite is being moved from circular orbit at 400.0 km altitude to one at 250.0 km altitude....

## Question:

A satellite is being moved from circular orbit at 400.0 km altitude to one at 250.0 km altitude. What is the {eq}\Delta V_2 {/eq} needed to go from the transfer orbit to the final circular orbit?

## Hohmann Transfer

The {eq}\Delta v {/eq} required for the Hohmann transfer can be computed using the formula;

{eq}{\displaystyle \Delta v_{1}={\sqrt {\frac {\mu }{r_{1}}}}\left({\sqrt {\frac {2r_{2}}{r_{1}+r_{2}}}}-1\right),} {/eq}

{eq}{\displaystyle \Delta v_{2}={\sqrt {\frac {\mu }{r_{2}}}}\left(1-{\sqrt {\frac {2r_{1}}{r_{1}+r_{2}}}}\right),} {/eq}

where {eq}{\displaystyle r_{1}} {/eq} and {eq}{\displaystyle r_{2}} {/eq} are respectively the radii of the departure and arrival circular orbits.

{eq}\mu = Gm {/eq} is the standard gravitational parameter.

Wherein, G is the universal gravitational constant and m is the mass of the earth.

## Answer and Explanation:

Initial Height {eq}r_1 = 400 \ km = 400 \times 10^3 \ m {/eq}

Final Height {eq}r_2 = 250 \ km = 250 \times 10^3 \ m {/eq}

we are looking for {eq}\Delta v_2 {/eq} then from Hohmann Transfer formula, we have,

{eq}{\displaystyle \Delta v_{2}={\sqrt {\frac {\mu }{r_{2}}}}\left(1-{\sqrt {\frac {2r_{1}}{r_{1}+r_{2}}}}\right),} \\ {\displaystyle \Delta v_{2}={\sqrt {\frac {Gm }{(250 \times 10^3 \ m)}}}\left(1-{\sqrt {\frac {2(400 \times 10^3 \ m)}{(400 \times 10^3 \ m)+(250 \times 10^3 \ m)}}}\right),} \\ wherein, \\ Gm = ( 6.67 \times 10^{-11} ) ( 5.97 \times 10^{24} ) = 39.8199 \times 10^{13} \frac {m^3}{s^2} \\ \\ {\displaystyle \Delta v_{2}={\sqrt {\frac {39.8199 \times 10^{13} \frac {m^3}{s^2} }{(250 \times 10^3 \ m)}}}\left(1-{\sqrt {\frac {2(400 \times 10^3 \ m)}{(400 \times 10^3 \ m)+(250 \times 10^3 \ m)}}}\right),} \\ {\displaystyle \Delta v_{2}= ({39,909.8484}) \left(1- 1.1094 \right),} \\ \Delta v_{2} = -4,366.15 \frac ms {/eq}

The {eq}\Delta v_2 \ is \ -4,366.15 \frac ms {/eq} which is needed to go from the transfer orbit to the final circular orbit.

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