Astronautics, Spring quarter 2003, HW 4, UCI

by Nasser M. Abbasi

1 problem 3.5 from BMW book

Analysis:

Since $Vc2$ is given as $DU- 0.5TU$ , and $Vc1 = 1DU ∕TU$ , then service orbit must be the larger orbit (outside orbit), since the smaller the distance from the attracting body the larger the velosity of the orbiting body).

First find $r1$ and $r2$

Since $-- V = ∘ μ-= ⇒ r = -μ-=⇒ r = 1-= 1DU c1 r1 1 V2c1 1 12$

Similarly $μ 1 r2 = V2c2-=⇒ r2 = 0.52 = 4DU$

From geometry, for the transfer orbit, $2at = r1 + r2 =⇒ at = 5DU$

But $ξ = - -μ =⇒ ξ = - 0.2(DU-)2 t at t TU$

Now, velocity in the transfer orbit is given by $∘ --(-----) Vt = 2 μr + ξt$ hence at point 1, $r = r1 = 1$ DU, so we get $∘ -(1-----)- Vt1 = 2 1 - 0.2 =⇒ Vt1 = 1.264911DU ∕TU$

Similarly, $∘ -(------)- V = 2 μ-+ ξ t2 r2 t$ hence at point 2, $r = 4 2$ DU, so we get $∘ ---------- V = 2 (1- 0.2) =⇒ V = 0.316227 DU- t2 4 t2 TU$

So, $ΔV1 = |Vc1 - Vt1| = |1 - 1.264911 | = 0.264911$ DU/TU

$ΔV2 = |Vc2 - Vt2| = |0.5- 0.316227| = 0.183773$ DU/TU

hence, minumum $ΔV = ΔV1 + ΔV2 =$ ____________________________ $0.448684$ DU/TU

2 problem 3.8 from BMW book

Compute the minimum $ΔV$ required to transfer between 2 coplaner elliptical orbits which have their major axes aligned. The parameters are:

$rp1 = 1.1$ DU. $rp2 = 5$ DU
$e1 = 0.290$ DU. $e2 = 0.412$ DU

Assume both preigrees lie on the same side of the earth.

Assumptions:

Method: First find $V1$ and $V2$ , the velositites for ellips 1 at its perigree and for ellips 2 at its apegee.

Next find $ξt$ , the energy for the transfer orbit. From this, find $V1t$ and $V2t$ , the velosities in the transfer orbit at point 1 and point 2 respectively.