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## Q. 6.12

Suppose in the previous example that part of the plane change, Δi, takes place at B,the perigee of the Hohmann transfer ellipse, and the remainder, $28^{º}$ − Δi, occurs at the apogee C. What is the value of Δi which results in the minimum $Δv_{total}$?

## Verified Solution

We found in Example 6.11 that if Δi = 0, then $Δv_{total}= 5.3803 km/s$,whereas $Δi = 28^{◦}$ made $Δv_{total}= 7.6307 km/s$. Here we are to determine if there is a value of Δi between $0^{◦} and 28^{◦}$ that yields a $Δv_{total}$ which is smaller than either of those two.
In this case a plane change occurs at both B and C. Recall that the most efficient strategy is to combine the plane change with the speed change, so that the delta-vs at those points are (Equation 6.21)

$\Delta v=\sqrt{v^{2}_{1}+v^{2}_{2}-2v_{1}v_{2}\cosδ}$                        (6.21)

$\Delta v_{B}=\sqrt{v^{2}_{B_{1}} +v^{2}_{B_{2}} -2v_{B_{1}}v_{B_{2}}cos\Delta i}$ $=\sqrt{7.7258^{2} + 10.152^{2} − 2 · 7.7258 · 10.152 · \cos\Delta i}$ $=\sqrt{162.74 − 156.86 \cos\Delta i}$

and

$\Delta v_{C}=\sqrt{v^{2}_{C_{2}} +v^{2}_{C_{3}} -2v_{C_{2}}v_{C_{3}}cos (28^{◦}-\Delta i)}$$=\sqrt{1.6078^{2} + 3.0747^{2} − 2 · 1.6078 · 3.0747 · \cos(28^{◦}-\Delta i)}$

$=\sqrt{12.039 − 9.8871\cos(28^{◦}-\Delta i)}$

Thus,

$\Delta v_{total }=\Delta v_{B}+\Delta v_{C}$

$=\sqrt{162.74 − 156.86\cos \Delta i}+\sqrt{12.039 − 9.8871\cos (28^{◦}-\Delta i)}$                    (a)

To determine if there is a Δi which minimizes $Δv_{total}$ , we take its derivative with respect to Δi and set it equal to zero:

$\frac{d\Delta v_{total }}{d\Delta i}=\frac{78.43\sin \Delta i}{\sqrt{162.74-156.86\cos \Delta i} }-\frac{4.9435\sin (28^{◦}-\Delta i)}{\sqrt{12.039-9.8871\cos (28^{◦}-\Delta i)} }=0$

This is a transcendental equation which must be solved iteratively. The solution, as the reader may verify, is

$\Delta i= 2.1751^{◦}$                   (b)

That is, an inclination change of $2.1751^{◦}$ should occur in low-earth orbit, while the rest of the plane change, $25.825^{◦}$, is done at GEO. Substituting (b) into (a) yields

$\Delta v_{vtotal }= 4.2207 km/s$

This is 21 percent less than the smallest $Δv_{total}$computed in Example 6.11.