A spacecraft is launched on a mission to Mars starting from a 300-km circular parking orbit. Calculate (a) the delta-v required, (b) the location of perigee of the departure hyperbola, and (c) the amount of propellant required as a percentage of the spacecraft mass before the delta-v burn, assuming

Question

A spacecraft is launched on a mission to Mars starting from a 300-km circular parking orbit. Calculate (a) the delta-v required, (b) the location of perigee of the departure hyperbola, and (c) the amount of propellant required as a percentage of the spacecraft mass before the delta-v burn, assuming a specific impulse of 300 s.

Accepted Answer

From Tables A.1 and A.2, we obtain the gravitational parameters for the sun and the earth,

[latex]\mu_{ sun }=1.327\left(10^{11} ight) km ^3 / s ^2[/latex]

[latex]\mu_{ ext {earth }}=398,600 km ^3 / s ^2[/latex]

Table A.1 Astronomical data for the sun, the planets, and the moon

Object	Radius (km)	Mass (kg)	Sidereal rotation period	Inclination of equator to orbit plane	Semimajor axis of orbit (km)	Orbit eccentricity	Inclination of orbit to the ecliptic plane	Orbit sidereal period
Sun	696000	[latex]1.989 imes 10^{30}[/latex]	25.38d	7.25°	-	-	-	-
Mercury	2440	[latex]330.2 imes 10^{21}[/latex]	58.56d	0.01°	[latex]57.91 imes 10^{6}[/latex]	0.2056	7.00°	87.97d
Venus	6052	[latex]4.869 imes 10^{24}[/latex]	a"}">243d^a	177.4°	[latex]108.2 imes 10^{6}[/latex]	0.0067	3.39°	224.7d
Earth	6378	[latex]5.974 imes 10^{24}[/latex]	23.9345h	23.45°	[latex]149.6 imes 10^{6}[/latex]	0.0167	0.00°	365.256d
(Moon)	1737	[latex]73.48 imes 10^{21}[/latex]	27.32d	6.68°	[latex]384.4 imes 10^{3}[/latex]	0.0549	5.145°	27.322d
Mars	3396	[latex]641.9 imes 10^{21}[/latex]	24.62h	25.19°	[latex]227.9 imes 10^{6}[/latex]	0.0935	1.850°	1.881y
Jupiter	71,490	[latex]1.899 imes 10^{27}[/latex]	9.925h	3.13°	[latex]778.6 imes 10^{6}[/latex]	0.0489	1.304°	11.86y
Saturn	60,270	[latex]568.5 imes 10^{24}[/latex]	10.66h	26.73°	[latex]1.433 imes 10^{9}[/latex]	0.0565	2.485°	29.46y
Uranus	25,560	[latex]86.83 imes 10^{24}[/latex]	a"}">17.24h^a	97.77°	[latex]2.872 imes 10^{9}[/latex]	0.0457	0.772°	84.01y
Neptune	24,764	[latex]102.4 imes 10^{24}[/latex]	16.11h	28.32°	[latex]4.495 imes 10^{9}[/latex]	0.0113	1.769°	164.8y
(Pluto)	1187	[latex]13.03 imes 10^{21}[/latex]	a"}">6.387d^a	122.5°	[latex]5.906 imes 10^{9}[/latex]	0.2488	17.16°	247.9y
[latex]^a[/latex]Retrograde.

Table A.2 Gravitational parameter (μ) and sphere of influence (SOI) radius for the sun, the planets, and the moon

Celestial body	μ (km³/s²)	SOI radius (km)
Sun	132,712,440,018	-
Mercury	22,032	112,000
Venus	324,859	616,000
Earth	398,600	925,000
Earth’s moon	4905	66,100
Mars	42,828	577,000
Jupiter	126,686,534	48,200,000
Saturn	37,931,187	54,800,000
Uranus	5,793,939	51,800,000
Neptune	6,836,529	86,600,000
Pluto	871	3,080,000

and the orbital radii of the earth and Mars,

[latex]R_{ ext {earth }}=149.6\left(10^6 ight) km[/latex]

[latex]R_{ ext {Mars }}=227.9\left(10^6 ight) km[/latex]

(a) According to Eq. (8.35), the hyperbolic excess speed is

[latex]v_{\infty}=\sqrt{\frac{\mu_{ ext {sun }}}{R_{ ext {earth }}}}\left(\sqrt{\frac{2 R_{ Mars }}{R_{ ext {earth }}+R_{ Mars }}}-1 ight)=\sqrt{\frac{1.327\left(10^{11} ight)}{149.6\left(10^6 ight)}}\left(\sqrt{\frac{2 \cdot 227.9\left(10^6 ight)}{149.6\left(10^6 ight)+227.9\left(10^6 ight)}-1} ight)[/latex]

from which

[latex]v_{\infty}=2.943 km / s[/latex]

The speed of the spacecraft in its 300-km circular parking orbit is given by Eq. (8.41),

[latex]v_c=\sqrt{\frac{\mu_{ ext {earth }}}{r_{ ext {earth }}+300}}=\sqrt{\frac{398,600}{6678}}=7.726 km / s[/latex]

Finally, we use Eq. (8.42) to calculate the delta-v required to step up to the departure hyperbola

[latex]\Delta v=v_c\left(\sqrt{2+\left(\frac{v_{\infty}}{v_c} ight)^2}-1 ight)=7.726\left(\sqrt{2+\left(\frac{2.943}{7.726} ight)^2}-1 ight)[/latex]

Δv = 3.590 km/s

(b) Perigee of the departure hyperbola, relative to the earth’s orbital velocity vector, is found using Eq. (8.43),

[latex]\beta=\cos ^{-1}\left(\frac{1}{1+\frac{r_p v_{\infty}{ }^2}{\mu_{ ext {earth }}}} ight)=\cos ^{-1}\left(\frac{1}{1+\frac{6678 \cdot 2.943^2}{368,600}} ight)[/latex]

β = 29.16°

Fig. 8.12 shows that the perigee can be located on either the sunlit or the dark side of the earth. It is likely that the parking orbit would be a prograde orbit (west to east), which would place the burnout point on the dark side.
(c) From Eq. (6.1), we have

[latex]\frac{\Delta m}{m}=1-\exp \left(-\frac{\Delta v}{l_{ ext {sp }} g_0} ight)[/latex]

Substituting [latex]\Delta v=3.590 km / s , I_{ sq }=300 s ext {, and } g_0=9.81\left(10^{-3} ight) km / s ^2[/latex], this yields

[latex]\frac{\Delta m}{m}=0.705[/latex]

That is, prior to the delta-v maneuver over, 70% of the spacecraft mass must be propellant.

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