In Example 8.8, calculate the delta-v required to place the spacecraft in an elliptical capture orbit around Mars with a periapsis altitude of 300 km and a period of 48 h. Sketch the approach hyperbola.

Question

Accepted Answer

From Tables A.1 and A.2, we know that

[latex]r_{ ext {Mars }}=3380 km[/latex]

[latex]\mu_{ Mars }=42,830 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]	243[latex]d^{a}[/latex]	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 times 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]	17.24[latex]h^{a}[/latex]	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]	6.387[latex]d^{a}[/latex]	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

The radius to periapsis of the arrival hyperbola is the radius of Mars plus the periapsis of the elliptical capture orbit,

[latex]r_p[/latex] = 3380 + 300 = 3680 km

According to Eq. (8.40) and Eq. (b) of Example 8.8, the speed of the spacecraft at periapsis of the arrival hyperbola is

[latex]\left.v_p ight)_{ ext {Arrival }}=\sqrt{\left.\left[v_{\infty} ight)_{ ext {Arrival }} ight]^2+\frac{2 \mu_{ ext {Mars }}}{r_p}}=\sqrt{2.8852^2+\frac{2 \cdot 42,830}{3680}}=5.621 km / s[/latex]

To find the speed [latex]\left.v_p ight)_{ ext { ellipse }}[/latex] at periapsis of the capture ellipse, we use the required period (48 h) to determine the ellipse’s semimajor axis, using Eq. (2.83),

[latex]a_{ ext {ellipse }}=\left(\frac{T \sqrt{\mu_{ ext {Mars }}}}{2 \pi} ight)^{3 / 2}=\left(\frac{48 \cdot 3600 \cdot \sqrt{42,830}}{2 \pi} ight)^{3 / 2}=31,880 km[/latex]

From Eq. (2.73), we obtain

[latex]e_{ ext {ellipse }}=1-\frac{r_p}{a_{ ext {ellipse }}}=1-\frac{3680}{31,880}=0.8846[/latex]

Then Eq. (8.59) yields

[latex]\left.v_p ight)_{ ext {ellipse }}=\sqrt{\frac{\mu_{ ext {Mars }}}{r_p}\left(1+e_{ ext {ellipse }} ight)}=\sqrt{\frac{42,830}{3680}(1+0.8846)}=4.683 km / s[/latex]

Hence, the delta-v requirement is

[latex]\left.\left.\Delta v=v_p ight)_{ ext {Arrival }}-v_p ight)_{ ext {ellipse }}=0.9382 km / s[/latex]

The eccentricity of the approach hyperbola is given by Eq. (8.38),

[latex]e=1+\frac{\left.r_p\left(v_{\infty} ight)_{ ext {Arival }} ight)^2}{\mu_{ ext {Mars }}}=1+\frac{3680 \cdot 2.8852^2}{42,830}=1.715[/latex]

Assuming that the capture ellipse is a polar orbit of Mars, then the approach hyperbola is as illustrated in Fig. 8.30. Note that Mars’ equatorial plane is inclined 25° to the plane of its orbit around the sun. Furthermore, the vernal equinox of Mars lies at an angle of 85° from that of the earth.

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