Calculate the radius of the earth’s sphere of influence.

Question

Accepted Answer

In Table A.1 we find

[latex]m_{ ext {earth }}=5.974\left(10^{24} ight) kg[/latex]

[latex]m_{ ext {sun }}=1.989\left(10^{30} ight) kg[/latex]

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

Substituting these data into Eq. (8.34) yields

[latex]r_{ ext {soI }}=149.6 imes 10^6\left(\frac{5.974 imes 10^{24}}{1.989 imes 10^{24}} ight)^{2 / 5}=925 imes 10^6 km[/latex]

[latex]\frac{r_{ SOI }}{R}=\left(\frac{m_p}{m_s} ight)^{2 / 5}[/latex] (8.34)

Since the radius of the earth is 6378 km,

[latex]r_{ SOI }[/latex] = 145 earth radii

Relative to the earth, its sphere of influence is very large. However, relative to the sun it is tiny, as illustrated in Fig. 8.7.

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^{\alpha}[/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 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]	17.24[latex]h^{\alpha}[/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^{\alpha}[/latex]	122.5°	[latex]5.906 imes 10^{9}[/latex]	0.2488	17.16°	247.9y

Question 8.3: Calculate the radius of the earth’s sphere of influence.