(a) Express the eccentricity ε of the elliptic orbit described by a satellite about a planet in terms of the distances r0 and r1 corresponding, respectively, to the perigee and apogee of the orbit. (b) Use the result obtained in Part a and the data given in Problem 12.109, where ER = 149.6 × 10^6

Question

(a) Express the eccentricity ε of the elliptic orbit described by a satellite about a planet in terms of the distances [latex]{ r }_{ 0 }[/latex] and [latex]{ r }_{ 1 }[/latex] corresponding, respectively, to the perigee and apogee of the orbit. (b) Use the result obtained in Part a and the data given in Problem 12.109, where [latex]{ R }_{ E }=149.6 imes { 10 }^{ 6 } \ km[/latex], to determine the approximate maximum distance from the sun reached by comet Hyakutake.

Accepted Answer

(a) We have [latex]\quad \quad \quad \frac{1}{r}=\frac{G M}{h^2}(1+\varepsilon \cos heta)[/latex] Eq. (12.39′)

At A, θ = 0: [latex]\quad \quad\frac{1}{r_0}=\frac{G M}{h^2}(1+\varepsilon)[/latex]

or [latex]\quad \quad \frac{h^2}{G M}=r_0(1+\varepsilon)[/latex]

At B, θ = 180° : [latex]\quad \quad\frac{1}{r_1}=\frac{G M}{h^2}(1-\varepsilon)[/latex]

or [latex]\quad \quad \frac{h^2}{G M}=r_1(1-\varepsilon)[/latex]

Then [latex]\quad \quad r_0(1+\varepsilon)=r_1(1-\varepsilon)[/latex]

or [latex]\quad \quad \quad \quad \quad \quad \varepsilon=\frac{r_1-r_0}{r_1+r_0}\blacktriangleleft [/latex]

(b) From above, [latex]\quad \quad \quad r_1=\frac{1+ \varepsilon }{1- \varepsilon } r_0[/latex]

where [latex]\quad \quad r_0=0.230 R_E[/latex]

Then [latex]\quad \quad r_1=\frac{1+0.999887}{1-0.999887} imes 0.230\left(149.6 imes 10^9 m ight)[/latex]

or [latex]\quad \quad \quad \quad \quad \quad r_1=609 imes 10^{12} m \blacktriangleleft [/latex]

Note: [latex]r_1=4070 R_E \quad ext { or } \quad r_1=0.064 ext { lightyears. }[/latex]

Question 12.119: (a) Express the eccentricity ε of the elliptic orbit describ...

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