Based on observations made during the 1996 sighting of comet Hyakutake, it was concluded that the trajectory of the comet is a highly elongated ellipse for which the eccentricity is approximately ε=0.999887. Knowing that for the 1996 sighting the minimum distance between the comet and the sun was

Question

Based on observations made during the 1996 sighting of comet Hyakutake, it was concluded that the trajectory of the comet is a highly elongated ellipse for which the eccentricity is approximately [latex] \varepsilon=0.999887[/latex]. Knowing that for the 1996 sighting the minimum distance between the comet and the sun was [latex]0.230{ R }_{ E }[/latex], where [latex]{ R }_{ E }[/latex] is the mean distance from the sun to the earth, determine the periodic time of the comet.

Accepted Answer

For Earth's orbit about the sun,

[latex] u_{0}=\sqrt{\frac{G M}{R_{E}}}, \quad au_{0}=\frac{2 \pi R_{E}}{ u_{0}}=\frac{2 \pi R_{E}^{3 / 2}}{\sqrt{G M}} \quad ext { or } \quad \sqrt{G M}=\frac{2 \pi R_{E}^{3 / 2}}{ au_{0}}[/latex] (1)

For the comet Hyakutake,

[latex]\begin{aligned}\frac{1}{r_{0}} & =\frac{G M}{h^{2}}=(1+\varepsilon), \quad \frac{1}{r_{1}}=\frac{G M}{h^{2}}(1+\varepsilon), \quad r_{1}=\frac{1+\varepsilon}{1-\varepsilon} r_{0} \a & =\frac{1}{2}\left(r_{0}+r_{1} ight)=\frac{r_{0}}{1-\varepsilon}, \quad b=\sqrt{r_{0} r_{1}}=\sqrt{\frac{1+\varepsilon}{1-\varepsilon}} r_{0} \h & =\sqrt{G M r_{0}(1+\varepsilon)} \ au & =\frac{2 \pi a b}{h}=\frac{2 \pi r_{0}^{2}(1+\varepsilon)^{1 / 2}}{(1-\varepsilon)^{3 / 2} \sqrt{G M r_{0}(1+\varepsilon)}} \& =\frac{2 \pi r_{0}^{3 / 2}}{\sqrt{G M(1-\varepsilon)^{3 / 2}}}=\frac{2 \pi r_{0}^{3 / 2} au_{0}}{2 \pi R_{E}^{3}(1-\varepsilon)^{3 / 2}} \& =\left\lgroup\frac{r_{0}}{R_{E}} ight group^{3 / 2} \frac{1}{(1-\varepsilon)^{3 / 2}} au_{0} \& =(0.230)^{3 / 2} \frac{1}{(1-0.999887)^{3 / 2}} au_{0}=91.8 imes 10^{3} au_{0}\end{aligned}[/latex]

Since [latex]\quad\quad au_{0}=1 \mathrm{\ yr}, au=\left(91.8 imes 10^{3} ight)(1.000)[/latex]

[latex] au=91.8 imes 10^{3} \mathrm{\ yr}\blacktriangleleft[/latex]

Question 12.109: Based on observations made during the 1996 sighting of comet...

This Question has been Answered.

Already have an account?

Login

Related Answered Questions