(III) The comet Hale–Bopp has an orbital period of 2400 years. (a) What is its mean distance from the Sun? (b) At its closest approach, the comet is about 1.0 AU from the Sun (1 AU = distance from Earth to the Sun). What is the farthest distance? (c) What is the ratio of the speed at the closest point to the speed at the farthest point.

The mean distance of comet from the sun is always is equal to the average of closest and farthest distance of comet.

Part (a)

The mean distance of comet can be calculated as,

\(\begin{aligned}{r_{{\rm{comet}}}} &= {r_{{\rm{Earth}}}}{\left( {\frac{{{T_{{\rm{comet}}}}}}{{{T_{{\rm{Earth}}}}}}} \right)^{\frac{2}{3}}}\\{r_{{\rm{comet}}}} &= 1\;{\rm{AU}}{\left( {\frac{{2400\;{\rm{years}}}}{{1\;{\rm{years}}}}} \right)^{\frac{2}{3}}}\\{r_{{\rm{comet}}}} &= 179.25\;{\rm{AU}}\end{aligned}\)

Thus, the mean distance from the sun is \(179.25\;{\rm{AU}}\).

Part (b)

The comet at its closest is \({r_{\min }} = 1\;{\rm{AU}}\).

The mean distance of comet is the average of closest and farthest distance.

\(\begin{aligned}{r_{{\rm{comet}}}} &= \frac{{{r_{\min }} + {r_{\max }}}}{2}\\179.25\;{\rm{AU}} &= \frac{{1\;{\rm{AU}} + {r_{\max }}}}{2}\\{r_{\max }} &= 357.5\;{\rm{AU}}\end{aligned}\)

Thus, the farthest distance of comet is \(357.5\;{\rm{AU}}\).

Part (c)

The ratio of speed at closest to the farthest point can be calculated as,

\(\begin{aligned}\frac{{{v_{\max }}}}{{{v_{\min }}}} &= \frac{{{r_{\max }}}}{{{r_{\min }}}}\\\frac{{{v_{\max }}}}{{{v_{\min }}}} &= \frac{{357.5\;{\rm{AU}}}}{{1\;{\rm{AU}}}}\\\frac{{{v_{\max }}}}{{{v_{\min }}}} &= 357.5\;{\rm{AU}}\end{aligned}\)

Thus, the ratio of speed is \(357.5\;{\rm{AU}}\).