Question

A comet (assumed to be in an elliptical orbit around the sun) is at a distance of $0.4\, AU$ from the sun at the perihelion. If the time period of the came t is $125$ yr. What is the aphelion distance?
($AU$ : Astronomical Unit)

• A. 50 AU
• B. 25 AU
• C. 49.6 AU
• D. 24.6 AU

Answer 1

Answer: (C)

For camet, Mean orbital radius,
$r_{1}=\frac{0.4+x}{2}AU$
Now, using $T^{2}\propto r^{3}$, we get
$\left(\frac{T_{1}}{T_{2}}\right)^{2}=\left(\frac{r_{1}}{r_{2}}\right)^{3} \ldots\left(i\right)$
Let $T_{2}$ and $r_{2}$ are time period and mean orbital radius for earth.
Then, $T_{2} = 1\, yr$ and $r_{2} = 1AU$ Also given,
$T_{1} =125 \,yr $
Substituting values in Eq. (i), we get
$\left(\frac{125}{1}\right)^{2}=\left(\frac{0.4+x}{2\left(1\right)}\right)^{3} $
$\Rightarrow \left(0.4+x\right)^{3}=\left(125\right)^{2}\left(2^{3}\right) $
$\Rightarrow \left(0.4+x\right)^{3}=\left(5^{6}\right)\left(2^{3}\right)$
$\Rightarrow 0.4+x=\left(5^{2}\right)\left(2\right)$
$ \Rightarrow x=50-0.4=49.6 \,AU$