Question

A comet (assumed to be in an elliptical orbit around the sun) is at a distance of $0.4AU$ 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
${\left(\frac{T_1}{T_2}\right)}^2={\left(\frac{r_1}{r_2}\right)}^3\dots \left(i\right)$
Let $T_2$ and $r_2$ are time period and mean orbital radius for earth.
Then, $T_2=1yr$ and $r_2=1AU$ Also given,
$T_1=125yr$
Substituting values in Eq. (i), we get
${\left(\frac{125}{1}\right)}^2={\left(\frac{0.4+x}{2\left(1\right)}\right)}^3$
$\Rightarrow {\left(0.4+x\right)}^3={\left(125\right)}^2\left(2^3\right)$
$\Rightarrow {\left(0.4+x\right)}^3=\left(5^6\right)\left(2^3\right)$
$\Rightarrow 0.4+x=\left(5^2\right)\left(2\right)$
$\Rightarrow x=50-0.4=49.6AU$