The ratio of the orbital radii of the planets is 4:1.
The ratio of their periods is 8:1.
Planetary bodies circle the Sun according to Kepler's three laws. They explain how planets orbit the Sun in elliptical fashion, how they traverse the same amount of space in a given length of time regardless of where they are in their orbit, and how their orbital periods are related to the size of their orbits (its semi-major axis).
An object's orbital radius is the typical distance it travels around a bigger object. An illustration would be that the Sun and Earth are typically 150 million kilometres apart. The orbital radius is also this.
Let the mass of the planet be m and the mass of the star be M.
r = radius of ordit
v = speed
We know that
gravitational attraction = centripital force
GMm/r^2 = mv^2/r
[tex]v = \sqrt{GM/r}[/tex]
Hence
v ∝ 1/[tex]\sqrt{r}[/tex]
a) Here v₁/v₂ = v/2v = [tex]\sqrt{\frac{r2}{r1\\\ }[/tex]
squuaring both sides
1/4 = r2/r1
Thus r1/r2 = 4/1
Hence ratio of orbital radii is 4:1.
b) As per keplers law of planetary motion,
T^2 ∝ r^3
[tex]\frac{T_{v} ^{2} }{T_{2v} ^{2} } = \frac{r_{v} ^{3} }{r_{2v} ^{3} }[/tex]
[tex]\frac{r_{v} ^{3} }{r_{2v} ^{3} } = 4^{3} = 64[/tex]
[tex]\frac{T_{v} }{T_{2v}} = \sqrt{64} = 8[/tex]
Thus ratio of time periods is 8:1.
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