Answer:
Explanation:
From the information given, by applying Kepler's 3rd law,
![T^2 \alpha a^3](https://tex.z-dn.net/?f=T%5E2%20%5Calpha%20%20a%5E3)
where;
T = period
a = semi major axis
T = 356 days (for earth)
a = 1 AU = 1.496 ![\times 10^8 \ km](https://tex.z-dn.net/?f=%5Ctimes%2010%5E8%20%5C%20km)
Therefore, ![T^2 = ca^3](https://tex.z-dn.net/?f=T%5E2%20%3D%20ca%5E3)
![c= \dfrac{365^2}{(1.496 \times 10^8)^3}](https://tex.z-dn.net/?f=c%3D%20%5Cdfrac%7B365%5E2%7D%7B%281.496%20%5Ctimes%2010%5E8%29%5E3%7D)
c = 3.9791 ![\times 10^{20} \ day^2/km^3](https://tex.z-dn.net/?f=%5Ctimes%2010%5E%7B20%7D%20%5C%20day%5E2%2Fkm%5E3)
However, if the body in the solar system has a period of 10.759.22 days, then, a =?
∴
![T^2 = ca^3](https://tex.z-dn.net/?f=T%5E2%20%3D%20ca%5E3)
![a3 = \dfrac{10759.22^2}{3.9791 \times 10^{-20}}](https://tex.z-dn.net/?f=a3%20%3D%20%5Cdfrac%7B10759.22%5E2%7D%7B3.9791%20%5Ctimes%2010%5E%7B-20%7D%7D)
![a^3 = 2.9092 \times 10^{27}](https://tex.z-dn.net/?f=a%5E3%20%3D%202.9092%20%5Ctimes%2010%5E%7B27%7D)
![a= \sqrt[3]{2.9092 \times 10^{27}}](https://tex.z-dn.net/?f=a%3D%20%5Csqrt%5B3%5D%7B2.9092%20%5Ctimes%2010%5E%7B27%7D%7D)
a = 1.4275 ![\times 10^9 \ km](https://tex.z-dn.net/?f=%5Ctimes%2010%5E9%20%5C%20km)
However, the velocity for a perihelion = 10.18 km/s
Using the formula
to calculate the radius, we have:
G = ![6.674 \times 10^{-11}](https://tex.z-dn.net/?f=6.674%20%5Ctimes%2010%5E%7B-11%7D)
M = ![1.989\times 10^{30} \ kg](https://tex.z-dn.net/?f=1.989%5Ctimes%2010%5E%7B30%7D%20%5C%20kg)
r = perihelion
![v ^2= GM ( \dfrac{2}{r}-\dfrac{1}{a})](https://tex.z-dn.net/?f=v%20%5E2%3D%20GM%20%28%20%5Cdfrac%7B2%7D%7Br%7D-%5Cdfrac%7B1%7D%7Ba%7D%29)
![(10.18 \times 10^3) ^2= 6.674 \times 10^{-11} \times 1.989 \times 10^{30} ( \dfrac{2}{r}-\dfrac{1}{1.425 \times 10^{12}})](https://tex.z-dn.net/?f=%2810.18%20%5Ctimes%2010%5E3%29%20%5E2%3D%206.674%20%5Ctimes%2010%5E%7B-11%7D%20%5Ctimes%201.989%20%5Ctimes%2010%5E%7B30%7D%20%20%28%20%5Cdfrac%7B2%7D%7Br%7D-%5Cdfrac%7B1%7D%7B1.425%20%5Ctimes%2010%5E%7B12%7D%7D%29)
![7.8068 \times 10^{-13}= \dfrac{2}{r}-\dfrac{1}{1.425 \times 10^{12}}](https://tex.z-dn.net/?f=7.8068%20%5Ctimes%2010%5E%7B-13%7D%3D%20%5Cdfrac%7B2%7D%7Br%7D-%5Cdfrac%7B1%7D%7B1.425%20%5Ctimes%2010%5E%7B12%7D%7D)
![\dfrac{2}{r} = 1.4824 \times 10^{-12}](https://tex.z-dn.net/?f=%5Cdfrac%7B2%7D%7Br%7D%20%3D%201.4824%20%5Ctimes%2010%5E%7B-12%7D)
![r = \dfrac{2}{1.4824 \times 10^{-12}}](https://tex.z-dn.net/?f=r%20%3D%20%5Cdfrac%7B2%7D%7B1.4824%20%5Ctimes%2010%5E%7B-12%7D%7D)
![r = 1.349 \times 10^{12}](https://tex.z-dn.net/?f=r%20%3D%201.349%20%5Ctimes%2010%5E%7B12%7D)
Similarly, the perihelion is expressed by the equation,
r = a(1 - e)
where ;
e= eccentricity
∴
![1.349 \times 10^{12} = 1.425 \times 10^{12} ( 1 - e)](https://tex.z-dn.net/?f=1.349%20%5Ctimes%2010%5E%7B12%7D%20%3D%201.425%20%5Ctimes%2010%5E%7B12%7D%20%28%201%20-%20e%29)
![1.349 \times 10^{12} - 1.425 \times 10^{12}= - 1.425 \times 10^{12} (e)](https://tex.z-dn.net/?f=1.349%20%5Ctimes%2010%5E%7B12%7D%20%20-%20%201.425%20%5Ctimes%2010%5E%7B12%7D%3D%20-%20%201.425%20%5Ctimes%2010%5E%7B12%7D%20%28e%29)
![-7.6\times 10^{10}= - 1.425 \times 10^{12} (e)](https://tex.z-dn.net/?f=-7.6%5Ctimes%2010%5E%7B10%7D%3D%20-%20%201.425%20%5Ctimes%2010%5E%7B12%7D%20%28e%29)
![\dfrac{-7.6\times 10^{10}}{- 1.425 \times 10^{12}}= (e)](https://tex.z-dn.net/?f=%5Cdfrac%7B-7.6%5Ctimes%2010%5E%7B10%7D%7D%7B-%20%201.425%20%5Ctimes%2010%5E%7B12%7D%7D%3D%20%20%28e%29)
e ( eccentricity) = 0.0533
Aphelion radius in natural miles, r = a( 1+ e)
![r = 1.425 \times 10^{12} ( 1 + 0.0533)](https://tex.z-dn.net/?f=r%20%3D%201.425%20%5Ctimes%2010%5E%7B12%7D%20%28%201%20%2B%200.0533%29)
![r = 1.50 \times 10^{12} \ m](https://tex.z-dn.net/?f=r%20%3D%201.50%20%5Ctimes%2010%5E%7B12%7D%20%5C%20m)
to nautical miles, we have:
![r = 1.50 \times 10^{12} \times 0.00054 \ nautical \ mile](https://tex.z-dn.net/?f=r%20%3D%201.50%20%5Ctimes%2010%5E%7B12%7D%20%5Ctimes%200.00054%20%20%5C%20nautical%20%5C%20mile)
radius of aphelion
nautical miles
In respect to the value of a( i.e ![1.4275 \times 10^9 \ km)](https://tex.z-dn.net/?f=1.4275%20%5Ctimes%20%2010%5E9%20%5C%20km%29)
the body of the solar system is Saturn