To calculate the gravitational force between the Sun and Jupiter, we use Newton's law of gravitation, which states:
[tex]\[
F = G \frac{m_1 m_2}{r^2}
\][/tex]
where:
- [tex]\( F \)[/tex] is the gravitational force,
- [tex]\( G \)[/tex] is the gravitational constant, [tex]\( 6.67430 \times 10^{-11} \, \text{m}^3 \text{kg}^{-1} \text{s}^{-2} \)[/tex],
- [tex]\( m_1 \)[/tex] is the mass of the Sun, [tex]\( 2 \times 10^{30} \, \text{kg} \)[/tex],
- [tex]\( m_2 \)[/tex] is the mass of Jupiter, [tex]\( 1.9 \times 10^{27} \, \text{kg} \)[/tex],
- [tex]\( r \)[/tex] is the distance between the Sun and Jupiter, [tex]\( 1.8 \times 10^8 \, \text{km} \)[/tex].
First, we must convert the distance from kilometers to meters:
[tex]\[
r = 1.8 \times 10^8 \, \text{km} = 1.8 \times 10^8 \times 10^3 \, \text{m} = 1.8 \times 10^{11} \, \text{m}
\][/tex]
Now we substitute the given values into the formula:
[tex]\[
F = 6.67430 \times 10^{-11} \, \frac{2 \times 10^{30} \times 1.9 \times 10^{27}}{(1.8 \times 10^{11})^2}
\][/tex]
Calculate the numerator first:
[tex]\[
m_1 m_2 = 2 \times 10^{30} \times 1.9 \times 10^{27} = 3.8 \times 10^{57} \, \text{kg}^2
\][/tex]
Now calculate the denominator:
[tex]\[
(1.8 \times 10^{11})^2 = 3.24 \times 10^{22} \, \text{m}^2
\][/tex]
Putting it all together:
[tex]\[
F = \frac{6.67430 \times 10^{-11} \times 3.8 \times 10^{57}}{3.24 \times 10^{22}}
\][/tex]
Perform the multiplication in the numerator and division by the denominator:
[tex]\[
F = \frac{2.536234 \times 10^{47}}{3.24 \times 10^{22}}
\][/tex]
[tex]\[
F \approx 7.827882716049382 \times 10^{24} \, \text{N}
\][/tex]
So the gravitational force between the Sun and Jupiter is:
[tex]\[
\boxed{7.827882716049382 \times 10^{24} \, \text{N}}
\][/tex]
Thus, the correct force calculated using Newton's law of gravitation, given the masses and distance between the Sun and Jupiter, confirms that the previously given force [tex]\(4.17 \times 10^{23} \, \text{N}\)[/tex] was incorrect. The actual correct value is approximately [tex]\(7.83 \times 10^{24} \, \text{N}\)[/tex].
