To solve this problem, we need to calculate the gravitational force between the Sun and Jupiter using Newton's law of universal gravitation. The formula for gravitational force [tex]\( F \)[/tex] between two masses [tex]\( m_1 \)[/tex] and [tex]\( m_2 \)[/tex] separated by a distance [tex]\( r \)[/tex] is given by:
[tex]\[ F = \frac{G \cdot m_1 \cdot m_2}{r^2} \][/tex]
Where:
- [tex]\( G \)[/tex] is the gravitational constant, approximately [tex]\( 6.67430 \times 10^{-11} \, \text{N} \cdot (\text{m/kg})^2 \)[/tex]
- [tex]\( m_1 \)[/tex] is the mass of the Sun, [tex]\( 1.99 \times 10^{30} \, \text{kg} \)[/tex]
- [tex]\( m_2 \)[/tex] is the mass of Jupiter, [tex]\( 1.90 \times 10^{27} \, \text{kg} \)[/tex]
- [tex]\( r \)[/tex] is the distance between the Sun and Jupiter, which needs to be converted from kilometers to meters.
First, we convert the distance from kilometers to meters.
[tex]\[ r = 7.79 \times 10^8 \, \text{km} \times 10^3 \, \left(\frac{\text{m}}{\text{km}}\right) = 7.79 \times 10^{11} \, \text{m} \][/tex]
Now, we can substitute these values into the gravitational force formula:
[tex]\[ F = \frac{6.67430 \times 10^{-11} \, \text{N(m/kg)}^2 \cdot 1.99 \times 10^{30} \, \text{kg} \cdot 1.90 \times 10^{27} \, \text{kg}}{(7.79 \times 10^{11} \, \text{m})^2} \][/tex]
After calculation, the gravitational force [tex]\( F \)[/tex] is:
[tex]\[ F \approx 4.1585074673596537 \times 10^{23} \, \text{N} \][/tex]
To express this force to three significant figures, we round the number appropriately:
[tex]\[ F \approx 4.159 \times 10^{23} \, \text{N} \][/tex]
Therefore, the gravitational force between the Sun and Jupiter, to three significant figures, is:
[tex]\[ \boxed{4.159 \times 10^{23} \, \text{N}} \][/tex]
