To verify the distance between the probe and the center of Venus, we follow these steps using the gravitational force formula:
The formula for the 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]\[ F = G \cdot \frac{m_1 \cdot m_2}{r^2} \][/tex]
where:
- [tex]\( F \)[/tex] is the gravitational force between the objects,
- [tex]\( G \)[/tex] is the gravitational constant ([tex]\( 6.67 \times 10^{-11} \, \text{N} \cdot \text{m}^2 / \text{kg}^2 \)[/tex]),
- [tex]\( m_1 \)[/tex] is the mass of Venus ([tex]\( 4.87 \times 10^{24} \, \text{kg} \)[/tex]),
- [tex]\( m_2 \)[/tex] is the mass of the probe (655 kg),
- [tex]\( r \)[/tex] is the distance between the two objects (which we need to verify as [tex]\( 1 \times 10^6 \, \text{m} \)[/tex]).
Given:
- [tex]\( F = 2.58 \times 10^3 \, \text{N} \)[/tex],
- [tex]\( G = 6.67 \times 10^{-11} \, \text{N} \cdot \text{m}^2 / \text{kg}^2 \)[/tex],
- [tex]\( m_1 = 4.87 \times 10^{24} \, \text{kg} \)[/tex],
- [tex]\( m_2 = 655 \, \text{kg} \)[/tex].
Plugging these values into the gravitational force equation, we get:
[tex]\[ 2.58 \times 10^3 = 6.67 \times 10^{-11} \cdot \frac{4.87 \times 10^{24} \cdot 655}{r^2} \][/tex]
To solve for [tex]\( r \)[/tex], we rearrange the equation:
[tex]\[ r^2 = \frac{6.67 \times 10^{-11} \cdot 4.87 \times 10^{24} \cdot 655}{2.58 \times 10^3} \][/tex]
[tex]\[ r^2 = \frac{2.11919035 \times 10^{17}}{2.58 \times 10^3} \][/tex]
[tex]\[ r^2 = 8.21472829 \times 10^{13} \][/tex]
Taking the square root of both sides to solve for [tex]\( r \)[/tex]:
[tex]\[ r \approx \sqrt{8.21472829 \times 10^{13}} \][/tex]
[tex]\[ r \approx 9.065 \times 10^6 \, \text{m} \][/tex]
Upon review, it seems the distance calculation from the initial Python code was incorrect. With further investigation, it reveals the probe should be approximately [tex]\( 9.065 \times 10^6 \, \text{m} \)[/tex] from the center of Venus, not [tex]\( 1 \times 10^6 \, \text{m} \)[/tex].
