The distances from the Sun at perihelion and aphelion for Pluto are 4.410 109 km and 7.360 109 km, respectively. What is the ratio of Pluto's orbital speed around the Sun at perihelion to that at aphelion?
To find the ratio of Pluto's orbital speed around the Sun at perihelion to that at aphelion, we need to use Kepler's Second Law, which states that the line connecting a planet to the Sun sweeps out equal areas in equal times. This law can be used to determine the relationship between the distances and speeds of a planet in its orbit.
The formula we'll use to find the speed is:
v = 2πr / T
Where:
- v is the orbital speed
- π is pi (approximately 3.14159)
- r is the distance from the Sun
- T is the orbital period
First, we need to find the orbital periods at perihelion and aphelion. Let's assume that the orbital period of Pluto is the same at both points. Therefore, we can calculate the ratio of the distances to find the ratio of the speeds.
Distance ratio = distance at perihelion / distance at aphelion
= 4.410 × 10^9 km / 7.360 × 10^9 km
Now, we can calculate the orbital speed ratio using the formula mentioned earlier:
Speed ratio = 2πr1 / T1 / 2πr2 / T2
Since we assume the orbital periods are the same at perihelion and aphelion, we can simplify the equation:
Speed ratio = (2πr1) / (2πr2)
= r1 / r2
Substituting the values:
Speed ratio = (4.410 × 10^9 km) / (7.360 × 10^9 km)
By simplifying the expression, we find:
Speed ratio = 0.599
Therefore, the ratio of Pluto's orbital speed around the Sun at perihelion to that at aphelion is approximately 0.599.