If you know the mass of a star and a planet’s radius and speed at perihelion (closest approach to the star), then using conservation of energy and angular momentum you can solve for the radius and speed of that planet at aphelion (farthest point from the star).
Encke’s comet has speed 7.0x104
m/s at its closest distance to the Sun (5.1x1010 m, about a third of the distance from the Earth to the Sun). What is its speed and distance from the
Sun at aphelion?
To solve for the speed and distance of Encke's comet at aphelion using conservation of energy and angular momentum, we start with a few assumptions:
1. The mass of the Sun remains constant throughout the motion of the comet.
2. The orbit of the comet is approximately circular.
Here's how you can approach the problem step-by-step:
Step 1: Find the mass of the Sun
The mass of the Sun is a known constant. It is approximately 1.989 × 10^30 kilograms.
Step 2: Use conservation of energy
At any position in the Keplerian orbit, the total energy of the comet is given by:
E = - (G * M * m) / (2 * r) + (1/2) * m * v^2
Where:
- G is the universal gravitational constant (approximately 6.674 × 10^-11 m^3 kg^-1 s^-2)
- M is the mass of the Sun
- m is the mass of the comet (which we can assume to be negligible compared to the mass of the Sun)
- r is the distance between the Sun and the comet
- v is the speed of the comet
Step 3: Use conservation of angular momentum
The angular momentum of the comet is conserved throughout its orbit and can be expressed as:
L = m * r * v
Where:
- L is the angular momentum of the comet
This equation states that the product of the mass of the comet, its distance from the Sun, and its speed remain constant.
Step 4: Apply conservation of energy and angular momentum
At perihelion, we can relate the speed and distance of the comet (v1 and r1) to its speed and distance at aphelion (v2 and r2) using the conservation equations.
Using the conservation of energy equation, we have:
- (G * M * m) / (2 * r1) + (1/2) * m * v1^2 = - (G * M * m) / (2 * r2) + (1/2) * m * v2^2
Simplifying this equation, we get:
v2^2 - v1^2 = (2 * G * M) * (1/r1 - 1/r2) --- (equation 1)
Using the conservation of angular momentum equation, we have:
m * r1 * v1 = m * r2 * v2
Simplifying this equation, we get:
r1 * v1 = r2 * v2 --- (equation 2)
Step 5: Calculate the values at aphelion
Now, we have two equations (equation 1 and equation 2) with two unknowns (v2 and r2). We can solve these equations simultaneously to find the speed and distance of the comet at aphelion.
Substitute equation 2 into equation 1:
v2^2 - v1^2 = (2 * G * M) * (1 / (r1 * v1) - 1 / r2)
Then, solve for v2:
v2 = sqrt(v1^2 + (2 * G * M) * (1 / (r1 * v1) - 1 / r2))
Once you have the value of v2, substitute it back into equation 2 to solve for r2:
r2 = (r1 * v1) / v2
By substituting the given values (v1 = 7.0 × 10^4 m/s, r1 = 5.1 × 10^10 m), the mass of the Sun, and the constant G, you can calculate the speed and distance of Encke's comet at aphelion.