If a comet has a perihelion distance of 1.AU and an orbital period of 125 Earth years.. What is the comet's aphelion? Cannot seem to find the correct calculation
If a comet has a perihelion distance of 1.AU and an orbital period of 125 Earth years.. What is the comet's aphelion? Cannot seem to find the correct calculation
1 AU = ~92,960,242 miles = 4.908300778x10^11 feet
The period of the comet about the Sun derives from
T = 2Pisqrt(a^3/µ) where T = the period in seconds, a = the semi-major axis of the orbit in feet and µ = the gravitational constant of the Sun, 4.68772x10^21
For T = 125 earth years T = 125(365.25(24(3600) = 3,994,700,000 seconds
The perihelion radius from the Sun's cen
ter 1AU = ~92,960,242 miles = 4.908300778x10^11 feet
Therefore
3,994,700,000 = 2(3.14)sqrt[a^3/4.68772x10^21)] from which a^3 = 1,847,693,311 or a = 1,227,090,599x 10^13 feet
The aphelion radius is therefore, 2a - r(per).
thank you
To calculate the comet's aphelion distance, we can use Kepler's laws of planetary motion. Kepler's second law states that a planet (or comet) sweeps out equal areas in equal intervals of time. This means that the area swept by the comet during its orbit is constant.
A comet's orbital period can give us the ratio of the areas swept out by the comet when it is at perihelion (closest to the Sun) and aphelion (farthest from the Sun).
Given:
- Perihelion distance (r1) = 1 AU
- Orbital period (T) = 125 Earth years
We need to find the aphelion distance (r2).
Kepler's third law states that the square of the orbital period is proportional to the cube of the mean distance from the Sun. Mathematically, we can express it as:
T^2 ∝ r^3
Using this relationship, we can find the ratio of the perihelion distance to the aphelion distance:
(r1/r2)^3 = T^2
Let's substitute the given values into the equation and solve for r2:
(1 AU / r2)^3 = (125 years)^2
(1 / r2)^3 = 125^2
Taking the cube root of both sides, we have:
1 / r2 = 125^(2/3)
Raising both sides to the -1 power, we get:
r2 = 1 / (125^(2/3))
Using a calculator, we can evaluate this value:
r2 ≈ 3.275 AU
Therefore, the comet's aphelion distance is approximately 3.275 AU.
To find the comet's aphelion, we can use Kepler's laws of planetary motion, specifically his second law, which states that a planet (or comet) sweeps out equal areas in equal times.
First, we need to understand what perihelion and aphelion are. Perihelion is the point in the orbit of a celestial object (such as a planet or comet) where it is nearest to the Sun, while aphelion is the point where it is farthest from the Sun.
Given that the comet's perihelion distance is 1 AU, we know that at perihelion, the comet is 1 AU away from the Sun. Now, let's assume that the comet's orbit is elliptical.
Kepler's second law tells us that as the comet moves in its elliptical orbit, it will sweep out equal areas in equal times. This means that the speed of the comet changes as it moves through its orbit. When it is closer to the Sun (at perihelion), it moves faster, and when it is farther away (at aphelion), it moves slower.
Since we know the orbital period of the comet is 125 Earth years, we can determine the average speed of the comet in its orbit. Let's use Earth's average orbital distance from the Sun, which is 1 AU, as a reference point.
The average speed of the comet in its orbit can be calculated by dividing the total distance traveled by the comet (2 times the semimajor axis of the orbit) by the time it takes to complete one orbit (orbital period).
Total distance traveled = 2 × semimajor axis of the orbit
Total distance traveled = 2 × 1 AU = 2 AU
Average speed = Total distance traveled / Orbital period
Average speed = 2 AU / 125 years
Now, let's find the time it takes for the comet to travel from its perihelion to its aphelion. Since the comet's average speed is constant throughout its orbit, the time taken to travel from perihelion to aphelion will be half of its orbital period (62.5 years).
During this time, the comet will sweep out equal areas, applying Kepler's second law. This means that the area swept out by the comet in its orbit between perihelion and aphelion is equal to the area swept out between aphelion and perihelion.
For our calculation, we can consider this area to be a triangle. The base of the triangle is the distance between perihelion and aphelion, and the height of the triangle is the average distance of the comet from the Sun (which is the semimajor axis).
Using the formula for the area of a triangle:
Area = (base × height) / 2
We can rearrange the formula to solve for the base (distance between perihelion and aphelion):
base = (2 × Area) / height
We know the base is the difference between the comet's aphelion distance (which we want to find) and perihelion distance. So we can rewrite the formula as:
aphelion distance = base + perihelion distance
Now let's plug in the values we know:
perihelion distance = 1 AU
orbital period = 125 years
average speed = 2 AU / 125 years
base = (2 × Area) / height
height = semimajor axis
Since we know the perihelion distance is 1 AU, the semimajor axis can be solved using the formula:
semimajor axis = (perihelion distance + aphelion distance) / 2
Now we have all the necessary information to find the comet's aphelion distance. By substituting the known values into the formulas, you can calculate the aphelion distance.