jupiter orbits the sun twice in 23.72 years. What is the length of its semi major axis as it orbits the sun in AU?
A) 563 AU
B) 5.2 AU
C) 115.5 AU
D) 4.9 AU
I need help
Well, if you want help, you've come to the right place, my friend! Let me assist you with that question.
To determine the length of Jupiter's semi-major axis as it orbits the sun in Astronomical Units (AU), we can use Kepler's Third Law. This law states that the square of a planet's orbital period (in Earth years) is proportional to the cube of its semi-major axis (in AU).
So, if Jupiter orbits the sun twice in 23.72 years, we can write the equation as:
(23.72 years)^2 = (Jupiter's semi-major axis in AU)^3
Now, to find the value of Jupiter's semi-major axis, we just need to solve the equation. And here comes the punchline, are you ready?
Drumroll, please...
The correct answer is C) 115.5 AU!
Why? Because when you square 23.72, it's approximately 563.0884. And when you take the cube root of that, you get a little over 115.5. Voilà!
Hope that helps, and remember, astronomy can be a "planet" of fun!
To find the length of Jupiter's semi-major axis as it orbits the Sun in astronomical units (AU), we can use Kepler's third law:
T^2 = (4π^2 / GM) * a^3
Where:
- T is the orbital period of Jupiter in years
- G is the gravitational constant (6.67430 × 10^-11 m^3 kg^-1 s^-2)
- M is the mass of the Sun (approximately 1.989 × 10^30 kg)
- a is the semi-major axis of Jupiter's orbit in AU
We are given that Jupiter orbits the Sun twice in 23.72 years.
Plugging the given values into the formula:
(23.72)^2 = (4π^2 / (6.67430 × 10^-11 × 1.989 × 10^30)) * a^3
Simplifying:
a^3 = [(23.72)^2 * (6.67430 × 10^-11 × 1.989 × 10^30)] / (4π^2)
Taking the cube root of both sides to find a:
a = cuberoot([(23.72)^2 * (6.67430 × 10^-11 × 1.989 × 10^30)] / (4π^2))
Using a calculator, we find that the approximate value of a is 5.2 AU.
Therefore, the correct answer is B) 5.2 AU.
To find the length of Jupiter's semi-major axis as it orbits the Sun in Astronomical Units (AU), we can use Kepler's Third Law of Planetary Motion. This law states that the square of the orbital period of a planet (in years) is proportional to the cube of its average distance from the Sun (in AU).
Let's solve this step by step:
1. We are given that Jupiter orbits the Sun twice in 23.72 years. This means that the square of the orbital period is (23.72)^2 = 562.7184.
2. We need to find the cube of its average distance from the Sun. Let's denote the semi-major axis (average distance) as 'a'. So, we need to find 'a^3'.
3. Using Kepler's Third Law, we can write: (Orbital Period)^2 = (Semi-major axis)^3
Substituting the values, we have: 562.7184 = a^3.
4. To solve for 'a', we take the cube root of both sides of the equation: (∛562.7184) = ∛(a^3), which simplifies to a = ∛562.7184.
5. Evaluating this expression, we find that a ≈ 8.8574 AU.
Now, let's compare this value to the given options:
A) 563 AU - This option does not match the calculated value.
B) 5.2 AU - This option does not match the calculated value.
C) 115.5 AU - This option does not match the calculated value.
D) 4.9 AU - This option does not match the calculated value.
None of the given options match the calculated value of approximately 8.8574 AU. Therefore, none of the provided options are correct.