Give step-by-step solution with explanation and final answer:
Law #3: The period (in Earth years) of a planet squared is equal Let's look at an example. The Earth takes one year to complete its orbit and the semi - major axis of its orbit is 1 AU. P squared is 1 x 1 = 1 and a cubed is to the semi - major axis (in Astronomical Units or AU) of the planet's orbit cubed. 1x1x1 = 1. The third law works for these values. Astronomers can determine the period of a planet or any other celestial object in orbit around the Sun by observing it carefully. Kepler's third law allows them to compute the semi - major axis from the periods they observe. Nothing else is needed. The third law established the scale of the solar system before astronomers had an independent way of determining the true dimensions of the orbits of the planets. For the planets and asteroid in the table below, compute the semi - major axes using Kepler's third law. You will need to find the cube root of the period squared. Then plot and label the objects on the figure at right. This will illustrate their approximate relative distances from the Sun. Object P (yr) (P2) = a (AU) Mercury 0.24 Venus 0.62 Earth 1.0 p2 Mars 1.88 Asteroid Ceres 4.6 Jupiter 11.86 Saturn 29.5 Uranus 84 Neptune 165 Did you know that you can use Google to find the cube root of a number? In the search box, type cube root(number). Then press the Enter key. For example, to find the cube root of 64, you would type cube root(64).
Question:
Give step-by-step solution with explanation and final answer:
Law #3: The period (in Earth years) of a planet squared is equal Let's look at an example. The Earth takes one year to complete its orbit and the semi - major axis of its orbit is 1 AU. P squared is 1 x 1 = 1 and a cubed is to the semi - major axis (in Astronomical Units or AU) of the planet's orbit cubed. 1x1x1 = 1. The third law works for these values. Astronomers can determine the period of a planet or any other celestial object in orbit around the Sun by observing it carefully. Kepler's third law allows them to compute the semi - major axis from the periods they observe. Nothing else is needed. The third law established the scale of the solar system before astronomers had an independent way of determining the true dimensions of the orbits of the planets. For the planets and asteroid in the table below, compute the semi - major axes using Kepler's third law. You will need to find the cube root of the period squared. Then plot and label the objects on the figure at right. This will illustrate their approximate relative distances from the Sun. Object P (yr) (P2) = a (AU) Mercury 0.24 Venus 0.62 Earth 1.0 p2 Mars 1.88 Asteroid Ceres 4.6 Jupiter 11.86 Saturn 29.5 Uranus 84 Neptune 165 Did you know that you can use Google to find the cube root of a number? In the search box, type cube root(number). Then press the Enter key. For example, to find the cube root of 64, you would type cube root(64).
This is AI generated Response.
Asked by: Rahul
Created at: 2026-01-17 06:08:19
Final Answer:
RenderedRaw Text
Did you like this answer?
# Step-by-Step Solution: Kepler's Third Law and Semi-Major Axes
## Kepler's Third Law (Simplified for Our Solar System)
Kepler's Third Law states:
> The square of the orbital period (in Earth years) of a planet is equal to the cube of the semi-major axis (in astronomical units, AU).
Mathematically:
\[
P^2 = a^3
\]
where
- \( P \) = orbital period (years)
- \( a \) = semi-major axis (AU)
We can rearrange to solve for \( a \):
\[
a = \sqrt[3]{P^2}
\]
---
## Step 1: Compute \( P^2 \) for Each Object
| Object | \( P \) (yr) | \( P^2 \) |
|----------------|--------------|--------------|
| Mercury | .24 | .0576 |
| Venus | .62 | .3844 |
| Earth | 1.00 | 1.00 |
| Mars | 1.88 | 3.5344 |
| Ceres (asteroid)| 4.6 | 21.16 |
| Jupiter | 11.86 | 140.6596 |
| Saturn | 29.5 | 870.25 |
| Uranus | 84 | 7056 |
| Neptune | 165 | 27,225 |
---
## Step 2: Compute \( a = \sqrt[3]{P^2} \) for Each Object
Calculate the cube root for each \( P^2 \):
| Object | \( P^2 \) | \( a = \sqrt[3]{P^2} \) (AU) |
|----------------|--------------|------------------------------|
| Mercury | .0576 | .38 |
| Venus | .3844 | .73 |
| Earth | 1.00 | 1.00 |
| Mars | 3.5344 | 1.52 |
| Ceres | 21.16 | 2.77 |
| Jupiter | 140.6596 | 5.20 |
| Saturn | 870.25 | 9.54 |
| Uranus | 7056 | 19.19 |
| Neptune | 27,225 | 30.08 |
---
## Step 3: Final Answers
### Table with Calculated Semi-Major Axes
| Object | Period \( P \) (years) | Semi-Major Axis \( a \) (AU) |
|----------------|-----------------------|------------------------------|
| Mercury | .24 | .38 |
| Venus | .62 | .73 |
| Earth | 1.00 | 1.00 |
| Mars | 1.88 | 1.52 |
| Ceres | 4.6 | 2.77 |
| Jupiter | 11.86 | 5.20 |
| Saturn | 29.5 | 9.54 |
| Uranus | 84 | 19.19 |
| Neptune | 165 | 30.08 |
---
## Step 4: Explanation
To find the semi-major axis for each object:
1. Square the orbital period (\( P \)) to get \( P^2 \).
2. Take the cube root of \( P^2 \) to solve for \( a \) in AU.
**Example Calculation (for Mars):**
- \( P = 1.88 \)
- \( P^2 = 1.88^2 = 3.5344 \)
- \( a = \sqrt[3]{3.5344} \approx 1.52 \) AU
---
## Conclusion
By applying Kepler's Third Law, we've calculated the semi-major axes of several solar system objects. These values illustrate their relative distances from the Sun in astronomical units (AU).
> **Note:** The semi-major axis is the average distance from the Sun to the object, measured in AU, where 1 AU is the average distance from the Earth to the Sun.
Question:
Law #3: The period (in Earth years) of a planet squared is equal Let's look at an example. The Earth takes one year to complete its orbit and the semi - major axis of its orbit is 1 AU. P squared is 1 x 1 = 1 and a cubed is to the semi - major axis (in Astronomical Units or AU) of the planet's orbit cubed. 1x1x1 = 1. The third law works for these values. Astronomers can determine the period of a planet or any other celestial object in orbit around the Sun by observing it carefully. Kepler's third law allows them to compute the semi - major axis from the periods they observe. Nothing else is needed. The third law established the scale of the solar system before astronomers had an independent way of determining the true dimensions of the orbits of the planets. For the planets and asteroid in the table below, compute the semi - major axes using Kepler's third law. You will need to find the cube root of the period squared. Then plot and label the objects on the figure at right. This will illustrate their approximate relative distances from the Sun. Object P (yr) (P2) = a (AU) Mercury 0.24 Venus 0.62 Earth 1.0 p2 Mars 1.88 Asteroid Ceres 4.6 Jupiter 11.86 Saturn 29.5 Uranus 84 Neptune 165 Did you know that you can use Google to find the cube root of a number? In the search box, type cube root(number). Then press the Enter key. For example, to find the cube root of 64, you would type cube root(64).
This is AI generated Response.
Asked by: Rahul
Created at: 2026-01-17 06:09:43
Final Answer:
RenderedRaw Text
Did you like this answer?
# Step-by-Step Solution: Kepler's Third Law and Semi-Major Axes
## Overview of Kepler's Third Law
Kepler's Third Law states:
> The square of the orbital period (in Earth years) of a planet is equal to the cube of the semi-major axis (in astronomical units, AU).
Mathematically, it can be expressed as:
\[
P^2 = a^3
\]
Where:
- \( P \) = orbital period (years)
- \( a \) = semi-major axis (AU)
To find the semi-major axis, we rearrange the equation:
\[
a = \sqrt[3]{P^2}
\]
---
## Step 1: Calculate \( P^2 \) for Each Object
We will first compute the square of the orbital period for each object listed.
| Object | \( P \) (years) | \( P^2 \) |
|----------------|------------------|------------------|
| Mercury | 0.24 | 0.0576 |
| Venus | 0.62 | 0.3844 |
| Earth | 1.00 | 1.00 |
| Mars | 1.88 | 3.5344 |
| Ceres | 4.6 | 21.16 |
| Jupiter | 11.86 | 140.6596 |
| Saturn | 29.5 | 870.25 |
| Uranus | 84 | 7056 |
| Neptune | 165 | 27225 |
---
## Step 2: Calculate \( a = \sqrt[3]{P^2} \) for Each Object
Next, we will find the cube root of \( P^2 \) for each object to determine the semi-major axis.
| Object | \( P^2 \) | \( a = \sqrt[3]{P^2} \) (AU) |
|----------------|------------------|------------------------------|
| Mercury | 0.0576 | 0.38 |
| Venus | 0.3844 | 0.73 |
| Earth | 1.00 | 1.00 |
| Mars | 3.5344 | 1.52 |
| Ceres | 21.16 | 2.77 |
| Jupiter | 140.6596 | 5.20 |
| Saturn | 870.25 | 9.54 |
| Uranus | 7056 | 19.19 |
| Neptune | 27225 | 30.08 |
---
## Step 3: Summary of Final Answers
### Table of Semi-Major Axes
| Object | Period \( P \) (years) | Semi-Major Axis \( a \) (AU) |
|----------------|-----------------------|------------------------------|
| Mercury | 0.24 | 0.38 |
| Venus | 0.62 | 0.73 |
| Earth | 1.00 | 1.00 |
| Mars | 1.88 | 1.52 |
| Ceres | 4.6 | 2.77 |
| Jupiter | 11.86 | 5.20 |
| Saturn | 29.5 | 9.54 |
| Uranus | 84 | 19.19 |
| Neptune | 165 | 30.08 |
---
## Step 4: Explanation of Calculation
To find the semi-major axis for each object:
1. Square the orbital period (\( P \)) to get \( P^2 \).
2. Calculate the cube root of \( P^2 \) to determine \( a \) in astronomical units (AU).
### Example Calculation (Mars):
- Orbital Period \( P = 1.88 \)
- Calculate \( P^2 = 1.88^2 = 3.5344 \)
- Then, \( a = \sqrt[3]{3.5344} \approx 1.52 \) AU
---
## Conclusion
Using Kepler's Third Law, we have successfully calculated the semi-major axes of various celestial objects in our solar system. This method provides insight into their relative distances from the Sun, measured in astronomical units (AU).
VIPSolutions focuses on clear, step‑by‑step explanations so you can learn quickly. Whether you need an AI‑generated walkthrough or a short hint to get unstuck, each solution is organized for fast reading and easy review later.
Search similar questions, compare approaches, and bookmark the best answers for revision. Our goal is simple: quick, reliable study help that feels natural—not noisy.
VIPSolutions