Use Kepler's Third Law And The Orbital Motion Of Earth To Determine The Mass Of The Sun. The Average Distance Between Earth And The Sun Is $1.496 \times 10^{11} \text{ M}$. Earth's Orbital Period Around The Sun Is 365.26 Days.A. $6.34

Introduction

Kepler's third law is a fundamental concept in astronomy that relates the orbital period of a planet to its average distance from the Sun. By applying this law to the Earth's orbit, we can determine the mass of the Sun. In this article, we will explore the concept of Kepler's third law, calculate the mass of the Sun using the Earth's orbital period and average distance, and discuss the significance of this calculation.

Kepler's Third Law

Kepler's third law states that the square of a planet's orbital period is directly proportional to the cube of its average distance from the Sun. Mathematically, this can be expressed as:

$$T^2 \propto r^3$$

where $T$ is the orbital period and $r$ is the average distance from the Sun.

Derivation of Kepler's Third Law

To derive Kepler's third law, we can start with the equation of motion for a planet in a circular orbit:

$$\frac{GMm}{r^2} = \frac{mv^2}{r}$$

where $G$ is the gravitational constant, $M$ is the mass of the Sun, $m$ is the mass of the planet, and $v$ is the orbital velocity.

Rearranging this equation, we get:

$$v^2 = \frac{GM}{r}$$

The orbital period $T$ is related to the orbital velocity $v$ by:

$$T = \frac{2\pi r}{v}$$

Substituting the expression for $v^2$ into this equation, we get:

$$T^2 = \frac{4\pi^2 r^3}{GM}$$

This is Kepler's third law, which shows that the square of the orbital period is directly proportional to the cube of the average distance from the Sun.

Calculating the Mass of the Sun

To calculate the mass of the Sun, we can use Kepler's third law and the given values for the Earth's orbital period and average distance. The average distance between the Earth and the Sun is $1.496 \times 10^{11} \text{ m}$, and the Earth's orbital period is 365.26 days.

First, we need to convert the orbital period from days to seconds:

$$T = 365.26 \text{ days} \times 24 \text{ hours/day} \times 3600 \text{ s/hour} = 3.1557 \times 10^7 \text{ s}$$

Next, we can plug in the values for $T$ and $r$ into Kepler's third law:

$$T^2 = \frac{4\pi^2 r^3}{GM}$$

Rearranging this equation to solve for $M$, we get:

$$M = \frac{4\pi^2 r^3}{GT^2}$$

Substituting the values for $r$ and $T$, we get:

$$M = \frac{4\pi^2 (1.496 \times 10^{11} \text{ m})^3}{(6.674 \times 10^{-11} \text{ N m}^2 \text{ kg}^{-2})(3.1557 \times 10^7 \text{ s})^2}$$

Evaluating this expression, we get:

$$M = 1.989 \times 10^{30} \text{ kg}$$

This is the mass of the Sun, calculated using Kepler's third law and the Earth's orbital period and average distance.

Significance of the Calculation

The calculation of the mass of the Sun using Kepler's third law and the Earth's orbital period and average distance is significant because it demonstrates the power of astronomical observations to determine the fundamental properties of celestial objects. By applying Kepler's third law to the Earth's orbit, we can determine the mass of the Sun with high accuracy, which is essential for understanding the behavior of our solar system.

Conclusion

In conclusion, Kepler's third law is a fundamental concept in astronomy that relates the orbital period of a planet to its average distance from the Sun. By applying this law to the Earth's orbit, we can determine the mass of the Sun. The calculation of the mass of the Sun using Kepler's third law and the Earth's orbital period and average distance is significant because it demonstrates the power of astronomical observations to determine the fundamental properties of celestial objects.

References

• Kepler, J. (1609). Astronomia Nova. Johann Planck.
• Newton, I. (1687). Philosophiæ Naturalis Principia Mathematica. Joseph Streater.
• NASA. (2020). Kepler's Third Law. Retrieved from https://www.nasa.gov/keplers-third-law

Appendix

Derivation of Kepler's Third Law

To derive Kepler's third law, we can start with the equation of motion for a planet in a circular orbit:

$$\frac{GMm}{r^2} = \frac{mv^2}{r}$$

Rearranging this equation, we get:

$$v^2 = \frac{GM}{r}$$

The orbital period $T$ is related to the orbital velocity $v$ by:

$$T = \frac{2\pi r}{v}$$

Substituting the expression for $v^2$ into this equation, we get:

$$T^2 = \frac{4\pi^2 r^3}{GM}$$

This is Kepler's third law, which shows that the square of the orbital period is directly proportional to the cube of the average distance from the Sun.

Calculation of the Mass of the Sun

To calculate the mass of the Sun, we can use Kepler's third law and the given values for the Earth's orbital period and average distance. The average distance between the Earth and the Sun is $1.496 \times 10^{11} \text{ m}$, and the Earth's orbital period is 365.26 days.

First, we need to convert the orbital period from days to seconds:

$$T = 365.26 \text{ days} \times 24 \text{ hours/day} \times 3600 \text{ s/hour} = 3.1557 \times 10^7 \text{ s}$$

Next, we can plug in the values for $T$ and $r$ into Kepler's third law:

$$T^2 = \frac{4\pi^2 r^3}{GM}$$

Rearranging this equation to solve for $M$, we get:

$$M = \frac{4\pi^2 r^3}{GT^2}$$

Substituting the values for $r$ and $T$, we get:

$$M = \frac{4\pi^2 (1.496 \times 10^{11} \text{ m})^3}{(6.674 \times 10^{-11} \text{ N m}^2 \text{ kg}^{-2})(3.1557 \times 10^7 \text{ s})^2}$$

Evaluating this expression, we get:

$$M = 1.989 \times 10^{30} \text{ kg}$$

Introduction

In our previous article, we explored the concept of Kepler's third law and used it to calculate the mass of the Sun. In this article, we will answer some frequently asked questions about Kepler's third law and the mass of the Sun.

Q: What is Kepler's third law?

A: Kepler's third law is a fundamental concept in astronomy that relates the orbital period of a planet to its average distance from the Sun. It states that the square of a planet's orbital period is directly proportional to the cube of its average distance from the Sun.

Q: How is Kepler's third law derived?

A: Kepler's third law is derived from the equation of motion for a planet in a circular orbit. By rearranging this equation, we can show that the square of the orbital period is directly proportional to the cube of the average distance from the Sun.

Q: What is the significance of Kepler's third law?

A: Kepler's third law is significant because it allows us to determine the mass of the Sun using the orbital period and average distance of a planet. This is essential for understanding the behavior of our solar system.

Q: How do we calculate the mass of the Sun using Kepler's third law?

A: To calculate the mass of the Sun using Kepler's third law, we need to know the orbital period and average distance of a planet. We can then plug these values into Kepler's third law and solve for the mass of the Sun.

Q: What are the units of the mass of the Sun?

A: The units of the mass of the Sun are kilograms (kg).

Q: What is the value of the mass of the Sun?

A: The value of the mass of the Sun is approximately 1.989 x 10^30 kg.

Q: How accurate is the calculation of the mass of the Sun using Kepler's third law?

A: The calculation of the mass of the Sun using Kepler's third law is highly accurate. The value of the mass of the Sun calculated using Kepler's third law is consistent with the value obtained using other methods.

Q: Can Kepler's third law be applied to other planets in our solar system?

A: Yes, Kepler's third law can be applied to other planets in our solar system. By using the orbital period and average distance of a planet, we can calculate its mass using Kepler's third law.

Q: What are some limitations of Kepler's third law?

A: One limitation of Kepler's third law is that it assumes a circular orbit. In reality, the orbits of planets are elliptical, which can affect the accuracy of the calculation. Additionally, Kepler's third law assumes that the mass of the Sun is constant, which is not always the case.

Conclusion

In conclusion, Kepler's third law is a fundamental concept in astronomy that relates the orbital period of a planet to its average distance from the Sun. By using Kepler's third law, we can calculate the mass of the Sun with high accuracy. We hope that this Q&A article has provided a better understanding of Kepler's third law and the mass of the Sun.

References

• Kepler, J. (1609). Astronomia Nova. Johann Planck.
• Newton, I. (1687). Philosophiæ Naturalis Principia Mathematica. Joseph Streater.
• NASA. (2020). Kepler's Third Law. Retrieved from https://www.nasa.gov/keplers-third-law

Appendix

Derivation of Kepler's Third Law

To derive Kepler's third law, we can start with the equation of motion for a planet in a circular orbit:

$$\frac{GMm}{r^2} = \frac{mv^2}{r}$$

Rearranging this equation, we get:

$$v^2 = \frac{GM}{r}$$

The orbital period $T$ is related to the orbital velocity $v$ by:

$$T = \frac{2\pi r}{v}$$

Substituting the expression for $v^2$ into this equation, we get:

$$T^2 = \frac{4\pi^2 r^3}{GM}$$

This is Kepler's third law, which shows that the square of the orbital period is directly proportional to the cube of the average distance from the Sun.

Calculation of the Mass of the Sun

To calculate the mass of the Sun, we can use Kepler's third law and the given values for the Earth's orbital period and average distance. The average distance between the Earth and the Sun is $1.496 \times 10^{11} \text{ m}$, and the Earth's orbital period is 365.26 days.

First, we need to convert the orbital period from days to seconds:

$$T = 365.26 \text{ days} \times 24 \text{ hours/day} \times 3600 \text{ s/hour} = 3.1557 \times 10^7 \text{ s}$$

Next, we can plug in the values for $T$ and $r$ into Kepler's third law:

$$T^2 = \frac{4\pi^2 r^3}{GM}$$

Rearranging this equation to solve for $M$, we get:

$$M = \frac{4\pi^2 r^3}{GT^2}$$

Substituting the values for $r$ and $T$, we get:

$$M = \frac{4\pi^2 (1.496 \times 10^{11} \text{ m})^3}{(6.674 \times 10^{-11} \text{ N m}^2 \text{ kg}^{-2})(3.1557 \times 10^7 \text{ s})^2}$$

Evaluating this expression, we get:

$$M = 1.989 \times 10^{30} \text{ kg}$$

This is the mass of the Sun, calculated using Kepler's third law and the Earth's orbital period and average distance.