Kepler's Laws, law of periods

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In summary, the period of revolution for the two stars in this binary-star system can be calculated using the equation T^2=(4pi^2/GM)*r^3, where M is the total mass of the two stars. However, in this case where both stars have the same mass, a more accurate calculation would be to use the gravitational parameter μ = 2*G*M.
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kari82
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In a certain binary-star system, each star has the same mass as our Sun, and they revolve about their center of mass. The distance between them is the same as the distance between Earth and the Sun. What is their period of revolution in years?
T^2=(4pi^2/GM)*r^3
I know that the mass of the sun is 1.9x10^30. I also found in my book that the distance between the Earth and the sun is 1.5 x 10^11. If I plug those numbers into the equation and I solve for the ratio between T/1y in seconds, I don't get the correct answer, which is 0.7 years. I'm assuming my mistake has to do with the distance between these two planets. Can anyone help me please?
 
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The problem is that the given expression for the period assumes that one of the masses is dominant (M >> m) and that the mass m of the smaller body can be ignored. Here both objects have the same mass.

The gravitational parameter μ = G*M is an approximation for when m << M. When this is not the case, take μ = G*(M + m), which in this instance will be G*(M + M), or μ = 2*G*M, since they are of equal mass M.
 

What is Kepler's Law of Periods?

Kepler's Law of Periods states that the square of the orbital period of a planet is directly proportional to the cube of the semi-major axis of its orbit. In simpler terms, this means that the farther a planet is from the sun, the longer its orbital period will be.

Who is Johannes Kepler and why are his laws important?

Johannes Kepler was a German mathematician, astronomer, and astrologer who is best known for his laws of planetary motion. These laws, which were developed in the early 1600s, revolutionized our understanding of the solar system and helped pave the way for future advancements in astronomy.

How did Kepler come up with his laws?

Kepler's laws were based on the meticulous observations of his mentor, Tycho Brahe, who had collected detailed data on the positions of planets in the sky. Kepler used this data to develop his three laws, which were later confirmed by the work of Sir Isaac Newton.

What is the difference between Kepler's Law of Periods and Newton's Law of Universal Gravitation?

Kepler's Law of Periods describes the relationship between the orbital period and distance of a planet from the sun, while Newton's Law of Universal Gravitation explains the force of gravity between two objects and how it affects their motion. Kepler's laws are based on observations, while Newton's law is a mathematical formula derived from his theory of gravity.

Are Kepler's laws still relevant today?

Yes, Kepler's laws are still relevant today and are used to predict the motion of planets and other celestial bodies in our solar system. They are also used as a basis for understanding the motion of objects in other solar systems and galaxies. However, they have been refined and expanded upon by subsequent scientific discoveries and theories.

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