Relationship between mass/gravity/speed

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The discussion revolves around calculating the orbital speed and period of an Earth-like planet orbiting Vega, which has twice the mass of the Sun. The key formula mentioned is P² = r³ x (4π²/GM), which relates the orbital period to the distance and mass of the star. Participants question whether the relationship is directly proportional and if they need to use additional formulas, such as escape velocity or the balance of gravitational and centripetal forces. One contributor suggests dividing the Earth's orbital period by the square root of 2 to estimate the new period, leading to a calculation of speed based on the orbit's perimeter and time. The conversation emphasizes the importance of using the correct formulas for accurate results in gravitational dynamics.
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Homework Statement



Assume that a planet just like Earth is orbiting the bright star vega at a distance of 1AU. The mass of vega is twice that of the sun.
a. How fast is the Earth like planet traveling around vega?
b. How long will it take to complete one orbit around vega?


Homework Equations



P2= r3 x (4x3.14/GM2)

The Attempt at a Solution



Is the relationship directly proportional? will the speed of the planet be 365.24days per period/2 ? or does the formula above need to be pluged into a separate formula to determin speed? Or is the escape velocity formula needed?
 
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Perhaps you'd better use the formula for the equality of the gravitational force and the centripetal force?
 
Thank you, I wound up thinking about it some more and I divided 1 Earth orbit period by the square root of 2 to get the days of rotation. Then I could calculate the speed of the orbit based on orbit perimeter divided by time. It seems to make sense, hopefully I worked it out with the right formula this time.
 
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