How long would our year be if our Sun were two fifths its present mass and the radius of the Earth’s orbit were six times its present value?

period^2= constant*r^3/M

T^2/1year=6^3/1* 1/(2/5)

T^2= 36*6*5/2

period= 6sqrt15 * 1year

check my thinking.
http://hyperphysics.phy-astr.gsu.edu/hbase/kepler.html#c6

The gravity force would be less by a factor of (2/5)*(1/6)^2 = 1/90

The centripetal force would have to decrease my the same factor. That means
V^2/R becomes 1/90 of the former value.
Call V1 and R1 the original values and V2 and R2 the new values
V2^2/R2 = (1/90)V1^2/R1
V2^2 = (1/90)(R2/R1)*V1 = 1/15
V2 = 0.258 V1
The period of an orbit is proportional to R/V. It will increase by a factor
6/0.258 = 23.3

That is the new orbital period in years.

dwls thank you very much

Ah, if our Sun were two fifths of its present mass and the radius of the Earth's orbit were six times its current size, our year would, well, let's just say it would be quite an eventful year! We would have no choice but to rename it the "Year of the Eternal Wait." You see, everything would slow down considerably. Days would seem to drag on forever, and you'd have plenty of time to finish all those New Year's resolutions. On the bright side, think of all the extra time you'll have to enjoy the circus! So, buckle up, my friend, because this year will truly be a test of patience.

To determine how long our year would be under the given conditions, we need to understand the factors that influence the length of a year in our solar system.

The length of a year is determined by the time it takes for a planet to complete one orbit around its star (in our case, the Sun). This time is directly related to the distance of the planet from its star and the mass of the star.

Let's break down the problem and find a solution step by step:

1. Determine the initial length of a year:
The length of a year in our solar system is approximately 365.25 days. This is the time it takes for the Earth to complete one orbit around the Sun.

2. Calculate the new distance from the Sun:
Given that the radius of Earth's orbit is six times its present value, we can multiply the current distance by 6 to find the new distance. However, we need to be careful here because the length of a year is inversely proportional to the distance from the star. As the distance increases, the year lengthens. So, the new distance will be 1/6th of the current distance from the Sun.

3. Determine the new mass of the Sun:
The problem states that the Sun's mass is reduced to two-fifths of its present value. To find the new mass, we multiply the current mass by 2/5.

4. Calculate the new length of a year:
Based on Kepler's third law of planetary motion, the period of an orbiting planet (T) squared is proportional to the cube of the semi-major axis (a) of its orbit. Mathematically, T^2 is proportional to a^3.
Therefore, we can set up the following equation:
(New Year Length)^2 = (New Distance from the Sun)^3 / (New Mass of the Sun)

Plugging in the values we calculated earlier, we get:
(New Year Length)^2 = (1/6)^3 / (2/5)

To simplify further, we calculate:
(New Year Length)^2 = 1/216 / 2/5
(New Year Length)^2 = 5/216

Finally, taking the square root of both sides gives us the new length of a year:
New Year Length ≈ √(5/216)

By performing the above steps, we can find the approximate value for the new length of a year if the Sun were two-fifths its present mass and the radius of the Earth's orbit were six times its present value.