How would the speed of Earth’s orbit around the sun change if Earth’s distance from the sun increased by 4 times?
The velocity required to keep the earth in a circulor orbit of radius 4 x(93,000,000) miles derives from
V = sqrt[µ/r] where µ = the Sun's gravitational constant = 4.68772x10^21 ft.^3/sec.^2 and r = the radius of the increased orbit in feet = 4.9104x10^11 feet.
Therefore, the orbital velocity becomes
V = sqrt[4.68772x10^21ft.^3/sec.^2/4.901x10^11] = 48,853 ft./sec. = mph. = 9.25 mph. compared to 97,706fps and 18.5 mph in its existing orbit
More symplictically, V = 97,700sqrt[r/4r] = 97,706[1/4) = 97,706/2 = 48,853 fps or 9.25 mph.
To determine how the speed of Earth's orbit around the sun would change if its distance from the sun increased by 4 times, we can use Kepler's Third Law of planetary motion.
Kepler's Third Law states that the square of the period of revolution (T) of a planet is directly proportional to the cube of the semi-major axis of its orbit (a). Mathematically, this can be represented as:
T^2 ∝ a^3
Where T is the period of revolution and a is the semi-major axis.
Let's assume that initially, the distance of Earth from the sun is 'd.'
If the distance increases by 4 times, the new distance (d') would become 4d.
Now, we can set up a proportion to compare the periods of revolution:
(T')^2 / T^2 = (a')^3 / a^3
Since we are comparing the changes, we can assign values:
T' is the new period of revolution (what we're trying to find)
T is the current period of revolution (roughly 365.25 days, or 1 year)
a' is the new semi-major axis (4d)
a is the current semi-major axis (d)
Plugging these values into the equation, we get:
(T')^2 / (365.25 days)^2 = (4d)^3 / d^3
Simplifying further:
(T')^2 / (365.25)^2 = 4^3
(T')^2 / (365.25)^2 = 64
Taking the square root of both sides:
T' / 365.25 = sqrt(64)
T' / 365.25 = 8
Now, we can isolate T':
T' = 8 * 365.25
T' ≈ 2922 days
So, if the distance between Earth and the sun increases by 4 times, the new period of revolution (T') would be approximately 2922 days, or roughly 8 years.
In conclusion, the speed of Earth's orbit around the sun would decrease if its distance from the sun increased by 4 times because it would take more time for Earth to complete one revolution around the sun.