Estimate the time difference between the longest day and the shortest day of a year in seconds if you lived on the Earth's equator with the assumptions below.
Note: this is not the difference between solstices as we are adjusting the earth's rotation axis to be in the orbital plane.
Details and assumptions
The furthest distance from the Sun to Earth is df=1.521×108 km.
The shortest distance from the Sun to Earth is ds=1.47×108 km.
To simplify the problem, assume that the Earth's axis is always perpendicular to the plane of its trajectory around the Sun.
The Sun always shines on half of the Earth.
There are 24 hours in a day and 365.25 days in a year.
I assume you are being asked about the change in the rate of a clock in a gravitational field (of the sun, in thic case).
There is a General Relativity formula for it. (Due to Schwarzschild, I believe)
delta T = T*(delta V)/c*2
V is the gravitational potential at distance d from the sun. It will be differnt at the perihelion and aphelion distances.
T is the time period you are comparing (24 hours)
answer=???????????
Never mind the General Relativity approach I suggested. Time goes slower at the lower orbit because of Special relativity . Use the two different satellite speeds to deduse the time shift for one orbit. You want the extra credit.. so you do the numbers
To estimate the time difference between the longest day and the shortest day of the year in seconds, we need to consider the effect of the Earth's distance from the Sun on the length of daylight.
First, let's find the average distance (davg) between the Earth and the Sun. We can use the given furthest distance (df) and shortest distance (ds) to calculate it:
davg = (df + ds) / 2
Substituting the given values:
davg = (1.521×10^8 + 1.47×10^8) / 2 km
Next, we need to calculate the difference in the distance between the Earth's equator at the furthest and shortest points from the Sun.
Δd = df - ds
Substituting the given values:
Δd = (1.521×10^8 - 1.47×10^8) km
Now, we can calculate the angular difference (θ) between the furthest and shortest points from the Sun using the formula:
θ = arctan(Δd / davg)
We will use the average distance (davg) and the distance difference (Δd) calculated above.
Now, we can estimate the time difference (Δt) between the longest and shortest days using the formula:
Δt = 24 * 60 * 60 * θ / 360
Here, we convert the angular difference from degrees to hours (24 hours in a day) and then to seconds (60 minutes in an hour, 60 seconds in a minute).
Let's plug in the values and calculate the time difference:
davg = (1.521×10^8 + 1.47×10^8) / 2 km
Δd = (1.521×10^8 - 1.47×10^8) km
θ = arctan(Δd / davg)
Δt = 24 * 60 * 60 * θ / 360
After performing the calculations, we will have an estimate of the time difference between the longest and shortest days of the year on the Earth's equator in seconds.