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.
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2.57 s exactly
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Sorry my mistake... Try 1.58 s
To estimate the time difference between the longest day and the shortest day of a year on the Earth's equator, we need to consider the effect of the Earth's distance from the Sun.
Here's how you can estimate the time difference:
1. Determine the angular velocity of the Earth's rotation:
The Earth completes one full rotation in approximately 24 hours (or 86,400 seconds). The angular velocity, ω, can be calculated using the formula:
ω = 2π / T
where T is the period of rotation in seconds.
2. Calculate the distance traveled by a point on the equator during the time difference:
Since the Earth rotates about its axis once in 24 hours, a point on the equator will travel a distance equal to the circumference of the Earth, 2πr, during this time.
Using the mean radius of the Earth, which is approximately 6,371 km or 6,371,000 meters, we can calculate the total distance traveled in meters.
3. Calculate the difference in Earth-Sun distances:
Subtract the shortest distance from the longest distance:
Δd = df - ds
4. Calculate the time taken to cover the Earth-Sun distance difference:
We need to convert the Earth-Sun distance difference to meters and use the speed of light as an approximation for the speed of travel from the Earth to the Sun.
Let's assume the speed of light, c, is approximately 299,792,458 meters per second. Calculate the time taken to travel the difference in distance:
Δt = Δd / c
5. Calculate the time difference:
Divide the distance traveled on the equator by the angular velocity of the Earth's rotation to get the time difference:
Δt_diff = (total distance traveled on the equator) / ω
6. Convert the time difference to seconds:
The previous calculation gives the time difference in units of (time taken for one full rotation). Convert this to seconds by multiplying by the period of rotation:
Δt_diff_seconds = Δt_diff * T
Now, let's plug in the given values and calculate the estimated time difference:
T = 24 hours = 86,400 seconds
Earth's mean radius, r = 6,371,000 meters
df = 1.521×10^8 km = 1.521×10^11 meters
ds = 1.47×10^8 km = 1.47×10^11 meters
c = 299,792,458 meters per second
1. Calculate the angular velocity:
ω = 2π / T = 2π / 86,400 ≈ 7.27×10^-5 radians per second
2. Calculate the distance traveled on the equator:
Total distance = 2πr = 2π × (6,371,000) ≈ 40,074,000 meters
3. Calculate the difference in Earth-Sun distances:
Δd = df - ds = 1.521×10^11 - 1.47×10^11 = 5.1×10^9 meters
4. Calculate the time taken to cover the Earth-Sun distance difference:
Δt = Δd / c = (5.1×10^9) / 299,792,458 ≈ 17 seconds
5. Calculate the time difference:
Δt_diff = (total distance traveled on the equator) / ω = 40,074,000 / (7.27×10^-5) ≈ 5.51×10^8 seconds
6. Convert the time difference to seconds:
Δt_diff_seconds = Δt_diff * T ≈ (5.51×10^8) * 86,400 ≈ 4.76×10^13 seconds
Therefore, the estimated time difference between the longest day and the shortest day of a year, with the given assumptions, would be approximately 4.76×10^13 seconds.