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.
Let's call the angular velocity with which the Earth rotates around its axis omega and the average angular velocity with which it rotates around the Sun, alpha. Then the average angular velocity at which the Sun moves in the sky is omega - alpha and this is equal to 2 pi/day. So, we have
omega - 2 pi/year = 2 pi/day -->
omega = 2pi/day - 2 pi/year (1)
omega stays the constant due to conservation of angular momentum of the Earth w.r.t. its center of mass. Due to the Eart's eliptical orbit, alpha does not stay constant, 2 pi/year is just the average over an entire year. We ahve:
2pi/day = omega + alpha
whhere you compute omega from (1). This allows you to solve for the length of a day if you know alpha. Now, alpha can be computed from conservation of angular momentum of the Earth w.r.t to the Sun. We have that:
alpha r^2 = constant, so if the distance to the Sun increases then alpha decreases.
Dude Can`t Understand Please Help
Try 16.13 seconds. Is it correct?
To estimate the time difference between the longest day and the shortest day of a year on the Earth's equator, we can first calculate the difference in solar irradiance reaching the equator at the furthest and shortest distances from the Sun. Then, we can use this information to estimate the time difference.
1. Find the difference in solar irradiance:
Assuming that the Earth's axis is always perpendicular to the plane of its trajectory around the Sun and that the Sun always shines on half of the Earth, we can use the inverse square law to calculate the difference in solar irradiance between the furthest and shortest distances. The inverse square law states that the intensity of radiation decreases with the square of the distance.
I1 = I0 * (d0 / df)^2
I2 = I0 * (d0 / ds)^2
Where:
- I1 and I2 are the solar irradiance values at the furthest and shortest distances, respectively.
- I0 is the solar irradiance at the average distance between the Earth and the Sun (d0).
- df is the furthest distance from the Sun to Earth (1.521×10^8 km).
- ds is the shortest distance from the Sun to Earth (1.47×10^8 km).
2. Calculate the time difference:
Assuming that the Earth rotates at a constant rate and completes one full rotation in 24 hours, we can use the ratio of the difference in solar irradiance to the average solar irradiance to estimate the time difference between the longest and shortest days.
Δt = (I1 - I2) / I0 * 24 hours
Where:
- Δt is the estimated time difference between the longest and shortest days in hours.
3. Convert the time difference to seconds:
To convert the time difference from hours to seconds, we can multiply it by 3600 seconds:
Δt_seconds = Δt * 3600 seconds
Now we can substitute the values and calculate the estimated time difference.
Using the given values, where d0 = average distance from the Sun to Earth (149.6×10^6 km):
I0 = 1 (since we assume this as the reference value)
d0 = 149.6×10^6 km
df = 1.521×10^8 km
ds = 1.47×10^8 km
I1 = I0 * (d0 / df)^2 = 1 * (149.6×10^6 / 1.521×10^8)^2
I2 = I0 * (d0 / ds)^2 = 1 * (149.6×10^6 / 1.47×10^8)^2
Δt = (I1 - I2) / I0 * 24 hours
Δt_seconds = Δt * 3600 seconds
Let's calculate the estimated time difference.