Calculating the Equation of Time: Understanding the Earth's Orbital Velocity

[atomqwerty]

Homework Statement

The orbital velocity of the Earth is not constant. Calculate the difference between the mean solar time and true solar time, The equation of time.

Homework Equations

Don't know how to start this one! :S

The Attempt at a Solution

Idem...

Thanks a lot

 

[Andrew Mason]

Homework Statement

The orbital velocity of the Earth is not constant. Calculate the difference between the mean solar time and true solar time, The equation of time.

Homework Equations

Don't know how to start this one! :S

You might start with the definition of a solar day. A solar day is determined by the time between consecutive positions of the sun in the sky eg. time between solar noons. How does the solar day vary with the Earth's orbital speed? (hint: think of Kepler's laws of equal areas swept in equal times).

AM

 

[atomqwerty]

(hint: think of Kepler's laws of equal areas swept in equal times).

AM

So I can relate the difference between the two times by considering the eccentricity of the Earth's orbit, isn't it? I'm going to try to figure it out!
thanks

 

[Andrew Mason]

So I can relate the difference between the two times by considering the eccentricity of the Earth's orbit, isn't it? I'm going to try to figure it out!
thanks

Start by assuming what it would be like if the Earth was in a perfectly circular orbit. (The Earth rotates counterclockwise viewed from the north pole and it orbits the sun in a counterclockwise direction looking at the orbital plane from the direction of the north pole).

What would the difference be between the time to complete exactly one complete rotation of the Earth on its axis and the time between two solar noons?

AM

 

[atomqwerty]

(*)The difference would be that while a sideral day is completed in 23h56'4", a solar day needs 24h to be completed, this is, 3'56" aprox. This difference can't be constant, because of Earth's orbital velocity, that changes along the year.
With Kepler's 2nd Law, we can write the angular velocity L=mrv=mr'v' as a constant, where m is tha Earth's mass, r and r' represent the radiovector sun-earth in two given days. I could use the ellipse formula to give an expression for the time between two given days, keeping in mind (*). Am I in the correct way?
thank you

A

 

[Andrew Mason]

(*)The difference would be that while a sideral day is completed in 23h56'4", a solar day needs 24h to be completed, this is, 3'56" aprox.

Can you show how this is calculated? What is the exact angle that the Earth moves through in one solar day, assuming a circular motion? (how is this related to the average angle per solar day for an elliptical motion?).

This difference can't be constant, because of Earth's orbital velocity, that changes along the year.
With Kepler's 2nd Law, we can write the angular velocity L=mrv=mr'v' as a constant, where m is tha Earth's mass, r and r' represent the radiovector sun-earth in two given days. I could use the ellipse formula to give an expression for the time between two given days, keeping in mind (*). Am I in the correct way?

You might be. How is the angle that the Earth travels through in a solar day related to the Earth's distance from the sun? How much does this distance vary through out the year? How much does the angle vary? How much time does this variation represent?

AM

 

[atomqwerty]

Can you show how this is calculated? What is the exact angle that the Earth moves through in one solar day, assuming a circular motion? (how is this related to the average angle per solar day for an elliptical motion?).

Considering orbital velocity as a constant in circular motion, this will be 360deg/365days = 59'10.68" each day and 59'0.98" for a sideral day. Heres is the problem: I don't know how to relate this numbers with elliptical motion by using the eccentricity, that must be necessarily the element that makes the difference.

How is the angle that the Earth travels through in a solar day related to the Earth's distance from the sun?
How much does this distance vary through out the year?
How much does the angle vary?
How much time does this variation represent?
AM

If we write the angle z as tan z = d/r, being d the arc length between two consecutive days and r the sun-earth distance, then is this last variable r which changes [eccentricity e=sqrt(1-a^2/b^2)], being larger in the aphelion (b) that in one equinnox (a). By knowing e, whe may calculate angle z through the year, and thus orbital velocity, dz/dt, that leads us to time t.

 

[Andrew Mason]

Considering orbital velocity as a constant in circular motion, this will be 360deg/365days = 59'10.68" each day and 59'0.98" for a sideral day.

A year, defined as the time it takes for the Earth to next return to the same position in its orbit about the sun, is 365.25 days (day = one sidereal day = 1436 minutes). This means that the Earth would have to turn an additional .9856 of a degree (59'8.25") for the sun to be at the same position. That would take .9856x1436/360 = 3.93 minutes. So an average solar day is about 3.93 minutes longer than an actual (sidereal) day.

Heres is the problem: I don't know how to relate this numbers with elliptical motion by using the eccentricity, that must be necessarily the element that makes the difference.

What you want to do is determine the range of a solar day.

The earth, in a circular orbit, would have to rotate a further .9856 of a degree (let's not use minutes and seconds) after completing a sidereal day in order to complete a solar day. Let's call this difference the extra angle of rotation - solar (EARS).

The eccentricity of the Earth's orbit is the difference between the shortest and longest radii divided by their sum:

\epsilon = \frac{r_{max} - r_{min}}{r_{max} + r_{min}}

For the Earth orbit \epsilon = .0167

Work out the ratio of minimum to maximum radius and then apply Kepler's law of equal areas in equal times to determine the range of the EARS.

AM