Mathematics Behind the Tilt of the Earth's Axis

Back to the Equation of Time

To derive this formula, it is important to know exactly what an equinox is since this is used in the formula.  An equinox is defined as the moment the sun passes directly over the equator. During these times, there are exactly 12 hours during the day and 12 hours during the night. This occurs twice a year, March 21st and September 23rd (These dates are for 2007 and will vary slightly depending on the year and these dates are for 2007).

For the calculation of the following angles, imagine a celestial sphere around Earth.

           

Let alpha (α) = 23.45� (the tilt of the Earth�s axis)

d = The distance from the center of the Earth to Actual Sun (in the x direction)

b = the distance from the center of the Earth to the Mean Sun (in the x direction)

c = If you are looking down at the celestial sphere, this is the distance from the center of the Earth to the Actual Sun (in the y direction). 

N = The number of days after January 1st

ε = the angle of the mean sun after N � 80 days (N � 80 days reflects the vernal equinox about 80 days into the year)

In other words, ε = (360� / 365.24 days) * (N � 80) = 0.985653 * (N- 80)

            If ε ≥ 270�, subtract 360� from ε

            If ε ≥ 90�, subtract 180� from ε

β = the angle of the true sun on N days after January 1st

We want to find angle β because this will allow us to calculate the difference between the true and mean Sun.

The Equation of Time (due to the tilt of the Earth�s axis) = (ε � β) * 3.989

We get 3.989 from computing the number of minutes it takes the Earth to rotate one degree.

= 1440 (minutes in one day) / 361� (rotated by Earth in one day) ≈ 3.989 min/degree

 

Applying Trigonometry, gives d = b cos α, b = sin ε, and c = cos ε

Since d = b cos α, substitution yields d = (sin ε) (cos α).

To find β, we need to take the tangent and set that equal to the opposite length over the adjacent length, which would be d/c.

So tan β = d / c = arctan (0.917408 tan ε)

Here is an example to show the difference between the true and mean Sun during a certain day of the year.

What is the Equation of Time (due to the tilt of the Earth�s axis) on May 25th? (May 25th is 145 days after January 1st.)

N = 145

ε = 0.985653 * (N � 80)

   = 0.985653* (145 � 80) ≈ 64.067445

Since ε < 90, we do not have to make any adjustments.

β = arctan (0.917408 tan ε)

   = arctan (0.917408 tan 64.067445) ≈ 62.07397

Equation of time (due to the tilt of the Earth�s axis) = (ε � β) * 3.989

 = (64.067445 � 62.07397) * 3.989 = 7.95197 = 7 minutes and 52 seconds