The earth has an oblique position. In its course around the sun the axis of the earth is not point pointing upwards but it has an inclination of about 23.4 degrees. We call this angle the tilt of the axis of the earth and in calculations we mostly call it by the greek letter *ε (*Epsilon).

The earth rotates around its own axis. That axis is the line between north pole and south pole, not the magnetic one but the one defined by rotation. The equator defines the plane that is created by the axial rotation of the earth. By definition the axis of the earth and the equator are separated by 90 degrees.

In the drawing we define the ecliptic as a horizontal gray plane. The ecliptic is the plane that is defined by the earths rotation around the sun and that is also indicated in the drawing.

At the top and the bottom you will find the North ecliptic pole and the South ecliptic pole. They indicate the axis that defines the movement of the ecliptic. But the earth rotates around another axis en that axis runts through the North celestial pole and the South celestial pole. The ecliptic pole and the celestial pole are separated by an angle of about 23.4 degrees.

The equator – defined in green as celestial equator – is the plane created by the rotation of the earth (the axial rotation around the North celestial pole and the South celestial pole). The pools of ecliptic and equator differ 23.4 degrees, therefore there is also a difference of 23.4 degrees between ecliptic and equator.

### Average obliquity and true obliquity

The moon influences the position of the axis of the earth, this influence is called the nutation. The calculation of Epsilon that we will describe is for the mean value. Because of the nutation minor differences will occur of about a few arc seconds. In the formulas I will omit the influence of nutation.

### Precision

It is quite common to use a constant value to define the obliquity of the earth axis, typically a value like 23.44. That is perfectly fine for calculations that require a standard precision and are performed more or less for the current timespan.

But if you need a higher precision, or if you need calculations for remote times, you will have to calculate Epsilon yourself.

The diagram gives the values for Epsilon for a period of 20.000 years. The value 0 at the horizontal line indicates the year 2000 CE. It is indicated with a red dot. About 8000 or 9000 years ago the value of Epsilon was 24.2 degrees, so a difference with the current value of 0.7 degrees.

10.000 years in he future the value will be about 22.5 degrees, that is about a full degree difference with the current value.

### Formula

*ε *(Epsilon) is about 23.447. If you are a programmer you can use the value as supplied by the Swiss Ephemeris via swe_houses_armc(), it is the value ‘eps’.

The following formula is provided by the Jet Propulsion Laboratory (NASA) and is also used by the International Astronomical Union.

*ε *= 23° 26′ 21.448″
− 46.815″ *T
*− 0.00059″ *T**^*2
+ 0.001813″*T**^*3

For factor T see Factor t and delta T. Technically you should also need to correct for delta T but the effect of this is extremely small and can safely be ignored.

After 2000 years, this formula will deviate from the real position by about 1 arc second. After 10.000 years the deviation will amount to about 10 arc seconds. For the current time the formula has a high precision.

If you need to calculate Epsilon for far remote times you can use the following formula:

*ε *= 23° 26′ 21.448″
− 4680.93″ *t
*− 1.55″ *t**^*2
+ 1999.25″ *t**^*3
− 51.38″ *t**^*4
− 249.67″ *t**^*5
− 39.05″ *t**^*6
+ 7.12″ *t**^*7
+ 27.87″ *t**^*8
+ 5.79″ *t**^*9
+ 2.45″ *t**^*10

Important: you need to calculate t by calculating factor T as described in Factor t and delta T. Then you calculate t by dividing factor T by 100 .

After thousand years this formula gives a maximal deviation of 0.2 arcseconds. After 10,000 years the deviation can add up to a few arcseconds.

You cannot use this formula before 8000 BCE or after 12000 CE

### Example calculations

We use the same date and time as in the example for calculating factor T:

November 2, 2016 (Gregorian calendar) 21:17:30 UT

Factor T is 0.168388423

**Simple formula:**

ε = 23° 26' 21.448” − 46.815″ T − 0.00059″ T^2 + 0.001813″ T^3

Convert to degrees using decimal values

23.439291111111 – 0.0130041666667 * T – 1.63888888889e-07 * T^2 + 5.959274797e-09 * T^3

Calculate values for T

23.439291111111 – 0.0130041666667 * 0.168388423 – 1.63888888889e-07 * 0.028354661 + 5.959274797e-09 * 0.00477459665056

Simplify

23.439291111111 - 0.00218975111631 - 4.64701388611e-09 + 2.84531334822e-11 = 23.4371013554

Converted to degrees, minutes and seconds

`23° 26' 13,56487944''`

**Long term formula:**

ε= 23° 26′ 21.448″ − 4680.93″t− 1.55″t^2 + 1999.25″t^3 − 51.38″t^4 − 249.67″t^5 − 39.05″t^6 + 7.12″t^7 + 27.87″t^8 + 5.79″t^9 + 2.45″t^10

Convert to degrees using decimal values

23.439291111111 - 1.30025833333 * t - 0.000430555555556 * t^2 + 0.555347222222 * t^3 - 0.0142722222222 * t^4 - 0.0693527777778 * t^5 - 0.0108472222222 * t^6 + 0.00197777777778 * t^7 + 0.00774166666667 * t^8 + 0.00160833333333 * t^9 + 0.000680555555556 * t^10

Calculate values for T

t is calculated for 10.000 years. Therefore you have to divide factor T by 100:

t = 0.168388423 / 100 = 0.00168388423

23.439291111111 - 1.30025833333 * 0.00168388423 - 0.000430555555556 * 2.8354661e-06 + 0.555347222222 * 4.77459665056e-09 - 0.0142722222222 * 8.03986800449e-12 - 0.0693527777778 * 1.3538206944e-14 - 0.0108472222222 * 2.27967731756e-17 + 0.00197777777778 * 3.83871268452e-20 + 0.00774166666667 * 6.463947753e-23 + 0.00160833333333 * 1.08845396848e-25 + 0.000680555555556 * 1.8328304726e-28

Simplify

23.439291111111 - 0.00218948450242 - 1.22082566619e-09 + 2.65155898711e-09 - 1.1474678278e-13 - 9.38912257592e-16 - 2.47281664535e-19 + 7.59212061286e-23 + 5.00417288114e-25 + 1.75059679568e-28 + 1.24734295034e-31 = 23.437101628

Converted to degrees, minutes and seconds

`23° 26' 13.56586091''`

The difference with the simple formula is only 1 thousand of an arc second and therefore irrelevant.

However, in remote times the differences will noticeable.

### References

- Wikipedia: https://en.wikipedia.org/wiki/Axial_tilt
:**Meeus, Jean**. Second edition. Richmond, Virginia, 1998.*Astronomical Algorithms*