# Kepler equation

The Kepler equation is a transcendent equation for calculating elliptical orbits of celestial bodies. It results from the first two Kepler laws , which Johannes Kepler published in 1609, and reads

${\ displaystyle M = Ee \ cdot \ sin E}$ .

It connects time with the location of the celestial object . ${\ displaystyle t}$${\ displaystyle \ mathrm {P}}$

For example, the Kepler equation can be used to determine the equation of time . Part of the task is to indicate the true anomaly of the earth on its elliptical orbit around the sun .

## Derivation

To the Kepler equation on an elliptical Kepler orbit
 Lengths: Points: ${\ displaystyle a \ !:}$ major semi-axis ${\ displaystyle \ mathrm {C} \ !:}$ Focus ${\ displaystyle b \ !:}$ small semi-axis ${\ displaystyle \ mathrm {S} \ !:}$ Focus ${\ displaystyle e \! \ cdot \! a \ !:}$ linear eccentricity ${\ displaystyle \ mathrm {Z} \ !:}$ Periapsis Angle: ${\ displaystyle T \ !:}$ true anomaly ${\ displaystyle \ mathrm {P} \ !:}$ object ${\ displaystyle E \ !:}$eccentric       anomaly ${\ displaystyle \ mathrm {X} \ !:}$Auxiliary point to the       object ${\ displaystyle M \ !:}$ medium anomaly ${\ displaystyle \ mathrm {Y} \ !:}$ fictional object

The second law of Kepler, the law of surfaces , follows from the conservation of angular momentum in the two-body problem , which in astronomy is also called the Kepler problem. Only a radial force acts here in the beam from the center of gravity to the celestial object . If this force also obeys a law (such as Newton's gravitational force ), i.e. if the total flow of force through all spherical surfaces is the same (i.e. independent of the spherical radius ), then the planetary orbit is a conic section , as the first Kepler law says. In the case of the periodic return of the celestial object, the case of the ellipse, is the Kepler equation${\ displaystyle \ mathrm {S}}$${\ displaystyle \ mathrm {P}}$${\ displaystyle 1 / r ^ {2}}$${\ displaystyle r}$

${\ displaystyle M = Ee \ cdot \ sin E}$

the statement of the area theorem poured into a calculation formula. It brings the time in the form of the mean anomaly (so called by Kepler) with the position of the celestial object on its orbit (Kepler ellipse "orbit") in the form of the true anomaly (so called by Kepler) , i. i. its angular distance from the periapsis (via the auxiliary variable of the eccentric anomaly ) in a clear formula-based context. ${\ displaystyle t}$ ${\ displaystyle M}$${\ displaystyle \ mathrm {P}}$ ${\ displaystyle T}$${\ displaystyle \ mathrm {Z}}$ ${\ displaystyle E}$

Here is the numerical eccentricity of the ellipse. ${\ displaystyle e}$

### Medium anomaly

The evenly passing time can be equated with the movement of a fictitious body ( in the figures) on a circular path with constant angular velocity . For this purpose, a “ circumference ” is placed around the Kepler ellipse as an auxiliary circle that runs around it. At the time both the true object and the periapsis are assumed. Both points have the same orbital period and are in the periapsis for every integral multiple and in the apoapsis for every half-integer multiple. ${\ displaystyle \ mathrm {Y}}$${\ displaystyle \ mathrm {Y}}$ ${\ displaystyle t_ {P}}$${\ displaystyle \ mathrm {Y}}$ ${\ displaystyle \ mathrm {P}}$ ${\ displaystyle \ mathrm {Z}}$

to equation ${\ displaystyle \ mathrm {(2)}}$

The current position of the point is specified as an angle (all following angles are represented with radians ) in the center of the auxiliary circle (and ellipse) in relation to the periapsis and referred to as the mean anomaly : ${\ displaystyle \ mathrm {Y}}$${\ displaystyle \ mathrm {C}}$${\ displaystyle \ mathrm {Z}}$ ${\ displaystyle M}$

${\ displaystyle \ mathrm {(1)} \ quad M = 2 \ pi {\ frac {t-t_ {P}} {U}}}$ .

Here is the orbit period and the mean angular velocity. At this point in time the celestial object is in the periapsis, where it has the smallest distance to its center of gravity . ${\ displaystyle U}$${\ displaystyle {2 \ pi} / U}$${\ displaystyle t_ {P}}$ ${\ displaystyle \ mathrm {S}}$

According to Kepler's second law, the beam of the body sweeps over the same area in the same period of time. Since the time share (in the circulation) is proportional to the share of the circle sector in the circumference, the share of the elliptical partial area in the ellipse is the same as that of the circle sector in the circumference: ${\ displaystyle {\ overline {\ mathrm {SP}}}}$${\ displaystyle \ mathrm {P}}$${\ displaystyle \ mathrm {SPZ}}$${\ displaystyle \ mathrm {CYZ}}$

${\ displaystyle \ mathrm {(2)} \ quad {\ frac {\ operatorname {area} \, \ mathrm {CYZ}} {\ operatorname {area} \, \ mathrm {SPZ}}} = {\ frac {\ pi a ^ {2}} {\ pi ab}} = {\ frac {a} {b}}}$ .

It is the semi-major axis of the ellipse, while the radius of the perimeter, the minor axis of the ellipse. The relation between the ellipse and the circumference is affine, i. In other words, the ellipse is in every parallel to the minor semiaxis the circumference "compressed" with this ratio. ${\ displaystyle a}$${\ displaystyle b}$${\ displaystyle b / a}$

to equation ${\ displaystyle \ mathrm {(4)}}$
to equation ${\ displaystyle \ mathrm {(3)}}$

### Eccentric anomaly

A projection of the point onto the circumference parallel to the semi-minor axis creates the auxiliary point , the angle of which in the center to the periapsis was called an eccentric anomaly by Kepler . The affinity establishes the following relationship: ${\ displaystyle \ mathrm {P}}$${\ displaystyle \ mathrm {X}}$${\ displaystyle \ mathrm {C}}$${\ displaystyle \ mathrm {Z}}$ ${\ displaystyle E}$

${\ displaystyle \ mathrm {(3)} \ quad \ operatorname {area} \, \ mathrm {SXZ} = {\ frac {a} {b}} \ operatorname {area} \, \ mathrm {SPZ}}$ .

After inserting equation into equation, it follows: ${\ displaystyle \ mathrm {(2)}}$${\ displaystyle \ mathrm {(3)}}$

${\ displaystyle \ mathrm {(4)} \ quad \ operatorname {area} \, \ mathrm {SXZ} = \ operatorname {area} \, \ mathrm {CYZ}}$ .

### Kepler equation

to the equations and${\ displaystyle \ mathrm {(5)}}$${\ displaystyle \ mathrm {(6)}}$
to equation ${\ displaystyle \ mathrm {(7)}}$

The equation implicitly finds the relationship between the eccentric anomaly (point ) and the mean anomaly (point ) , which fulfills Kepler's second law . An explicit relationship results from the following steps: ${\ displaystyle \ mathrm {(4)}}$${\ displaystyle \ mathrm {X}}$${\ displaystyle \ mathrm {Y}}$

If the driving beam travels the angle in a period and sweeps over the area , it sweeps over the angle and an area that is smaller by the factor up to that point in time : ${\ displaystyle {\ overline {\ mathrm {CY}}}}$${\ displaystyle U}$${\ displaystyle 2 \ pi}$${\ displaystyle \ pi a ^ {2}}$${\ displaystyle t}$${\ displaystyle M}$${\ displaystyle M / 2 \ pi}$

${\ displaystyle \ mathrm {(5)} \ quad \ displaystyle \ operatorname {area} \, \ mathrm {CYZ} = {\ frac {a ^ {2}} {2}} M}$ .

The analogous consideration for the driving beam over the angle results in: ${\ displaystyle {\ overline {\ mathrm {CX}}}}$${\ displaystyle E}$

${\ displaystyle \ mathrm {(6)} \ quad \ displaystyle \ operatorname {area} \, \ mathrm {CXZ} = {\ frac {a ^ {2}} {2}} E}$ .

The area consists of the partial areas and : ${\ displaystyle \ mathrm {CXZ}}$${\ displaystyle \ mathrm {CXS}}$${\ displaystyle \ mathrm {SXZ}}$

${\ displaystyle \ mathrm {(7)} \ quad \ displaystyle \ operatorname {area} \, \ mathrm {CXZ} = \ operatorname {area} \, \ mathrm {CXS} + \ operatorname {area} \, \ mathrm { SXZ}}$ .

The partial area (outlined in light blue in the illustration) is a straight-line bounded triangle with the base and the height  : ${\ displaystyle \ mathrm {CXS}}$${\ displaystyle e \ cdot a}$${\ displaystyle a \ cdot \ sin E}$

${\ displaystyle \ mathrm {(8)} \ quad \ displaystyle \ operatorname {area} \, \ mathrm {CXS} = {\ frac {ea \ cdot a \ sin E} {2}} = {\ frac {a ^ {2}} {2}} \, e \ sin E}$ .

${\ displaystyle e}$is the numerical eccentricity of the ellipse and the linear one , which indicates the distance between the center and the focal point . ${\ displaystyle ea = {\ sqrt {a ^ {2} -b ^ {2}}}}$

According to the equation , the partial area is the same size as the area whose value is given in the equation . ${\ displaystyle \ mathrm {SXZ}}$${\ displaystyle \ mathrm {(4)}}$${\ displaystyle \ mathrm {CYZ}}$${\ displaystyle \ mathrm {(5)}}$

Substituting the equations , and the equation becomes the equation ${\ displaystyle \ mathrm {(6)}}$${\ displaystyle \ mathrm {(8)}}$${\ displaystyle \ mathrm {(5)}}$${\ displaystyle \ mathrm {(7)}}$

${\ displaystyle \ mathrm {(9)} \ quad \ displaystyle {\ frac {a ^ {2}} {2}} E = {\ frac {a ^ {2}} {2}} e \ sin E + {\ frac {a ^ {2}} {2}} M}$ .

This finally results in the Kepler equation:

${\ displaystyle Ee \ cdot \ sin E = M}$ .

### Solution of the Kepler equation

The Kepler equation cannot be solved in closed form for the eccentric anomaly . Examples of how it can be used to determine from the mean anomaly : ${\ displaystyle E (t)}$${\ displaystyle E (t)}$${\ displaystyle M (t) = 2 \ pi {\ frac {t-t_ {P}} {U}}}$

1. ${\ displaystyle EM}$is an odd, with periodic function in . As such, it can be expanded into a Fourier series that is for all and converges, and that is ${\ displaystyle 2 \ pi}$${\ displaystyle M}$${\ displaystyle M \ in \ mathbb {R}}$${\ displaystyle e \ in \ mathbb {R}}$
${\ displaystyle F (M): = EM = e \ cdot \ sin E = 2 \ cdot \ sum _ {n = 1} ^ {\ infty} {\ frac {J_ {n} (ne)} {n}} \ sin (nM)}$
with a
first- order Bessel function . All other values can easily be calculated from the values for : ${\ displaystyle J_ {n}}$ ${\ displaystyle n}$
${\ displaystyle F (M ^ {\ prime})}$${\ displaystyle M ^ {\ prime} \ in [0, \ pi]}$${\ displaystyle F (M)}$
${\ displaystyle F (M) = s \ cdot F {\ bigl (} s \ cdot M ^ {\ prime} {\ bigr)}}$
with (
Gaussian brackets ), and , so that . ${\ displaystyle k: = {\ bigl \ lfloor} {\ tfrac {M} {\ pi}} {\ bigr \ rfloor} \ in \ mathbb {Z}}$${\ displaystyle s: = (- 1) ^ {k} \ in \ {1, -1 \}}$${\ displaystyle M ^ {\ prime}: = M - {\ bigl (} k + {\ tfrac {1-s} {2}} {\ bigr)} \ pi \ in [- \ pi, + \ pi]}$${\ displaystyle s \ cdot M ^ {\ prime} \ in [0, + \ pi]}$
2. A zero of the function ${\ displaystyle E}$
${\ displaystyle f (E) = Ee \ cdot \ sin EM}$
is a solution to the Kepler equation. The zero can be calculated numerically using the Newton method as follows:
${\ displaystyle E_ {n + 1} = E_ {n} - {\ frac {f (E_ {n})} {f '(E_ {n})}} = E_ {n} - {\ frac {E_ { n} -e \ sin (E_ {n}) - M} {1-e \ cos (E_ {n})}}}$ .
The initial value is suitable for most elliptical orbits . For eccentricities can be taken. ${\ displaystyle E_ {0} = M}$${\ displaystyle 0 {,} 8 ${\ displaystyle E_ {0} = \ pi}$
3. A more stable , but slower converging method is based on Banach's fixed point theorem :
${\ displaystyle E_ {0} = M, \ qquad E_ {n + 1} = M + e \ cdot \ sin E_ {n}}$ .
4. For small eccentricity , the following can also be approximated : ${\ displaystyle e}$${\ displaystyle E}$
${\ displaystyle E = M + e \ cdot \ sin M + {\ frac {1} {2}} e ^ {2} \ cdot \ sin 2M}$
The error here is of the order of magnitude . In the case of the
earth and its eccentricity , the error is after the 5th decimal place for a limited period of time  . ${\ displaystyle {\ mathcal {O}} (e ^ {3})}$${\ displaystyle e = 0 {,} 0167}$
5. A solution for in the manner of the Lagrangian inversion formula is the Maclaurin series into that for linearly converges. So is , then it converges for linear. The coefficients of the numerator polynomials in are recorded in the sequence A306557 in OEIS . ${\ displaystyle e <1}$${\ displaystyle M}$   ${\ displaystyle {\ begin {array} {ll} \ textstyle E = {\ frac {1} {1-e}} M \! \! \! \! & - {\ frac {e} {(1-e ) ^ {4}}} {\ frac {M ^ {3}} {3!}} + {\ Frac {(e + 9e ^ {2})} {(1-e) ^ {7}}} { \ frac {M ^ {5}} {5!}} - {\ frac {(e + 54e ^ {2} + 225e ^ {3})} {(1-e) ^ {10}}} {\ frac {M ^ {7}} {7!}} \\ & + {\ frac {(e + 243e ^ {2} + 4131e ^ {3} + 11025e ^ {4})} {(1-e) ^ { 13}}} {\ frac {M ^ {9}} {9!}} - {\ frac {(e + 1008e ^ {2} + 50166e ^ {3} + 457200e ^ {4} + 893025e ^ {5} )} {(1-e) ^ {16}}} {\ frac {M ^ {11}} {{11}!}} \\ & + {\ frac {(e + 4077e ^ {2} + 520218e ^ {3} + 11708154e ^ {4} + 70301925e ^ {5} + 108056025e ^ {6})} {(1-e) ^ {19}}} {\ frac {M ^ {13}} {{13}! }} \ mp \ cdots, \ end {array}}}$
${\ displaystyle | M | <\ cosh ^ {- 1} (e ^ {- 1}) - {\ sqrt {1-e ^ {2}}}}$ ${\ displaystyle 0 \ leq e \ lessapprox 0 {,} 031803066}$${\ displaystyle | M | \ leq \ pi}$
${\ displaystyle e}$

## Solution of some subtasks in the Kepler problem

### True anomaly

For a celestial body on a Kepler orbit, the location or the true anomaly must be specified for the point in time or for the associated mean anomaly . The eccentric anomaly is first determined using the Kepler equation (see above). From the latter the true anomaly follows according to one of the following relationships: ${\ displaystyle t}$${\ displaystyle M (t)}$${\ displaystyle T (t)}$${\ displaystyle E (t)}$${\ displaystyle T (t)}$

${\ displaystyle \ tan {\ frac {T} {2}} = {\ sqrt {\ frac {1 + e} {1-e}}} \ cdot \ tan {\ frac {E} {2}}}$

or

${\ displaystyle \ cos T = {\ frac {a \ cos E-ae} {a-ae \, \ cos E}} = {\ frac {\ cos Ee} {1-e \ cos E}}}$

Here is the linear eccentricity of the orbit ellipse. To resolve by , a distinction between the cases and is necessary. ${\ displaystyle ae = {\ sqrt {a ^ {2} -b ^ {2}}}}$${\ displaystyle T}$${\ displaystyle 0 \ leq E \ leq \ pi}$${\ displaystyle \ pi \ leq E \ leq 2 \ pi}$

Remarks
• The denominator of the second formula gives the distance from the celestial object to the focal point :${\ displaystyle r}$${\ displaystyle s}$
${\ displaystyle r = a-ae \, \ cos E}$
• The formulas can easily be solved for or , it results:${\ displaystyle \ tan {\ tfrac {E} {2}}}$${\ displaystyle \ cos E}$
${\ displaystyle \ tan {\ frac {E} {2}} = {\ sqrt {\ frac {1-e} {1 + e}}} \ cdot \ tan {\ frac {T} {2}}}$
and
${\ displaystyle \ cos E = {\ frac {a \ cos T + ae} {a + ae \ cos T}} = {\ frac {\ cos T + e} {1 + e \ cos T}}}$

There are numerous other relationships between the true anomaly of the eccentric anomaly and the mean anomaly , which have been developed in the long history of celestial mechanics . In particular, can the true anomaly - without going through the Kepler equation - directly from a special differential equation in calculating what is for numerical approximations of interest. ${\ displaystyle T,}$${\ displaystyle E}$${\ displaystyle M}$${\ displaystyle M}$

In particular, the true anomaly can also be approximated here by the mean anomaly for small eccentricities; the useful approximation is obtained ${\ displaystyle T}$${\ displaystyle M}$

${\ displaystyle T = M + 2e \ sin (M) + {\ frac {5} {4}} e ^ {2} \ sin (2M) + {\ mathcal {O}} (e ^ {3}). }$

The difference - is called the midpoint equation . ${\ displaystyle T}$${\ displaystyle M}$

### Medium anomaly

For a celestial body on a Kepler orbit with the true anomaly , the associated mean anomaly or the associated time must be specified. It is the opposite of the above task. ${\ displaystyle T}$${\ displaystyle M (T)}$${\ displaystyle t (T)}$

Starting from the eccentric anomaly arises to ${\ displaystyle T}$

${\ displaystyle E = 2 \ arctan _ {\ tfrac {T} {2}} \ left ({\ sqrt {\ frac {1-e} {1 + e}}} \ cdot \ tan {\ frac {T} {2}} \ right)}$ .

The position parameter index at returns the value of the arctangent that is closest to this ( ) (see arctangent with position parameter ). The Kepler equation gives the corresponding mean anomaly ${\ displaystyle {\ tfrac {T} {2}}}$${\ displaystyle \ arctan}$${\ displaystyle {\ tfrac {T} {2}}}$${\ displaystyle E}$

${\ displaystyle M (t) = E (t) - {\ frac {180 ^ {\ circ}} {\ pi}} \ cdot e \ cdot \ sin E (t)}$ .

From the linear equation for the path element it finally follows:

${\ displaystyle t = {\ frac {M-M_ {0}} {\ dot {M}}}}$
example

Passage times of the four apexes
of the Earth's orbit ellipses: The orbital elements valid for the earth are given under middle Kepler elements . The time used in the referring article is calculated in Julian centuries. It measures in days, so divide the linear coefficients of time by 36525 to get and . However, the very slow change in numerical eccentricity is neglected ( ). The zero point of time - and thus also of  - is January 1st, 2000, 12:00
UT . The true anomaly in the Earth's perihelion passage in 2000 is 360 ° (not zero!), In 2001 it is 720 °, etc. ${\ displaystyle T}$${\ displaystyle t}$${\ displaystyle T}$${\ displaystyle {\ dot {M}}}$${\ displaystyle {\ dot {e}}}$${\ displaystyle {\ dot {e}} = 0}$${\ displaystyle T}$${\ displaystyle t}$

 Perihelion 2000 Spring side parting Aphelion Autumn side parting Perihelion 2001 True anomaly ${\ displaystyle T / ^ {\ circ}}$ 360 450 540 630 720 time ${\ displaystyle t / {\ text {d}}}$ 2.511 91.883 185.140 278,398 367.770 Time interval ${\ displaystyle \ Delta t / {\ text {d}}}$ 89.372 93.258 93.258 89.372

The distance between the mean perihelion passages ( anomalistic year ) is The mean perihelion times calculated in this way can differ by several days from the real (especially moon-disturbed) value. ${\ displaystyle J_ {an} = 360 ^ {\ circ} / {\ dot {M}} = 365,260 {\ text {d}}.}$

The true anomaly indicates the direction of a celestial body on its Kepler orbit for a time . The corresponding distance - the orbit radius - can be calculated as follows: ${\ displaystyle t}$

${\ displaystyle r = r (T (t)) = r (t) = a \ cdot {\ frac {1-e ^ {2}} {1 + e \ cdot \ cos T}}}$
${\ displaystyle r \ !:}$ Distance (orbit radius)
${\ displaystyle a \ !:}$ major semi-axis of the ellipse
${\ displaystyle e \ !:}$ numerical eccentricity
${\ displaystyle T \ !:}$ true anomaly

### Track speed

The time change of the true anomaly corresponds to the angular velocity with respect to the center of gravity. The normal component of the speed follows directly from ${\ displaystyle \ omega}$

${\ displaystyle v _ {\ perp} = {\ dot {T}} \ cdot r.}$

The radial speed is the change in the radius of the orbit over time:

${\ displaystyle v_ {r} = {\ dot {r}}}$

Then follows for the orbital velocity or orbital velocity ${\ displaystyle v}$${\ displaystyle v ^ {2} = v _ {\ perp} ^ {2} + v_ {r} ^ {2}.}$

${\ displaystyle v = v (T (t), r (t)) = v (t) = {\ sqrt {({\ dot {T}} \ cdot r) ^ {2} + {\ dot {r} } ^ {2}}}}$
${\ displaystyle v \ !:}$ Track speed
${\ displaystyle T \ !:}$ true anomaly
${\ displaystyle r \ !:}$ Orbit radius

The path speed can be more easily derived from the area set using the hodograph : ${\ displaystyle {\ vec {\ dot {r}}}}$

${\ displaystyle v ^ {2} = {\ frac {C ^ {2}} {p}} \ left ({\ frac {2} {r}} - {\ frac {1} {a}} \ right) }$
${\ displaystyle C \ !:}$ specific angular momentum as the central parameter of movement
${\ displaystyle C = v _ {\ mathrm {max}} \ cdot r _ {\ mathrm {min}} = v _ {\ mathrm {min}} \ cdot r _ {\ mathrm {max}}}$
${\ displaystyle p \ !:}$ Half parameter as a characteristic path element
${\ displaystyle p = 2 \ cdot {\ frac {r _ {\ mathrm {min}} \ cdot r _ {\ mathrm {max}}} {r _ {\ mathrm {min}} + r _ {\ mathrm {max}}} } = {\ frac {b ^ {2}} {a}}}$
${\ displaystyle a \ !:}$ major semi-axis
${\ displaystyle b \ !:}$ small semi-axis
${\ displaystyle {\ frac {C ^ {2}} {p}} = G \ cdot M}$with gravitational constant and mass of the central body${\ displaystyle G}$${\ displaystyle M}$

From this follow the minimum and maximum speed in the apocenter and pericenter of an elliptical orbit:

${\ displaystyle v _ {\ mathrm {max}} ^ {2} = {\ frac {C ^ {2}} {p \ cdot a}} \ cdot {\ frac {1 + e} {1-e}} \ qquad v _ {\ mathrm {min}} ^ {2} = {\ frac {C ^ {2}} {p \ cdot a}} \ cdot {\ frac {1-e} {1 + e}}}$
${\ displaystyle e \ !:}$ numerical eccentricity

## Applying the Kepler equation to the equation of time

The quantitative, i.e. computational, treatment of the equation of time is essentially an application of the Kepler equation - namely for the portion of the equation of time resulting from the elliptical orbital motion of the earth. In particular, the location of the earth on its elliptical orbit (also Kepler orbit ) is determined at a given point in time.

### Definitions of the equation of time

First definition:

${\ displaystyle \ mathrm {(10)} \ quad \ mathrm {ZG} = \ mathrm {WOZ} - \ mathrm {MOZ}}$

The value of the true local time (WOZ) or mean local time (MOZ) corresponds to the respective position of the true or a fictitious mean sun in the sky. Since the time of day is related to the rotation of the earth around its axis, only the respective right ascension (not the declination ) of the sun (s) is of interest . In other words: Of the apparent annual movements of the true sun that occur at right angles to one another, only the one on the celestial equator is of interest , but not the periodic ascent and descent. The mean sun, which represents the uniformly passing time, revolves around the celestial equator. The equation of time is proportional to the difference between the right ascensions of the fictitious central sun and the real, true sun. ${\ displaystyle \ alpha _ {M}}$${\ displaystyle \ alpha}$

Second definition:

${\ displaystyle \ mathrm {(11)} \ quad {\ text {ZG}} = 4 (\ alpha _ {M} - \ alpha) \ quad [{\ text {min}}]}$

The factor 4 results from the fact that two celestial bodies with 1 ° right ascension difference pass the meridian at a time interval of 4 minutes. The order of the two subtraction terms has been reversed because the directions for the hour angle (WOZ and MONT correspond to it) and the right ascension are defined opposite to one another. ${\ displaystyle \ tau}$${\ displaystyle \ alpha}$

### method

The right ascension (equation ) of the sun to be determined at a certain point in time corresponds, in a heliocentric view, to the equatorial length of the earth, which can be easily calculated from its ecliptical length (second of the adjacent figures). With the help of the Kepler equation, the true anomaly (first of the adjacent figures) is determined, from which it is then determined by changing the reference point . ${\ displaystyle t}$${\ displaystyle \ alpha}$${\ displaystyle \ mathrm {(11)}}$${\ displaystyle \ lambda}$${\ displaystyle V}$${\ displaystyle \ lambda}$

### Application of the Kepler equation

Current anomalies of the earth (at time t) on its elliptical orbit around the sun:
V - true, M - mean, E - eccentric anomaly
B - sun, X - earth, Y - fictional earth, P - perihelion , A -  aphelion , K - 1st Jan. point
lower left: V and M as functions of time
Transition from the heliocentric elliptical earth orbit (left, with true earth X and fictitious earth Y) to the geocentric sun orbit (ecliptic circle, right, with true sun S and fictitious sun S ')
"Bringing down" the true sun on the equator: Determination of its right ascension angle α from its ecliptical
length angle λ S ″: central sun on the equator

Mean anomaly:

The mean anomaly formulated in general in equation reads in connection with the equation of time: ${\ displaystyle \ mathrm {(1)}}$

${\ displaystyle \ mathrm {(12)} \ quad M (t) = {\ frac {360 ^ {\ circ}} {J _ {\ text {an}}}} \ cdot (t-t_ {P})}$
${\ displaystyle J _ {\ text {an}}}$: anomalous year between two passages of the perihelion
${\ displaystyle t_ {P}}$: Timing of perihelion passage

In the case of perihelion, the mean anomaly has the following value:

${\ displaystyle \ mathrm {(13)} \ quad M_ {0} = - {\ frac {360 ^ {\ circ}} {J _ {\ text {an}}}} \ cdot t_ {P}}$

With the equation of time, it is common to publish the values ​​of a calendar year in the corresponding astronomical yearbook . January 1st 12:00 (UT) of the corresponding year is used as the zero point for , so that currently applies to about 2 to 3 days and from that for about 2 ° to 3 °. It has conveniently become common practice to publish the new value for in advance as a so-called annual constant. ${\ displaystyle t}$${\ displaystyle t_ {P}}$${\ displaystyle M_ {0}}$${\ displaystyle M_ {0}}$

With and from January 1st 12:00 (UT) equation (12) becomes  :${\ displaystyle M_ {0}}$${\ displaystyle t}$

${\ displaystyle M (t) = M_ {0} + {\ frac {360 ^ {\ circ}} {J _ {\ text {an}}}} \ cdot t}$

Kepler equation:

${\ displaystyle M (t) = E (t) - {\ frac {180 ^ {\ circ}} {\ pi}} \ cdot e \ cdot \ sin E (t)}$

With the mean anomaly corresponding to the given point in time and the eccentricity of the earth's orbit , the eccentric anomaly is determined with the aid of the Kepler equation . ${\ displaystyle M}$${\ displaystyle e}$${\ displaystyle E}$

True anomaly:

When dealing with the equation of time, the symbol (instead of the above) is mostly used for the true anomaly . ${\ displaystyle V}$${\ displaystyle T}$

The eccentric anomaly leads to the true anomaly in a purely geometrical view in the ellipse and its periphery (first of the adjacent figures) as follows : ${\ displaystyle E}$${\ displaystyle V}$

${\ displaystyle \ tan \ left ({\ frac {V (t)} {2}} \ right) = \ kappa \ cdot \ tan \ left ({\ frac {E (t)} {2}} \ right) }$
${\ displaystyle \ mathrm {(14)} \ quad \ kappa = {\ sqrt {\ frac {1 + e} {1-e}}}}$ ... an ellipse constant

The Kepler problem is solved with the determination of the true anomaly of the earth. The determination of the equation of time is concluded in the following.

### True anomaly of the earth → right ascension of the sun

True anomaly of the earth → ecliptical longitude of the earth → ecliptical longitude of the sun:

Seen from the earth, the movement of the earth around the sun is reflected in the apparent movement of the sun in the ecliptic, the intersection of the plane of the earth's orbit with the directional sphere around the earth as the center (see second of the adjacent figures). The ecliptical length of the earth and the ecliptical length of the sun are therefore synonyms with the symbol${\ displaystyle \ lambda.}$

The point of reference for the ecliptical longitude (and also the right ascension) is the vernal equinox , according to common custom . The ecliptical longitude of the sun is obtained by adding the angle between perihelion  P and the point corresponding to the vernal equinox  (F) to the angle related to the perihelion of the earth's orbit : ${\ displaystyle \ lambda (t)}$${\ displaystyle V (t)}$${\ displaystyle L}$

${\ displaystyle \ mathrm {(15)} \ quad \ lambda (t) = V (t) + L}$

The value of is negative. Among the almost constant basic sizes is the one that changes the most over time because of the slow approach between the vernal equinox or point  (F) and perihelion. It is therefore not only set anew every year as a so-called annual constant, but also permanently changed with the following equation: ${\ displaystyle L}$${\ displaystyle L}$${\ displaystyle L_ {0}}$

${\ displaystyle \ mathrm {(16)} \ quad L (t) = L_ {0} + {\ tfrac {0 {,} 0172 ^ {\ circ}} {J _ {\ text {tr}}}} \ cdot t}$

The vernal equinox and perihelion approach each other with the tropical year (time for two successive passages of the vernal equinox or the point  (F) ). Taking into account the equation , instead of the equation , write: ${\ displaystyle \ approx {\ tfrac {0 {,} 0172 ^ {\ circ}} {J _ {\ text {tr}}}}.}$ ${\ displaystyle J _ {\ text {tr}}}$${\ displaystyle \ mathrm {(16)}}$${\ displaystyle \ mathrm {(15)}}$

${\ displaystyle \ mathrm {(17)} \ quad \ lambda (t) = V (t) + L_ {0} + {\ tfrac {0 {,} 0172 ^ {\ circ}} {J _ {\ text {tr }}}} \ cdot t}$

The value of is negative. ${\ displaystyle L_ {0}}$

Ecliptical longitude of the sun → right ascension of the sun:

In addition to the ellipticity of the earth's orbit, the non-perpendicular position of the earth's axis and its change in direction relative to the sun cause the equation of time.

The right ascension of the sun can be z. B. with generally known transformation equations or with the following simple relation in the corresponding right-angled spherical triangle (see third of the adjacent figures) from the ecliptical length : ${\ displaystyle \ alpha}$${\ displaystyle \ lambda}$

${\ displaystyle \ mathrm {(18)} \ quad \ alpha (t) = \ arctan (\ tan \ lambda (t) \ cdot \ cos \ varepsilon)}$

${\ displaystyle \ varepsilon}$is the obliquity of the Earth's axis: . ${\ displaystyle \ varepsilon = 23 {,} 44 ^ {\ circ}}$

### Right ascension of the central sun

The movement of the central sun S ″ (third of the illustrations on the right) on the equator makes the evenly passing time clear as that of the fictitious earth orbiting on the earth's orbit (point Y). Its course is to be coupled as closely as possible to that of the true sun, so that it roughly “averages” its course. This was achieved with the following definition:

${\ displaystyle \ mathrm {(19)} \ quad \ alpha _ {M} (t) = L (t) + M (t)}$

If one neglects the change in time , the following also applies: ${\ displaystyle L}$

${\ displaystyle \ left (20 \ right) \ quad \ alpha _ {M} (t) = L_ {0} + M_ {0} + {\ tfrac {360 ^ {\ circ}} {J _ {\ text {tr }}}} \ cdot t}$

### Equation of time

The two right ascensions and required to apply the equation of time are found. ${\ displaystyle \ mathrm {(11)}}$${\ displaystyle \ alpha _ {M}}$${\ displaystyle \ alpha}$

${\ displaystyle \ mathrm {(11)} \ quad {\ text {ZG}} = 4 (\ alpha _ {M} - \ alpha) \ quad [{\ text {min}}]}$

### Calculation example

The equation of time for April 2, 2015, 12:00 UT (t = 91 days) is to be calculated.

The year constants 2015 are:

${\ displaystyle M_ {0} = - 2 {,} 3705 ^ {\ circ}}$
${\ displaystyle J_ {an} = 365 {,} 259991 {\ text {days}}}$
${\ displaystyle J_ {tr} = 365 {,} 242907 {\ text {days}}}$
${\ displaystyle e = 0 {,} 016703}$
${\ displaystyle \ varepsilon = 23 {,} 43734 ^ {\ circ}}$
${\ displaystyle L_ {0} = - 76 {,} 8021 ^ {\ circ}}$

The bills are:

${\ displaystyle M (t) = M_ {0} + {\ frac {360 ^ {\ circ}} {J _ {\ text {an}}}} \ cdot t = 87 {,} 3190 ^ {\ circ}}$
${\ displaystyle \ mathrm {(16)} \ quad L (t) = L_ {0} + {\ tfrac {0 {,} 0172 ^ {\ circ}} {J _ {\ text {tr}}}} \ cdot t = -76 {,} 7978 ^ {\ circ}}$
${\ displaystyle M (t) = E (t) - {\ frac {180 ^ {\ circ}} {\ pi}} \ cdot e \ cdot \ sin E (t) \ quad \ rightarrow \ quad E (t) = 88 {,} 2756 ^ {\ circ}}$
${\ displaystyle \ textstyle \ kappa = {\ sqrt {\ frac {1 + e} {1-e}}} = 1 {,} 0168445}$
${\ displaystyle \ tan \ left ({\ frac {V (t)} {2}} \ right) = \ kappa \ cdot \ tan \ left ({\ frac {E (t)} {2}} \ right) \ quad \ rightarrow \ quad V (t) = 89 {,} 2325 ^ {\ circ}}$
${\ displaystyle \ mathrm {(17)} \ quad \ lambda (t) = V (t) + L (t) = 12.4347 ^ {\ circ}}$
${\ displaystyle \ mathrm {(18)} \ quad \ alpha (t) = \ arctan (\ tan \ lambda (t) \ cdot \ cos \ varepsilon) = 11 {,} 4369 ^ {\ circ}}$
${\ displaystyle \ mathrm {(19)} \ quad \ alpha _ {M} (t) = L (t) + M (t) = 10 {,} 5212 ^ {\ circ}}$
${\ displaystyle \ mathrm {(11)} \ quad {\ text {ZG}} (t) = 4 {\ frac {\ text {min}} {^ {\ circ}}} \ cdot (\ alpha _ {M } (t) - \ alpha (t)) = - 3 {,} 6629 {\ text {min}} = - 3 {\ text {min}} {\ text {40}} {\ text {sec}}}$

On April 2nd, 2015, 12:00 UT, the equation of time has the value:

${\ displaystyle {\ text {ZG}} (t = 91 {\ text {days}}) = - 3 {\ text {min}} {\ text {40}} {\ text {sec}}}$

### Equation of time values ​​for the passage of marked orbital points

The equation of time values ​​for the passage of marked points through the earth on its orbit (or through the sun on the ecliptic) are independent of the calendar and thus of the annual constant : spring, summer, autumn and winter starting point, perihelion and aphelion. ${\ displaystyle M_ {0}}$

Time equation values ​​and times for the passage of marked path points *)
F-beginning S-beginning H-beginning W beginning Perihelion Aphelion
λ / ° 0 90 180 270 L 0 L 0  + 180
ZG / min −7.44 −1.74 +7.48 +1.70 −4.50 −4.50
t P / d **) 76.234 168.990 262.641 352,485 0 182.621

*) The values ​​apply to the year 2004 with L 0  = −76.99 ° and J tr  = 365.2428 days.
**) The times given refer to the passage of the perihelion, not, as in the example above, to January 1st, 12:00 UT.

Their calculation is easier than that for any point in time because the Kepler equation does not have to be solved. From the given ecliptical length one of the marked points is easy to find to the true (Eq. ) And further to the eccentric anomaly. The mean anomaly follows from the latter with the rearranged Kepler equation , i.e. the orbit point of the fictitious mean earth. The ecliptical length of the perihelion added to the latter (Eq. ) Is the searched mean right ascension ( minuend in the equation of time ). The true right ascension (subtrahend) is identical to the ecliptical length of the points spring to winter . The coordinate transformation (Eq. ) Only results in small value differences for the points perihelion and aphelion . ${\ displaystyle E = f (M)}$${\ displaystyle \ lambda}$${\ displaystyle \ mathrm {(15)}}$${\ displaystyle M = f (E)}$${\ displaystyle \ mathrm {(19)}}$${\ displaystyle \ alpha _ {M}}$${\ displaystyle \ mathrm {(11)}}$${\ displaystyle \ alpha}$${\ displaystyle \ lambda}$${\ displaystyle \ mathrm {(18)}}$

In the procedure of starting the calculation with a given ecliptical length or a given true anomaly, one obtains not only the equation of time but also the time that has passed since the earth's perihelion passage. This is the time that represents the mean anomaly and it is calculated from the intermediate result for the mean anomaly with the aid of the equation to be converted accordingly . ${\ displaystyle M}$${\ displaystyle \ mathrm {(12)}}$

This procedure is sometimes recommended for the general work determining time equation tables. You save yourself the time-consuming solving of the Kepler equation, but you can only find values ​​for the desired points in time by trial and error or, if the results are sufficiently dense, by interpolating .

## literature

• Andreas Guthmann: Introduction to celestial mechanics and ephemeris calculus. BI-Wiss.-Verl., Mannheim 1994, ISBN 3-411-17051-4 .
• Peter Colwell: Solving Kepler's equation over three centuries . Ed .: Willmann-Bell. Richmond, VA 1993, ISBN 0-943396-40-9 , pp. 202 .

## Individual evidence

1. J.-L. Lagrange, Sur leproblemème de Kepler , in Mémoires de l'Académie Royale des Sciences de Berlin , vol. 25, 1771, pp. 204-233
2. ^ Peter Colwell: Bessel functions and Kepler's equation (=  Amer. Math. Monthly . Volume 99 , no. 1 ). January 1992, p. 45-48 (English).
3. § II.6.67 Numerical Methods. Guthmann , p. 128 f.
4. § II.6.66 Series development of the eccentric anomaly. Guthmann , p. 125 ff.
5. ^ Siegfried Wetzel: The equation of time for non-astronomers. German Chronometry Society, Announcements No. 111, Fall 2007, Appendix 3.
6. R. Strebel: The Kepler equation. ( Memento of the original from August 13, 2011 in the Internet Archive ) Info: The archive link was inserted automatically and has not yet been checked. Please check the original and archive link according to the instructions and then remove this notice. October 2001, chap. 1.3 and 5.1.
7. Tasks to § II.5. Guthmann , p. 122 f.
8. a b 10th and 11th task to § II.5. Guthmann, p. 123.
9. a b § II.5.58 The Hodograph. Guthmann, p. 114 f.
10. The symbols used here are the same as in:
Sundial manual, calculation of the equation of time. Deutsche Gesellschaft für Chronometrie eV, Fachkreis Sonnenuhren, 2006, pp. 43–49.
11. Because of the leap day regulation in the calendar, both values ​​fluctuate slightly within the four-year period: Δt P ≈ ¾day, ΔM 0 ≈ ¾ °.
12. ^ Siegfried Wetzel: The equation of time for non-astronomers. ( Memento from April 7, 2014 in the Internet Archive ) German Society for Chronometry, Communications No. 111, Autumn 2007, Appendix 3.
13. Manfred Schneider: Heaven Mechanics, Volume II: System Models. BI-Wissenschaftsverlag, 1993, ISBN 3-411-15981-2 , p. 507.
14. Conversely, this relationship allows the ecliptical longitude and the spring point F as a reference point (both for and for ) to be reflected back onto the earth's orbit (see adjacent figure, right → left).${\ displaystyle \ lambda}$ ${\ displaystyle L}$${\ displaystyle \ alpha}$
15. Sign for angle difference and location in the adjacent figure in brackets, since angle and location for use on the earth's orbit are not defined.
16. a b c Sundial Manual, 3.3 Calculation of the equation of time. Deutsche Gesellschaft für Chronometrie eV, sundials specialist group, 1900.
17. These “Underlyings” apply to January 1, 2015 12:00 UT. Their slow change is ignored below throughout 2015. The change that has accumulated during this time is only reflected in the 2016 annual constants. The exception is . Equation contains permanent change . The extrapolation of the annual constants takes place with the base values ​​of the years 2000 and 1900 as follows (DGC manual, p. 47): ${\ displaystyle L_ {0}}$${\ displaystyle \ mathrm {(8)}}$${\ displaystyle L (t)}$
${\ displaystyle M_ {0} = 357 {,} 5256 ^ {\ circ} +35999 {,} 0498 ^ {\ circ} \ cdot T / 36525}$
${\ displaystyle J_ {tr} = (365 {,} 24219878 + 6 {,} 16 \ cdot 10 ^ {- 8} \ cdot J) {\ text {days}}}$
${\ displaystyle J_ {an} = (365 {,} 25964124 + 3 {,} 04 \ cdot 10 ^ {- 8} \ cdot J) {\ text {days}}}$
${\ displaystyle e_ {0} = 0 {,} 016709-4 {,} 2 \ times 10 ^ {- 7} \ times T / 36525}$
${\ displaystyle \ varepsilon _ {0} = 23 {,} 439291 ^ {\ circ} -0 {,} 013004 ^ {\ circ} \ cdot T / 36525}$
${\ displaystyle L_ {0} = 282 {,} 9400 ^ {\ circ} +1 {,} 7192 ^ {\ circ} \ cdot T / 36525}$
${\ displaystyle T}$is the number of days since January 1, 2000 12:00 UT; is the number of years since 1900. The angles and are to be calculated modulo 360 °, and they must be between −180 ° and + 180 °.${\ displaystyle J}$${\ displaystyle M_ {0}}$${\ displaystyle L_ {0}}$
18. The year constants (e.g. for 2015) are named here because they are only used for the year to which they refer. In addition, they also apply to dates in distant years (e.g. for 2050 or 1950) without any significant loss of precision in the equation of time. The time then assumes correspondingly high positive or negative values; the given calculation scheme remains applicable without any changes. When determining and , the secondary values ​​of the arctangent that are closest to or are to be used.${\ displaystyle t}$${\ displaystyle V}$${\ displaystyle \ alpha}$${\ displaystyle E / 2}$${\ displaystyle \ lambda}$
19. calculation is made with the ecliptical length L = L 0 of the perihelion, which is sufficiently accurate and, because of the unknown time t, not otherwise possible.
20. Heinz Schilt: To calculate the mean time for sundials. Writings of the Friends of Old Watches, 1990.