SOME NOTES ON THE EQUATION OF TIME

by Carlos Herrero

Version v2.1, February 2014

I. INTRODUCTION

Since ancient times humans have taken the Sun as areference for measuring time. This seems to be a natu-ral election, for the strong influence of the Sun on ourdaily life, with a perpetual succession of days and nights.However, it has also been observed long time ago (e.g.,ancient Babylonians) that our Sun is not a perfect timekeeper, in the sense that it sometimes seems to go faster,and sometimes slower. In particular, it is known thatthe time interval between two successive transits of theSun by a given meridian is not constant along the year.Of course, to measure such deviations one needs another(more reliable) way to measure time intervals.

In this context, since ancient times it has been definedthe equation of time to quantify deviations of the timedirectly measured from the Sun position respect to anassumed perfect time keeper. In fact, the equation oftime is the difference between apparent solar time andmean solar time (as yielded by clocks in modern times).At any given instant, this difference will be the same forevery observer on the Earth.

Apparent (or true) solar time can be obtained for ex-ample by measuring the current position (hour angle) ofthe Sun, as indicated (with limited accuracy) by a sun-dial. Mean solar time, for the same place, would be thetime indicated by a steady clock set so that over the yearits differences from apparent solar time average to zero(with zero net gain or loss over the year). Apparent timecan be ahead (fast) by as much as 16 min 33 s (around3 November), or behind (slow) by as much as 14 min 6s (around 12 February). The equation of time has zerosnear 15 April, 13 June, 1 September, and 25 December.It changes slightly from one year to the next.

The graph of the equation of time is closely approx-imated by the sum of two sine curves, one with a pe-riod of a year and another with a period of half a year.These curves reflect two effects, each causing a differentnon-uniformity in the apparent daily motion of the Sunrelative to the stars: the obliquity of the ecliptic, whichis inclined by about 23.44o relative to the plane of theEarth’s equator, and the eccentricity of the Earth’s orbitaround the Sun, which is about 0.017.

The equation of time has been used in the past to setclocks. Between the invention of rather accurate clocksaround 1660 and the advent of commercial time distri-bution services around 1900, one of two common land-based ways to set clocks was by observing the passage ofthe Sun across the local meridian at noon. The momentthe Sun passed overhead, the clock was set to noon, off-set by the number of minutes given by the equation of

αγ

λ δ

Ecliptic

Equator

S

N

ε

FIG. 1: Celestial sphere showing the position of the Sun onthe ecliptic. α, right ascension; δ, declination; λ, eclipticlongitude. γ indicates the vernal point, and ǫ is the obliquityof the ecliptic.

time for that date. (Another method used stellar ob-servations to give sidereal time, in combination with therelation between sidereal and solar time.) Values of theequation of time for each day of the year, compiled by as-tronomical observatories, were widely listed in almanacsand ephemerides. Now, it can be found in many places,in particular in numerous web pages, e.g., the so-calledProcivel in Rodamedia.com.

Note that the name “equation of time” can be mislead-ing, as it does not refer to any equation in the modernsense of this word (a mathematical statement that assertsthe equality of two expressions, often including quanti-ties yet to be determined, the unknowns). Here, the termequation is employed in its Medieval sense, taken fromthe Latin term aequatio (which means equalization oradjustment), and that was used for Ptolemy’s differencebetween mean and true solar time.

In the following we present some questions related tothe equation of time. For convenience, we will considermotion of the Earth around the Sun or motion of theSun as seen from Earth, depending on the discussionat hand. For example, when discussing orbital motionwe have in mind the movement of the Earth. However,when displaying the celestial sphere it is the Sun thatmoves on the ecliptic, as shown in Fig. 1.

These notes are organized as follows:

2

a F

rϕ

b

c

p

y

x

FIG. 2: Ellipse with notation for different distances and pa-rameters.

- In Sec. II we define the “artificial Suns” that are usedto (presumably) simplify the discussion on the motion ofthe true Sun and the definition of mean time.

- In Sec. III we discuss the two major contributionsto the equation of time: eccentricity of the orbit andobliquity of the ecliptic.

- In Sec. IV we present the mathematical details of ourcalculations, based only on Newton’s laws and ellipticorbits.

- The position of the Gregorian calendar on the Earth’sorbit is discussed in Sec. V.

- In Sec. VI we present a schematic way to calculaterather precisely the equation of time with some very basicassumptions and simplifications.

- In Sec. VII we compare the results of our calculationswith those found by using other approaches.

- In Sec. VIII we present the analemma.- In Sec. IX we introduce a simple correction to the

equation of time, due to the lunar perturbation.- Finally, we give some appendixes, including the no-

tation we have employed, a glossary of the main terms,and some mathematical formulas.

II. HOW MANY SUNS ?

A useful tool to study the equation of time is theso-called mean Sun, which is a mental artifact giving usa reliable time keeper, that should coincide with our bestclocks if the Earth rotation had a constant speed (whichunfortunately is not the case). This complication dueto the variable angular velocity will not be consideredhere, as in a first approximation is not relevant for ourcalculations. Its influence on the different time scalespresently used can be found in the glossary at the endof the text.

1 - True Sun (or apparent Sun). Suns there is only one,the others are mental artifacts to simplify the calculationsand mainly to understand the whole thing. The true Sunmoves on the ecliptic with nonuniform velocity, i.e., itgoes faster close to the perigee (perihelion for the Earth)

γ

Equator

S

N

P

Ecliptic

α S

SFT

S

MM

Λ

FIG. 3: Celestial sphere displaying the position of the (true)Sun ST and the fictitious (dynamical mean) Sun SF on theecliptic, as well as the mean Sun SM in the equator. P in-dicates perigee; γ, vernal point; αM , right ascension of themean Sun; Λ, ecliptic longitude of SF . Note that αM = Λ.

and slower near the apogee (aphelion for the Earth). Wewill call it ST . Its position is given by the true anomaly,ϕ, which is the angle between ST and the perigee; seeFig. 2.

2 - Fictitious Sun (or dynamical mean Sun). This isonly an intermediate tool between true Sun and meanSun. We will call it SF . It moves on the celestial spherefollowing the ecliptic with uniform motion. To be precise,SF is an imaginary body that moves uniformly on theecliptic with the mean angular velocity of the true Sun,and which coincides with ST at perigee and apogee. Theposition of SF is given by its ecliptic longitude Λ; seeFig. 3.

3 - Mean Sun. We call it SM . It moves uniformlyon the equator. Its position is measured by the meananomaly M on the equator, in contrast with SF forwhich the position is measured on the ecliptic. Tobe concrete, the mean Sun is supposed to move onthe equator in such a manner that it right ascension,αM , is equal to the ecliptic longitude of the (fictitious)dynamical mean Sun, Λ. SM is directly related to ourclocks, as it is used to define the solar mean time.

Now we put the three objects in motion:

1 - ST does not need to start, it has been moving formany years on the ecliptic.

2 - Now we humans wait until ST goes through theperigee, and then SF starts on the same position and thesame direction as ST . Since SF moves uniformly on theecliptic, at the beginning it will move slower than ST .

3 - We wait until SF arrives at the vernal point γ. Atthat moment, the mean Sun SM starts moving at γ withthe same velocity as SF , BUT ON THE EQUATOR.

3

Thus, SF and SM coincide twice each year (at theequinoxes). See Fig. 3.

Note that when we will simply speak about the Sun,we will obviously mean “true” Sun.

We will define the equation of time ∆t at a certainmoment as the difference between the right ascension ofthe mean Sun and that of the true Sun:

∆t = αM − α (1)

Equivalently, it is the difference between the hour anglesof the mean Sun and true Sun: ∆t = H − HM . Wenote that sometimes ∆t is defined as α − αM , but thisdefinition will not be used here. With our definition ofequation of time, ∆t > 0 means that the Sun crosses agiven meridian before SM , and ∆t < 0 indicates that ST

crosses it after SM . Thus, for ∆t > 0 the Sun is ahead(fast), and for ∆t < 0 it is behind (slow).

III. THE TWO MAJOR CONTRIBUTIONS TO

THE EQUATION OF TIME

This is a qualitative explanation of the origin of theequation of time. For a more rigorous calculation oneshould go to Sec. IV.

A. Eccentricity of the Earth’s orbit

As seen from Earth, the Sun appears to revolve oncearound the Earth through the background stars in oneyear. If the Earth orbited the Sun with a constant speed,in a circular orbit and on a plane perpendicular to theEarth’s axis, then the Sun would culminate (would crossa given meridian) every day at exactly the same time. Inthat hypothetical case, the Sun would be a rather goodtime keeper, similar to the UTC time given by modernatomic clocks (except for the small effect of the slowingrotation of the Earth). But the orbit of the Earth is anellipse, and thus:

(1) its orbital speed varies by about 3.4% betweenaphelion and perihelion (29.291 and 30.287 km/s), ac-cording to Kepler’s laws of planetary motion;

(2) its angular velocity changes accordingly, beingmaximum at the perihelion and minimum at the aphe-lion, and

(3) the Sun appears to move faster (in its annual mo-tion relative to the background stars) at perihelion (cur-rently around January 3) and slower at aphelion a halfyear later.

At these extreme points, this causes the apparent solarday to increase or decrease by about 7.9 s from its meanof 24 hours. This daily difference accumulates along thedays. As a result, the eccentricity of the Earth’s orbitcontributes a sine wave variation with an amplitude of

0 100 200 300

Mean anomaly, M (degrees)

-10

-5

0

5

10

Dif

fere

nce

M -

phi

FIG. 4: Contribution of the eccentricity of the Earth’s orbit tothe equation of time (given in minutes of time), as a functionof the mean anomaly M . It is assumed that the motion takesplace on the equator plane.

Earth

∆ϕ

∆ϕ

Sun

FIG. 5: Schematic representation of the Earth’s motionaround the Sun. The angle ∆ϕ swept by the Earth (as seenfrom the Sun) in one day is the same as the angle that theEarth has to rotate to complete a solar day, in addition to the360o corresponding to a sidereal day.

7.66 minutes and a period of one year to the equation oftime. The zero points are reached at perihelion (at thebeginning of January) and aphelion (beginning of July),while the maximum values are in early April (negative)and early October (positive). In Fig. 4 we plot this vari-ation along the year, as calculated by the method de-scribed in Sec. IV. The mean anomaly appearing in thisplot is defined as the angle from the periapsis to the dy-namical mean Sun.

A simple (although not rigorous) derivation of themaximum changes in the solar day length, caused by el-lipticity of the orbit, is the following (see Fig. 5). For anelliptical orbit, we know, from conservation of the angu-lar momentum, that [this is explained with more detail

4

below, see Eq. (19)]

C =1

2r2ϕ ≈ 1

2rv (2)

(r: distance Sun–Earth; v: velocity) since v ≈ rϕ (be-cause the eccentricity e ≪ 1) and C is a constant. Nowwe call ∆ϕ the change of ϕ in one solar day, i.e. betweentwo successive returns of the Sun to the local meridian.For small ∆ϕ (as happens for one day):

∆ϕ ≈ 2C

r2∆t (3)

Then, putting “P” for perihelion and “A” for aphelion,we have:

(∆ϕ)P

(∆ϕ)A=

r2A

r2P

=

(

1 + e

1 − e

)2

= 1.0691 (4)

(We have used e = 0.0167 for the Earth’s orbit). Thismeans that ∆ϕ varies in about 6.9% from its maximumto its minimum value, i.e. ± 3.45% with respect to themean day (24 hours). For the mean day, (∆ϕ)M =360o/365.2564 days = 0.9856o/day. This means that inaverage the Earth sweeps in its translational motion anangle of 0.9856o per day, which is exactly the angle thatthe Earth has to rotate in addition to 360o to completea whole solar day (see Fig. 5). This angle correspondsto a delay of 3.94 min of the mean solar day respect thesidereal day. The actual delay will change along the year,so that close to the perihelion it will be larger (the Earthmoves faster), and near the aphelion it will be shorter.Thus, the difference with respect to the mean day will beat perihelion and aphelion: ±0.0345× 3.94 min ≈ ± 8 s.

B. Obliquity of the ecliptic

If the Earth’s orbit were circular, the motion of theSun, as seen from the rotating Earth, would still not beuniform. This is a consequence of the tilt of the Earth’srotation axis with respect to its orbit, or equivalently, tothe obliquity of the ecliptic with respect to the equator.The projection of this motion onto the celestial equator,along which “clock time” is measured, is a maximum atthe solstices, when the yearly movement of the Sun isparallel to the equator and appears as a change in rightascension (the time derivative of the solar declination δ isthen zero, dδ/dt = 0 ). That projection takes a minimumat the equinoxes, when the Sun moves in a sloping direc-tion and |dδ/dt| is maximum, leaving less for the changein right ascension, which is the only component that af-fects the duration of the solar day. At the equinoxes, theSun is seen slowing down by up to 20.3 seconds every dayand at the solstices speeding up by a similar amount.

Concerning the equation of time, the obliquity of theecliptic contributes a sine wave variation with an ampli-tude of 9.87 minutes and a period of half a year. Thezero points of this sine wave are reached at the equinoxes

0 100 200 300

Mean anomaly, M (degrees)

-10

-5

0

5

10

Dif

fere

nce

alp

ha -

alp

ha_M

FIG. 6: Contribution of the obliquity of the ecliptic to theequation of time (measured in minutes of time), as a functionof the mean anomaly M . It is assumed that the orbit iscircular.

and solstices, while the extreme values appear at the be-ginning of February and August (negative), and the be-ginning of May and November (positive). In Fig. 6 weplot this variation along the year, as calculated by themethod described in Sec. IV.

As indicated above, the contribution of obliquity to thechange in duration of apparent solar days is maximum atthe solstices and equinoxes. For the latter, we now derivein a simple way this contribution, assuming that the orbitis circular. First note that the daily change in eclipticlongitude is: ∆λ = 360o/365.2564 = 0.9856o/day. Fromthe spherical triangle shown in Fig. 7, we have:

tan α = tanλ cos ǫ (5)

and the daily change in right ascension at the equinoxwill be:

∆α = tan−1[tan(∆λ) cos ǫ] , (6)

which gives ∆α = 0.9042o. This translates into a timeinterval δt = 3.6168 min = 3 min 37 s, which is the differ-ence between the corresponding solar day and a siderealday. (This is similar to the discussion above in Sec. III.A,see Fig. 5) Taking into account that for a mean day δt =3 min 56 s, we find that at the equinoxes the obliquitycontributes to shorten the solar day in about 20 s. InAppendix E we give some more details on the estimationof δt at different times.

Note: As a rule of thumb, if a given day the changeof solar right ascension ∆α is larger than the mean value(∆α)M , then the Sun takes more than 24 hours to returnto a meridian (Sun slower than clock). On the contrary,if ∆α < (∆α)M , the Sun turns to a meridian in less than24 hours (Sun faster than clock).

5

Ecliptic

εδ

λ

γEquator

α

FIG. 7: Spherical triangle showing the Sun’s position on thecelestial sphere. γ, vernal point; ǫ, obliquity of the ecliptic;α, right ascension; δ, declination; λ, ecliptic longitude.

IV. KEPLERIAN ELLIPTIC MOTION

A. Basic assumptions

The present calculations give a rather accurate valuefor the equation of time. However, they are based on thevery simple assumption that Sun and Earth are isolatedin the Universe without any external interactions. Thismeans:

- One neglects gravitational (for some purposes impor-tant) interactions with the Moon and other objects in theSolar System (mainly Jupiter). A correction due to thelunar perturbation is presented in Sec. IX.

- Sun and Earth are considered as point masses, whichwould be precise for spherical bodies with uniform massdistribution. This means that we neglect oblateness ofSun and Earth, as well as Earth deformations. In partic-ular, nutation of the Earth axis is not taken into accountin the calculations.

- The gravitational interaction is assumed to beNewtonian, i.e., we do not consider corrections due toGeneral Relativity.

These effects give rise to changes in the Earth orbit,such as precession of the equinoxes and precession of theperiapsis, that can be considered in an effective way bylocating the orbit according to the known position of thevernal point and perihelion for a given date.

B. Two-body problem

We consider a problem of two bodies interacting grav-itationally. To study the trajectory and dynamics, thebasic ingredients are the gravitation law (attraction forceproportional to the product of masses and inverse to thesquared distance) and the second law of motion, bothpostulated by Newton.

The force exerted by the Sun on the planet is:

Fsp = m rp = −Gm0m

r3r , (7)

and that exerted by the planet on the Sun is

Fps = m0 rs = Gm0m

r3r , (8)

where m0 and rs are the mass and position of the Sun; mand rp those of the planet, and r = rp − rs is the planetposition, as seen from the Sun. Also, r = |r|, and dotsindicate time derivatives.

Note that

Fsp + Fps = 0 , (9)

as should be, or

m rp + m0 rs = 0 , (10)

from where

MT R = A t + B , (11)

(A and B are integration constants) with the center-of-mass position:

R =m rp + m0 rs

m0 + m, (12)

and the total mass MT = m0 + m. The center-of-masshas inertial motion.

For the relative position r we have r = rp − rs, so thatwe find the following differential equation

r = −k2 r

r3, (13)

where we have defined

k2 = GMT = G(m0 + m) (14)

C. Equation of the trajectory

We note first that r‖r, so that r × r = 0, from where

d(r × r)

dt= r × r + r × r = r × r = 0 (15)

Thus, the cross product r × r is constant along the tra-jectory, and we will write:

r × r = 2C (16)

This means: First, that the angular momentum (parallelto C) is a constant of motion; second, that the relativemotion of the two bodies takes place on a plane normal tothe vector C (since r ·C = 0), and third, that the motionverifies the so-called “law of equal areas” (second Kepler’slaw of planetary motion). In fact, if we describe themotion in planar polar coordinates (r, ϕ) (with origin onthe focus), the velocity r has components r and rϕ in thedirections parallel and perpendicular to r, respectively.Then, we have for the velocity:

v2 = |r|2 = r2 + r2ϕ2 (17)

6

and from Eq. (16):

C = |C| =1

2|r × r| =

1

2r2ϕ (18)

Now note that the elementary area swept by the vectorradius r in the orbit plane is

dS =1

2r.rdϕ =

1

2r2dϕ (19)

so that

dS

dt= C (20)

and the “areal” velocity is a constant of motion. A moredetailed derivation of this equation and related conceptsare given in Appendix D.

We will now prove that the trajectory is a conic. Withthis purpose we calculate

r × 2C = −k2

r3r × (r × r) = −k2

r2(rr− rr) = k2 d

dt

(

r

r

)

,

(21)where we have used the fact that r · r = rr [see Eq. (C1)],as well as the formula (C3) for the vector triple product.

By integrating the above expression, and calling e aconstant vector, one finds

r× 2C = k2(

r

r+ e

)

(22)

Taking the dot product of both sides of this equationtimes r, we have for the left-hand side:

r · (r × 2C) = 2C · (r × r) = 4 C2 , (23)

with C = |C|, and for the right-hand side:

k2r ·

(

r

r+ e

)

= k2(r + re cosϕ) , (24)

where ϕ is the angle between e and r (that will be thepolar angle, taking e for the reference direction). Finally,from Eqs. (23) and (24)

r =p

1 + e cosϕ(25)

which is the equation of the relative trajectory in pla-nar polar coordinates (r, ϕ). This equation correspondsto a conic with eccentricity e and parameter (semi-latusrectum)

p =4C2

k2. (26)

Note that p is a function of the angular momentum, as itis proportional to C2. Depending on the value of e, theconic is:

- For e = 0, a circumference- For 0 < e < 1, an ellipse

- For e = 1, a parabola- For e > 1, a hyperbolaIn the case of planetary motion we have elliptic orbits,

i.e. the motion is bound with negative energy h < 0 (seebelow). For nonnegative h the motion is neither boundnor periodic (parabolic for h = 0 or hyperbolic for h > 0).

We are interested here in the case 0 < e < 1, whichcorresponds to elliptic trajectories, and therefore to plan-etary motion. In this case, Eq. (25) can be transformedto Cartesian coordinates by using the usual expressionsx = r cosϕ, y = r sinϕ (see Fig. 2). Thus, one has theequation

(x + c)2

a2+

y2

b2= 1 (27)

referred to one of the focal points (foci) of the ellipse (Fin Fig. 2), where the semi-major axis a and the semi-minor axis b are given by:

a =p

1 − e2, b =

p√1 − e2

(28)

and the eccentricity e = c/a, with c = pe/(1 − e2).

D. Dynamics and anomalies

We will now obtain a constant of motion that will allowus to calculate the velocity v = (r · r)1/2. We have:

1

2

dv2

dt=

1

2

d

dt(r · r) = r · r = −k2

r3r · r = k2 d

dt

(

1

r

)

(29)

(See Appendix C for the time derivative of 1/r). Byintegrating this equality, one has

v2 − 2k2

r= 2h , (30)

where h is a constant proportional to the energy of thesystem. In the l.h.s. of this equation one recognizes(apart from a constant) the kinetic and potential energyin the gravitational field.

Now, to find a direct relation between the distance rand the velocity v, we will use Eq. (17) along with therelations:

r =2 C e

psin ϕ , (31)

obtained as a time derivative of Eq. (25) (see AppendixC), and

r2ϕ2 =4C2

r2=

4C2

p2(1 + e cosϕ)2 (32)

This yields

v2 = r2 + r2ϕ2 =4C2e2

p2sin2 ϕ+

4C2

p2(1+ e cosϕ)2 (33)

7

or

v2 =k2

p(1+e2+2e cosϕ) = k2

(

21 + e cosϕ

p+

e2 − 1

p

)

.

(34)We find :

v2 = k2

(

2

r+

e2 − 1

p

)

. (35)

The energy constant h is, from Eqs. (30) and (35):

2h =k2

p(e2 − 1) =

k4

4C2(e2 − 1) . (36)

In particular, for elliptic motion, one has from Eq. (28)

p = a(1 − e2) , (37)

so that

v2 = 2h +2k2

r= k2

(

2

r− 1

a

)

(38)

Note that the constant C (which gives the “areal ve-locity”) can be written as

C =Se

P=

πab

P, (39)

Se being the area of the ellipse. Then (note that p =b2/a):

GMT = k2 =4C2

p=

4π2a2b2

P 2

a

b2= 4π2 a3

P 2(40)

with MT = m0 + m. This is the third Kepler’s law.For an elliptic keplerian motion it is usual to use, in-

stead of the orbital period P , the average angular velocity(called average motion) n:

n =2π

P(41)

Then, Eq. (40) can be rewritten as

k2 = n2a3 (42)

and we have

r2ϕ2 =4C2

r2=

pk2

r2=

n2a4(1 − e2)

r2, (43)

where we have employed Eqs. (18), (26), (37), and (42).Now, using Eq. (38) for the velocity, and taking into

account that v2 = r2 + r2ϕ2, we have:

r2 +n2a4(1 − e2)

r2= n2a3

(

2

r− 1

a

)

, (44)

from where one has

r2 =n2a2

r2[a2e2 − (a − r)2] (45)

and finally

±n dt =rdr

a√

a2e2 − (a − r)2(46)

We now replace the distance r by E (the Sun’seccentric anomaly) through the following change of vari-able

a − r = a e cosE (47)

dr = a e sinE dE (48)

so that Eq. (46) reduces to

±n dt = (1 − e cosE) dE (49)

Here we use the eccentric anomaly E as an intermediatevariable to connect the true anomaly ϕ with the meananomaly M (see below). For more details on the geo-metric meaning of E, see Appendix F and Fig. 20.

By integration of Eq. (49) we find

E − e sin E = n(t − T ) (50)

where T is an integration constant which coincides withthe instant of transit by the periapsis, as for t = T wehave E = 0 and r = a(1−e) = a−c (see Fig. 2). Eq. (50)is the so-called Kepler equation. The quantity

M = n(t − T ) (51)

is the mean anomaly, i.e., the angle from the periap-sis to the dynamical fictitious Sun (along the ecliptic,see Sec. II), or equivalently the angle from the point α0

to the dynamical mean Sun (along the equator). Thus,this equation gives us a relation between the eccentricanomaly E and the mean anomaly M . Note that M isnot the total mass of the system, which is called MT .

Going back to Eqs. (25) and (47), we have for ellipticmotion:

r =a(1 − e2)

1 + e cosϕ= a(1 − e cosE) (52)

where we have a relation between the eccentric anomalyE and the true anomaly ϕ. From the last two expressionsin Eq. (52), we can find out cosϕ by elementary algebra:

cosϕ =cosE − e

1 − e cosE(53)

and from here:

tan2 ϕ

2=

1 − cosϕ

1 + cosϕ=

(1 + e)(1 − cosE)

(1 − e)(1 + cosE)=

1 + e

1 − etan2 E

2(54)

and finally

tanϕ

2=

√

1 + e

1 − etan

E

2(55)

8

0 100 200 300

Mean anomaly, M (degrees)

-20

-10

0

10

20E

quat

ion

of

tim

e (

min

)

FIG. 8: Equation of time as a function of the mean anomalyM , as derived from the method given in Sec. IV.

or

ϕ = 2 tan−1

[

√

1 + e

1 − etan

E

2

]

(56)

which gives us the true anomaly ϕ as a function of theeccentric anomaly E.

Knowing ϕ we calculate the Sun’s longitude λ on theecliptic as

λ = λp + ϕ (57)

where λp is the ecliptic longitude of the periapsis (λp =283.084o on 1.1.2011). Note that the mean anomaly M isrelated to λp through the ecliptic longitude of the meanSun, Λ, i.e., M = Λ − λp.

Knowing λ, we calculate the right ascension α by relat-ing it to the angles ǫ and λ in the right spherical triangleshown in Fig. 7:

tan α = cos ǫ tan λ (58)

Finally, ∆t can be calculated from the difference ∆t =αM − α, where the right ascension of the mean Sun isgiven by αM = M + λp. This gives the curve shown inFig. 8.

V. POSITION OF THE CALENDAR ON THE

ORBIT

With the above calculations the problem of calculatingthe equation of time is formally solved, but now we haveto translate the mean anomaly M to a precise date inour Gregorian calendar. This can be done in two steps:

(1) Locate the mean perihelion at its precise date ofthe year we are interested in. Since the mean perihelion

P

P’

γ γ’

A

λp

FIG. 9: Schematic representation of the Earth’s orbit, indi-cating changes along the time. P, perihelion; A, aphelion; γ,vernal point; λp: ecliptic longitude of the perihelion. γ′ is afuture position of the vernal point due to equinox precession.The perihelion precesses from P to P’. The latter motion ismuch slower than that of γ. Note that the true Earth’s orbitis much less eccentric than that plotted here.

is our reference for angles on the elliptic orbit, it hasto be accurately located. Its position changes along theyears, and is referred to the vernal equinox by means ofits ecliptic longitude λp. In 2011, λp(2011) = 283.084o,and it increases at a rate of 0.017o per year:

λp = λp(2011) + 0.0170 ∆n (59)

where ∆n = ny − 2011, and ny is the year. This shiftis due to the combination of the equinox precession andthe precession of the Earth’s perihelion itself (see Fig. 9).It causes a change (advance) of the mean perihelion in24.83 min per year, which corresponds to the 0.017o shiftmentioned above.

The instant of the mean perihelion in our calendarchanges also due to the non-commensurability of the cal-endar years with the anomalistic year. Then, for yearsfollowing a common (not leap) year there appears a shift(ahead) of 6.233 hours (= 6 hours 14 min), and for yearsfollowing a leap year a retard of 24 - 6.233 hours = 17hours 46 min. (This is because inserting the 29th Febru-ary affects the following perihelion in the next January.Also note that these 6 hours 14 min include the 24.83min of the perihelion shift). Thus, in 2011 the mean per-ihelion was on 3 Jan at 19:54 UTC, in 2012 it occurredon 4 Jan at 2:08 UTC, in 2013 it took place on 3 Jan at8:22 UTC, and so on.

A practical rule:Given a year ny > 2011, the instant of the mean peri-

helion, T , is given (in hours) by

T = T2011 + 6.233 ∆n− 24 ix (60)

where ∆n = ny − 2011, and ix is the number of leapyears from 2011, given by the expression: ix = int[(ny −2009)/4)]. Here “int” means the integer part of the quo-tient, and thus one has ix = 0 for 2011 and 2012, 1 for2013–2016, 2 for 2017–2020, and so on. For complete-ness, we mention that this rule is valid also for 2009 and

9

2010. For earlier years one can use a similar rule, takingthe same reference of year 2011:

T = T2011 + 6.233 ∆n + 24 kx (61)

where kx = int[(2008 − ny)/4)] + 1, so that kx = 1 for2005–2008, 2 for 2001–2004, and so on.

Note that the actual perihelion may differ from themean perihelion by up to about 30 hours, mainly dueto the influence of the Moon, but we will not discussthis here. The important point to be emphasized hereis that to calculate the equation of time we have to“place” the whole orbit in the calendar as accurately aspossible (associate a precise date to each angle M), andthis can be only done with respect to the mean perihelion.

(2) The mean anomaly M is an angle that increasesuniformly (clock) from 0o at the mean perihelion to 360o

at the next perihelion. In fact, at the next perihelionthis angle is slightly larger than 360o due to perihelionprecession, and 360o corresponds to a tropical year (alittle shorter than the anomalistic year). Then, given adate, we translate it to an angle M by calculating thenumber of days N (not necessarily an integer) from themean perihelion, so that

M =N

365.2425× 360o (62)

where 365.2425 is the number of days in a tropical year(in fact, mean Gregorian year).

VI. A RECIPE TO CALCULATE THE

EQUATION OF TIME

A schematic and simple procedure to obtain an ap-proximate value for the equation of time is the following.It is based on Newton equations of motion, which giverise to conical (elliptical for planets) trajectories in thetwo-body problem (Sun and planet). In fact, this is asummary of the procedure presented in Sec. IV.

1) Give the time, i.e., the Sun’s mean anomaly M

M =N

365.2425× 360o , (63)

where N is the number of days from the periapsis (ingeneral, a non-integer number), and 365.2425 is thenumber of days in a mean Gregorian year [see Eq. (62)].

2) From M we calculate the eccentric anomaly Ethrough Kepler’s equation [see Eq. (50)]:

M = E − e sin E . (64)

This transcendental equation cannot be solved analyti-cally in E. A first and rather accurate approximationconsists in taking

E ≈ M + e sin M . (65)

Better approximations can be obtained in a recursiveway, by inserting the obtained value again in theequation: E1 = M + e sin M , E2 = M + e sin E1,E3 = M + e sin E2, ..., until obtaining the requiredprecision. This works well because e ≪ 1.

3) Calculate the true anomaly ϕ from the eccentricanomaly E [see Eq. (56)]:

ϕ = 2 tan−1

[

√

1 + e

1 − etan

E

2

]

(66)

4) Find the ecliptic longitude of the (true) Sun:

λ = λp + ϕ (67)

where λp is the ecliptic longitude of the periapsis (in year2011, λp = 283.084o; for the following years see Eq. (59)).

5) Calculate the right ascension α by relating it to theangles ǫ and λ in the right spherical triangle shown inFig. 7:

α = tan−1(cos ǫ tanλ) (68)

6) Obtain the right ascension αM of the mean Sun as:

αM = λp + M (69)

7) Finally, calculate the equation of time as:

∆t = αM − α (70)

and we obtain ∆t as a function of the mean anomaly M .

8) If we are interested in converting M into an actualdate in our Gregorian calendar, the number of days N hasto be added to the date of the perihelion in the year underconsideration. A practical rule to find rather preciselythe instant T of the mean perihelion is given in Sec. V.

VII. COMPARISON WITH OTHER METHODS

TO CALCULATE ∆t

A. Precision of our method

We remember that the calculations presented aboveare based on the assumption that Earth and Sun are ce-lestial bodies isolated in the Universe, and we solved thetwo-body problem with Newtonian gravitational interac-tion. We pointed out, however, the presence in the realworld of interactions with other celestial bodies (mainlythe Moon and other planets) which cause perturbationson the Earth’s orbit, such as precession of the equinoxesand precession of the perihelion. Also orbital parametersas the eccentricity e and the obliquity ǫ change with time,but this change is very slow for our present purpose, sothat they may be considered constant. For our calcula-tions on the equation of time, an important point is the

10

0 100 200 300

Days (year 2011)

-2

-1

0

1

2

Err

or (

diff

eren

ce C

- P

) (

s)

FIG. 10: Difference EC = (∆t)C −(∆t)P along the year 2011.Points are plotted at five-day intervals.

difference between the actual Earth’s perihelion (the in-stant when the distance Sun-Earth is a minimum) andthe mean perihelion, which is used in our calculations asif the Moon were not present at all.

To assess the precision and reliability of our calcu-lations based on a simple two-body problem, we willcompare our results with precise values of the equationof time, as those given in the Procivel tables (see Ro-damdia.com). Values presented in these tables are basedon the actual position of the Sun at each moment, asderived from reliable astronomical calculations.

Given an instant of time (date), we call error EC thedifference between the equation of time calculated fromour approximation, (∆t)C , and that given in the Prociveltables, (∆t)P :

EC = (∆t)C − (∆t)P (71)

This difference is shown in Fig. 10 for year 2011, wheredata points are presented every five days. It is remarkablethat the absolute error of the procedure described here isless than 2 s along the whole year. It is also remarkablethat the evolution of EC along the year can be separatedinto two main contributions: (1) a high-frequency oscil-lation with a period Th of about 30 days and amplitudeless than one second, and (2) a low-frequency backgroundof larger amplitude.

Contribution (1) is related to the influence of theMoon, since a careful analysis indicates that the periodTh coincides with the synodic period of the Moon. Takinginto account that the influence of the Moon has not beenconsidered in our calculations for the two-body problem,its gravitational interaction with the Earth should ap-pear as a residual source of error in our results. We notein passing that the Moon interaction causes maxima andminima in the sine-shape oscillation displayed in Fig. 10for first and third quarter Moon, respectively. This con-tribution vanishes for full and new Moon. This will be

0 500 1000

Days (since 1 Jan 2011)

-2

-1

0

1

2

Err

or (

diff

eren

ce C

-P)

(s)

FIG. 11: Difference EC = (∆t)C − (∆t)P from January 2011to December 2013.

discussed in more detail in Sec. IX.Concerning contribution (2) of the error EC , we cannot

give at this moment a precise reason for it, as it couldbe due to a sum of different contributions. An importantpoint is that we do not find at first sight any apparentperiodicity in this contribution along the years, as can beobserved in Fig. 11, where we have plotted EC for threeconsecutive years, from January 2011 to December 2013.

B. Sum of two independent contributions

In Sec. III we discussed the two main contributionsto the equation of time: eccentricity of the orbit andobliquity of the ecliptic. A simplification to calculateapproximately ∆t consists in adding both contributions,as if they could be considered totally independent. Thismeans that one calculates first the contribution of ec-centricity, assuming that the Sun moves on the equator(ǫ → 0), and then the contribution of obliquity assumingthat the orbit is circular (e → 0).

For this estimation, we have carried out the same cal-culations presented in Sec. IV, but with the correspond-ing simplifications:

(1) We neglect the obliquity of the ecliptic putting ǫ =0, and find the contribution of the eccentricity, (∆t)exc.The resulting contribution to the equation of time (thatcan be written in this case as αM −α = M −ϕ) is shownin Fig. 4 as a function of the mean anomaly M along oneyear. As expected, one finds a periodic behavior with aperiod of 2π (one year).

(2) We neglect the eccentricity of the orbit putting e =0, and obtain the contribution of the obliquity, (∆t)obl.This contribution is presented in Fig. 6 as a function ofthe mean anomaly M . One finds a periodic function,with a period of π (half a year).

The sum of both contributions has a shape similar to

11

0 100 200 300

Mean anomaly, M (degrees)

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8E

rror

(m

inut

es t

ime)

Sum of two contributions - "exact" value

FIG. 12: Difference between our ”precise” calculation for theequation of time, (∆t)C , and ∆t calculated as a sum of twoindependent contributions.

the more precise ∆t displayed in Fig. 8, but it differs fromthe latter by an amount that changes along the year, andis always less than 40 s (see Fig. 12).

C. Milne’s formula

A simple analytical formula, presented by R. M. Milnein 1921 (“Note on the Equation of Time”, The Mathe-matical Gazette 10, 372-375), gives an approximation forthe equation of time as a function of the mean anomalyM . To derive this approximation, we consider Eq. (58),which relates the right ascension α to the ecliptic longi-tude λ:

tan α = cos ǫ tan λ (72)

Remembering that

cos ǫ =1 − tan2 ǫ

2

1 + tan2 ǫ2

(73)

we have(

1 + tan2 ǫ

2

)

sinα cosλ =(

1 − tan2 ǫ

2

)

cosα sin λ

(74)or

sin(α − λ) = − tan2 ǫ

2sin(α + λ) . (75)

Now, since the right ascension α and the ecliptic longi-tude λ take similar values, the difference α − λ is small(compared to π; in fact, |α − λ| is always less than 3o),and Taylor expanding the sine function to first order, wehave:

α ≈ λ − sin(2λ) tan2 ǫ

2. (76)

We now look for a simplified relation between the trueanomaly ϕ and the mean anomaly M . To this end,we start from a relation between ϕ and the eccentricanomaly E. From Eq. (55) we have

tanϕ

2=

√

1 + e

1 − etan

E

2. (77)

First, we note that e ≪ 1, and an expansion of the func-tion containing e gives:

(

1 + e

1 − e

)1

2

= 1 + e + O(e2) (78)

and we will retain only terms up to first order in e. Then,we have:

tanϕ

2≈ tan

E

2+ e tan

E

2(79)

Since the difference ϕ−E is small (vs. π), we put ϕ/2 =E/2 + ∆, with ∆ a small parameter. Taylor expandingtan(ϕ/2) with the parameter ∆ we find

tanϕ

2= tan

E

2+

1

cos2 E2

∆ + O(∆2) (80)

Now, comparing Eqs. (79) and (80), and identifying lin-ear terms in e and ∆, we have

∆ =e

2sinE (81)

and finally

ϕ ≈ E + e sin E . (82)

With this result, and taking into account that E ≈ M +e sin M [see Eq. (65)], we find that the true anomaly ϕcan be written as a function of M :

ϕ ≈ M + 2 e sin M . (83)

Since λ = λp + ϕ, Eq. (76) transforms into

α ≈ λp + M + 2 e sin M − tan2 ǫ

2sin(2M + 2λp) . (84)

The right ascension of the mean Sun is αM = λp + M ,so that we have for the difference αM − α:

(∆t)Milne = −2 e sin M + tan2 ǫ

2sin(2M + 2λp) (85)

This equation gives approximately the equation of timeas a simple analytical function of the mean anomaly M ,with the orbital parameters e, ǫ, and λp. It has a clearphysical explanation as the sum of two terms, the firstone due to eccentricity of the orbit, and the second onedue to obliquity of the ecliptic. Introducing the values:e = 0.0167, ǫ = 23.44o, and ecliptic longitude of theperiapsis on 2011-1-1: λp = 283.084o, one finds

(∆t)Milne = −7.655 sin M + 9.863 sin(2M + 206.168)(86)

12

where ∆t is given in minutes of time and M in degrees.This equation was first derived by Milne, who wrote it interms of the ecliptic longitude of the (fictitious) dynam-ical mean Sun, Λ = λp + M .

The absolute error of this formula is less than 45 sthroughout the year, with its largest value 44.8 s at thebeginning of October.

D. A simple and rather accurate approximation

A very simple approximation, that is however moreaccurate than Milne’s method is the following. Giventhe mean anomaly M , the Sun’s ecliptic longitude canbe approximated as:

λ = λp + ϕ ≈ λp + M + 2 e sin M , (87)

where we have used Eq. (83). From λ, one calculates theright ascension of the true Sun, α, as [see Eq. (58)]:

tan α = cos ǫ tan λ . (88)

The right ascension of the mean Sun is αM = λp + M ,from where one finds the equation of time ∆t = αM −α.

The largest error of this approximation for ∆t is about6 s. This simple approximation is much more precise thanMilne’s formula. The main source of error in Milne’s for-mula is the approximation for the right ascension α givenin Eq. (76), which is obtained here directly from Eq. (88)without any approximation. The main approximationintroduced here is that given in Eq. (87) for λ, whichappears also in the derivation of Milne’s formula.

VIII. ANALEMMA

An analemma is a curve showing the angular offset ofa celestial body (usually the Sun) from its mean positionon the celestial sphere, as viewed from another celestialbody (usually the Earth). This name is commonly ap-plied to the figure traced in the sky when the position ofthe Sun is plotted at the same clock time each day over acalendar year from a fixed position on Earth. The result-ing curve resembles a lemniscate of Bernoulli. The wordanalemma comes from Greek and means “pedestal of asundial”, since it was originally employed to describe theline traced by the shadow of the sundial’s gnomon alongthe year.

The actual shape of the Sun’s analemma depends on:(1) general parameters for all observers on Earth, such asthe eccentricity e of the Earth’s orbit and the obliquityof the ecliptic, ǫ, and (2) particular parameters of theobserver, as his/her geographic latitude and the actualobservation time.

Viewed from an imaginary planet with a circular orbit(e = 0) and no axial tilt (equator parallel to ecliptic,ǫ = 0), the Sun would always appear at the same pointin the sky at the same time of day throughout the year,

-6 -4 -2 0 2 4 6

Azimuth (degrees)

10

20

30

40

50

60

70

80

90

Alti

tutd

e (

degr

ees)

1 Jan

1 June

1 Oct

1 Mar

Year 2011 - 12 UTC - Latitude 40o

FIG. 13: Analemma for year 2011 at 12 UTC as seen from aplace with longitude 0o and latitude 40o N.

and therefore the analemma would be a simple dot, ashappens on Earth for the mean Sun described above.For a celestial body with a circular orbit but appreciableaxial tilt, the analemma would be a figure like “8” withnorthern and southern lobes equal in size. For an objectwith an eccentric (i.e., non-circular) orbit but no axialtilt, the analemma would be a straight line (in fact, asegment). At noon this line would appear in the east-west direction at an altitude h = 90o − φ (φ: latitude).

In Fig. 13 we present an analemma as seen from theEarth’s northern hemisphere. It is a plot of the positionof the Sun at 12:00 UTC (noon) as seen from a pointon the Greenwich meridian (longitude 0o) and latitudeφ = 40o N during year 2011. The horizontal axis is theazimuth angle A in degrees (0o is facing south), and thevertical axis is the altitude h measured in degrees abovethe horizon. The first day of each month is shown asa circle, and the solstices and equinoxes are shown assquares. It can be seen that the equinoxes occur at al-titude h = 90o − φ = 50o, and the solstices appear ataltitudes h = 90o − φ ± ǫ, where ǫ is the axial tilt ofthe Earth (obliquity of the ecliptic, ǫ = 23o 26’). Notethat the analemma is plotted with its width exaggerated,to permit observing that it is asymmetrical. A diamondin Fig. 13 indicates the fixed position of the mean Sunthroughout the year at the selected clock time.

We calculate the analemma by using the Sun equatorialcoordinates (α, δ), and transforming them to horizontalcoordinates (A, h) for a given place on Earth at a giventime. The Sun’s right ascension α is obtained from theexpressions given above in Secs. IV and VI, i.e.,

α = tan−1(cos ǫ tanλ) (89)

and the declination δ is obtained from the expression

sin δ = sinλ sin ǫ (90)

obtained from the spherical triangle shown in Fig. 7.

13

-4 -2 0 2 4

Azimuth (degrees)

-30

-20

-10

0

10

20

30A

ltitu

de (

degr

ees)

1 Jan

1 Mar

1 May1 Aug

1 Oct

Year 2011 - 12 UTC - Latitude 90o

HORIZON

FIG. 14: Analemma for year 2011 at 12 UTC as seen from theNorth Pole. (Note that on the Pole the actual time is ratherirrelevant for the analemma’s shape).

Now, to convert to horizontal coordinates we use thespherical triangle displayed in Fig. 22. In this triangle

cos z = sin δ sin φ + cos δ cosφ cos H (91)

from where we find z ∈ [0, 180o]. Here H is the localhour angle, given by H = θ−α, with θ the local siderealtime (i.e., the hour angle of the vernal point γ). Once zis known, we use the expression:

sin H

sin z=

sin A

cos δ(92)

to obtain the azimuth A ∈ [0, 360o).The analemma is oriented with the smaller loop ap-

pearing north of the larger loop (see Fig. 13). At theNorth Pole, the analemma is totally upright (an 8 withthe small loop at the top), and one is able to see onlythe top half of it (the trajectory of the Sun in half ayear). This is displayed in Fig. 14, where the symbolshave the same meaning as in Fig. 13. We now movesouth and cross the Arctic Circle, then we can see thewhole analemma. If we look at it at noon, it is still up-right, and moves higher from the horizon as we go south.When we are on the equator, the analemma is overhead.If we continue going further south, it moves toward thenorthern horizon, and is now seen with the larger loop atthe top.

At noon the analemma appears rather “vertical” onthe sky. However, it appears inclined when observed atother day times. Imagine now that we are looking atthe analemma in the early morning or evening. Then itstarts to tilt to one side as we move southward from theNorth Pole. In Figs. 15 and 16 we show the analemmaas seen from a place with longitude 0o and latitude 40o

N at 9 UTC and 15 UTC, respectively. When we arriveat the equator, the analemma appears totally horizon-tal. If we continue going south, it still continues rotating

-80 -70 -60 -50 -40

Azimuth (degrees)

10

20

30

40

50

60

Alti

tude

(de

gree

s)

1 Jan

1 Apr

1 Mar

1 June

1 Aug

1 Oct

Year 2011 - 9 UTC - Latitude 40o

FIG. 15: Analemma for year 2011 at 9 UTC as seen from aplace with longitude 0o, latitude 40o N.

40 50 60 70 80

Azimuth (degrees)

10

20

30

40

50

60

Alti

tude

(de

gree

s)

1 Dec

1 Mar

1 June1 Aug

1 Oct

Year 2011 - 15 UTC - Latitude 40o

FIG. 16: Analemma for year 2011 at 15 UTC as seen from aplace with longitude 0o, latitude 40o N.

so that the small loop is beneath the large loop in thesky. When we cross the Antarctic Circle, the analemmaappears almost completely inverted, and it begins to dis-appear below the horizon, and finally only a part of thelarger loop is visible when we are on the South Pole.

If we look at the sky at an earlier (or later) time froma point with latitude 40o N, then we see only a part ofthe analemma, as the rest lies below the horizon. This isshown in Fig. 17 at 6 UTC.

The analemma can be used to find the dates of theearliest and latest sunrises and sunsets of the year, whichdo not occur on the dates of the solstices. An analemmain the eastern sky with its lowest point just above thehorizon corresponds to the latest sunrise of the year, sincefor all other points (dates) on the analemma, the sunrise

14

-120 -100 -80 -60

Azimuth (degrees)

-20

-10

0

10

20A

ltitu

de (

degr

ees)

1 Jan

1 Mar

1 June

1 Aug

1 Oct

Year 2011 - 6 UTC - Latitude 40o

HORIZON

FIG. 17: Analemma for year 2011 at 6 UTC as seen from aplace with longitude 0o, latitude 40o N.

occurs earlier. Therefore, the date when the Sun is at thislowest point is the date of the latest sunrise. Likewise,when the Sun is at the highest point on the analemma,near its top-left end, the earliest sunrise of the year willoccur. Similarly, the earliest sunset will occur when theSun is at its lowest point on the analemma when it isclose to the western horizon, and the latest sunset whenit is at the highest point.

Note that in many places in the web, one reads thatthe east-west component of the analemma is the equationof time, but this is not right. The equation of time is thedifference between solar time and mean time (or betweentrue and mean right ascension), but in the analemma wehave the local azimuth of the Sun, as given by the localcoordinates.

IX. LUNAR PERTURBATION ON THE

EARTH’S ORBIT

As shown above, there appears in our calculated equa-tion of time a modulation apparently due to the Lunarperturbation on the Earth’s orbit, as its period coincideswith the synodic period of the Moon.

To take into account this perturbation, we assume thatthe two-body calculations presented above refer to themotion of the system Earth-Moon around the Sun, i.e.they give the trajectory of the barycenter B of the Earth-Moon system. Now we have to calculate the positionof the Earth respect the barycenter B at each moment.We approach this problem by making some very basicapproximations:

- The actual three-body problem is replaced by a two-body problem (Sun plus barycenter B), and then weadd a correction due to the relative motion of Earth andMoon.

- The orbits of Earth and Moon around B are circular

0 500 1000 1500

Days (since 1 Jan 2011)

-2

-1

0

1

2

Err

or (

diff

eren

ce C

- P

) (

s)

FIG. 18: Difference EC = (∆t)C − (∆t)P from January 2011to December 2014, after correction of the Lunar perturbation,as described in the text..

- The Moon moves on the ecliptic plane.

These approximations, although very crude, areenough for the precision required here.

We then assume that the true anomaly ϕ calculatedin the two-body problem above refers to the barycenterB. We will call ϕ′ the true anomaly of the Earth. Formotion on a plane, we write (x′, y′) for the position ofthe Earth and (x, y) for the position of B. We have:

x′ = x + xT = r cosϕ + d cos θ (93)

y′ = y + yT = r sin ϕ + d sin θ (94)

where (xT , yT ) is the position of the Earth respect B. ris the distance Sun-B (assumed to be constant), d is thedistance Earth-B (also assumed to be constant), and θchanges from 0 to 2π in a sidereal month (period Ts =27.321662 mean days).

To give an initial position for the three-body system,we put θ = ϕ + π for the full Moon, where r′ = r − d(Earth closer to the Sun). Thus

θ = θ0 +2π

Ts(t − t0) (95)

with t0 the instant of a full Moon, and θ0 = ϕ0 + π.

In the calculations, we use the distance ratio R = r/d.Putting r = 149,598,000 km and d = 4678 km, we haveR = 31979.05 (note that B is inside the Earth).

Introducing the modified true anomaly ϕ′ instead of ϕto calculate the equation of time ∆t, we find data thatcompared with (∆t)P from Procivel tables give the resid-ual error shown in Fig. 18. Now the oscillations shownin Figs. 10 and 11 have disappeared almost completely.Taking into account the simplicity of the model, this canbe considered a good accomplishment.

15

Appendix A: Notation

Ellipse:- a : semi-major axis of the ellipse- b : semi-minor axis of the ellipse- e : eccentricity of the conical orbit. For the Earth’sorbit, e = 0.0167- c : linear eccentricity of the ellipse (semi-focal distance,c = ae)- p : parameter of the conic, semi-latus rectum[p = 4C2/k2 = a(1 − e2)]

Orbit:- t : dynamical time, the independent variable in thetheory (Newtonian, ideal)- P : orbital period- λp : ecliptic longitude of the periapsis (= 283.084o on2011-1-1)- Λ = λp + M , ecliptic longitude of the (fictitious)dynamical mean Sun- λ = λp + ϕ, Sun’s true longitude on the ecliptic- T : instant of transit by the periapsis- ǫ: obliquity of the ecliptic (angle between equatorialand ecliptic planes). Now ǫ = 23o 26’- v : velocity- n : average motion (average angular velocity)- α : right ascension of the (true) Sun- αM : right ascension of the mean Sun, αM = λp + M

Anomalies:- ϕ : Sun’s true anomaly- M : Sun’s mean anomaly- E : Sun’s eccentric anomaly

Constants:- G : Newton’s gravitational constant- m0 : Sun mass- m : planet mass- MT = m0 + m, total mass- k =

√GMT , constant for the calculations

- h : constant proportional to the mechanical energy(kinetic plus potential)- C : constant proportional to the angular momentum

Appendix B: Glossary

ORBIT

- ecliptic: plane of the Earth’s orbit around the Sun.In astronomical terms, it is the intersection of the celes-tial sphere with the ecliptic plane. This plane is differentfrom the invariable plane of the solar system, which isperpendicular to the vector sum of the angular momentaof all planets, Jupiter being the main contributor. Thepresent ecliptic plane is inclined to the invariable plane

by about 1.5o.

- obliquity of the ecliptic: angle between the celestialequator and the ecliptic. It is usually denoted as ǫ, andpresently it amounts to about 23o 26’.

- vernal point (Aries point): point on the celestialsphere where the ecliptic intersects with the celestialequator, and the Sun crosses the latter from south tonorth, i.e., its declination changes from negative topositive. This corresponds to the spring equinox in theNorthern hemisphere, and happens around the 20th- 21st of March. Due to axial precession, the vernalequinox moves slowly westward relative to the fixedstars, completing one revolution in about 25,770 years.

- axial precession: gravity-induced, slow and continu-ous change in the orientation of an astronomical body’srotational axis. For our planet, it refers to the gradualshift in the orientation of Earth’s axis of rotation, whichtraces out a pair of cones joined at their apices, in acycle of approximately 25,770 years. Earth’s precessionhas been historically called precession of the equinoxesbecause they move westward along the ecliptic relative tothe fixed stars, opposite to the motion of the Sun alongthe ecliptic. This precession amounts to 0.0140o/year or50.29”/year.

- nutation: a rocking, swaying, or nodding motion inthe axis of rotation of a largely axially symmetric object,such as a gyroscope, planet, or bullet in flight, or as anintended behavior of a mechanism.

- Earth’s nutation: nutation in Earth’s axis mainlydue to tidal forces of the Sun and the Moon, whichcontinuously change location relative to each other. Itcan be decomposed in several components, the largestone having a period of 18.6 years, the same as theprecession of the Moon’s orbital nodes. It reaches plusor minus 17” in longitude and 9” in obliquity. Allother terms are much smaller. The next-largest, witha period of 183 days (∼ half a year), has amplitudes1.3” and 0.6”, respectively. The periods of all termslarger than 0.0001” lie between 5.5 and 6798 days. Toexplain accurately Earth’s nutation, one has to accountfor deformations of the solid Earth.

- periapsis: closest position of a planet to the focus(Sun). For celestial bodies orbiting around the Sun, it iscalled perihelion.

- apoapsis: farthest position of a planet from the focus(Sun). For celestial bodies orbiting around the Sun, it iscalled aphelion.

- perihelion precession (or absidal precession): motionof the perihelion and the major axis of a planet’s ellipticalorbit within its orbital plane, partly in response to per-

16

turbations due to changing gravitational forces exertedby other planets. This causes the orbit to be not reallyan ellipse but a flower-petal shape. Because of apsidalprecession the ecliptic longitude of the Earth’s perihelionslowly increases. It takes about 112,000 years for theellipse to revolve once relative to the fixed stars.

As a consequence of this, the anomalistic year isslightly longer than the sidereal year, while the tropicalyear (which calendars attempt to track) is shorter due tothe precession of Earth’s rotational axis, so that the twoforms of precession add (see Fig. 9). Thus, it takes about21,000 years for the ellipse to revolve once relative to thevernal equinox, that is, for the perihelion to return tothe same date (given a calendar that tracks the seasonsperfectly). The dates of perihelion and aphelion advanceon this cycle an average of one day every 58 years. (Notethat all this refers to the mean perihelion, and does nottake into account short-period changes, as those causedby the presence of the Moon, and that may change theinstant of the actual perihelion in about ± 1 day).

TIME

- apparent solar time or true solar time: time givenby the daily apparent motion of the (true) Sun

- mean solar time: hour angle of the imaginarymean Sun. It is realized with the UT1 time scale.Due to the nonuniformity of the Earth’s angular ve-locity, other more regular procedures are now used tomeasure time, mainly based on atomic clocks (i.e., UTC).

- second: International System (SI) unit of time, de-fined as the duration of 9,192,631,770 periods of the ra-diation corresponding to the transition between the twohyperfine levels of the ground state of 133Cs (caesium).To be precise, this definition refers to a Cs atom at rest ata temperature of 0 K (absolute zero), and with zero exter-nal radiation effects (i.e., zero local electric and magneticfields).

The nuclear spin of 133Cs is I = 72 , and the electronic

spin in the ground state is 2S1/2, so that the total spinof the two lowest levels is F = 3 and 4. The groundstate corresponds to F = 3, which under a magneticfield splits into its seven mF sublevels (mF = −3, ..., 3).

- UT1: principal form of Universal Time. Concep-tually it is mean solar time at 0o longitude. However,precise measurements of the Sun are difficult, and UT1is computed in fact from observations of distant quasarsusing long baseline interferometry, laser ranging of theMoon and artificial satellites. UT1 is the same every-where on Earth, and is proportional to the rotation angleof the Earth with respect to distant quasars, specifically,the International Celestial Reference Frame (ICRF),neglecting some small adjustments. The observationsallow the determination of the Earth’s angle with respectto the ICRF, called the Earth Rotation Angle (ERA,

which serves as a modern replacement for GreenwichMean Sidereal Time).

- UTC (Coordinated Universal Time): atomic timescale that approximates UT1, introduced on 1 January1972 . It is the international standard on which civiltime is based. It ticks SI seconds, in step with TAI(International Atomic Time). It usually has 86,400SI seconds per day, but is kept within 0.9 seconds ofUT1 by the introduction of occasional intercalary leapseconds. Until now (January 2014) these leaps havealways been positive, with a day of 86401 seconds. Whenan accuracy better than one second is not required,UTC can be used as an approximation of UT1. Thedifference between UT1 and UTC is known as DUT1 (=UT1 - UTC). Weekly updated values of DUT1 with 0.1s precision are broadcast by several time signal services.

- leap second: positive or negative one-second ad-justment to the Coordinated Universal Time (UTC)scale that keeps it close to mean solar time. UTC ismaintained using atomic clocks, and is the basis forofficial time-of-day radio broadcasts for civil time. Tokeep the UTC time scale close to mean solar time, itis occasionally corrected by an intercalary adjustmentof one second (“leap” second), so that UTC remainswithin the range -0.9 s < DUT1 < +0.9 s. There aretwo reasons for this correction: (1) the rate of rotationof the Earth is not constant, due to tidal braking andredistribution of mass within the Earth, including oceansand atmosphere, and (2) the SI second (used for UTC)was already, when adopted, shorter than the second ofmean solar time. The timing of leap seconds is nowdetermined by the International Earth Rotation andReference Systems Service (IERS). Since June 1972,there have been 25 leap seconds (all positive) untilJanuary 2014. The most recent one happened on June30, 2012 at 23:59:60 UTC.

- TAI (International Atomic Time): high-precisionatomic coordinate-time standard based on proper timeon Earth’s geoid. It is the basis for Coordinated Uni-versal Time (UTC), which is used for civil timekeepingall over the Earth, and for Terrestrial Time, used inAstronomy. Since 2012-6-30 when the last leap secondwas added, TAI has been 35 seconds ahead of UTC.This is still true as of 2014-1-1. These 35 seconds resultfrom the initial difference of 10 seconds at the start of1972, plus 25 leap seconds introduced in UTC since1972. Thus, in January 2014, UTC = TAI – 35 s.

- TT (Terrestrial Time): modern astronomical timestandard defined by the International AstronomicalUnion, primarily for time-measurements of astronomicalobservations made from the surface of the Earth.Presently, the Astronomical Almanac uses TT for itstables of positions of the Sun, Moon and planets as seenfrom the Earth. TT continues Terrestrial Dynamical

17

Time (TDT), which in turn succeeded ephemeris time(ET). The unit of TT is the SI second, but it is notactually defined by atomic clocks. It is a theoreticalideal, which real clocks can only approximate. TT isdistinct from UTC, the time scale currently used as abasis for civil purposes, and indirectly underlies UTCvia International Atomic Time (TAI). To millisecondaccuracy, TT runs parallel to TAI, and can be approxi-mated as TT = TAI + 32.184 s. (This offset arises fromhistorical reasons.) Since TAI = UTC + 35 s, we haveTT = UTC + 67.2 s.

- Greenwich Mean Time (GMT): a term originallyreferring to mean solar time at the Royal Observatory inGreenwich, London. It is arguably the same as Coordi-nated Universal Time (UTC) and when this is viewed asa time zone the name Greenwich Mean Time is especiallyused by bodies connected with the United Kingdom,such as the BBC World Service, the Royal Navy, theMet Office, and others. Before the introduction of UTCon 1 January 1972 Greenwich Mean Time was the sameas Universal Time (UT), which is a standard astronom-ical concept used in many technical fields. The term“Greenwich Mean Time” is no longer used in Astronomy.

- Ephemeris time (ET): refers to time in connectionwith any astronomical ephemeris. A practical definitionwas proposed in 1948, to overcome the drawbacks of ir-regular fluctuations in mean solar time, and thus define auniform time based on Newtonian theory. ET is a (clas-sical) dynamical time scale, defined implicitly from theobserved positions of astronomical objects through thedynamical theory of their motion.

The unit and origin of ET are usually defined by adopt-ing a numerical expression for the geometric mean longi-tude of the Sun. For this purpose, it has been tradition-ally employed the Newcomb formula:

λm = 279o41′48.04” + 129, 602, 768.13” T + 1.089” T 2

where T is counted in Julian centuries of 36525 ephemerisdays. The origin of T is at the beginning of year 1900,when λm took the value 279o 41’ 48.04”. This instantof time is dated 1900 January 0, 12 h ET exactly (asdecided in 1958). Some corrections have been proposedto obtain more precisely the solar longitude. However, itwas agreed in 1961 that “The origin and rate of ephemeristime are defined to make the Sun’s mean longitude agreewith Newcomb’s expression”.

We note that ET can also refer to relativistic coor-dinate time scales, as that implemented by the JPLephemeris time argument Teph.

YEAR

- sidereal year: time taken by a planet to orbit theSun once with respect to the fixed stars. For the Earth,it amounts to 365.2564 mean days (365 d 6 h 9 min 9.76

s) (at the epoch J2000.0 = 2000 January 1 12:00:00 TT).

- tropical year: period of time for the ecliptic longitudeof the Sun to increase by 360o, i.e., length of time thatthe Sun takes to return to the same position in the cycleof seasons. The mean tropical year is 365.24219 meandays (365 days, 5 hours, 48 minutes, 45 seconds). In theGregorian calendar, one (mean) year = 265.2425 meandays. Because of the Earth’s axial precession and theassociated shift of vernal equinox, this year is about 20minutes shorter than the sidereal year. In fact, the ver-nal point precesses by ∆ω = 50.29”/year in the oppositedirection to the apparent Sun motion (∆ω = 360o/25770years), which converts into δt = 365.2422 ∆ω/360o =0.01417 days = 20.4 min.

- anomalistic year: time taken for the Earth tocomplete one revolution with respect to the apsides ofits orbit (perihelion and aphelion). It is usually definedas the time between perihelion passages. Its average du-ration is 365.25964 days (365 d 6 h 13 min 52.6 s) at theepoch J2011.0. The perihelion moves ∆ω = 360o/112000years = 0.00321o/year in the direction of the apparentSun motion, i.e., it approaches the vernal point. Then,an anomalistic year is longer than a sidereal year byδt = 365.2564/112000 = 0.00326 days = 4 min 42 s.The interaction between the anomalistic and tropicalcycle seems to be important in the long-term climatevariations on Earth, called the Milankovitch cycles.

- Besselian year: a year that begins at the momentwhen the mean longitude of the Sun is exactly 280o. Thismoment falls near the beginning of the correspondingGregorian year. The definition depends on a particulartheory of the orbit of the Earth around the Sun, thatof Newcomb (1895), which is now obsolete. For thatreason among others, the use of Besselian years has alsobecome obsolete.

- Julian year (average): 365.25 days of 86,400 seach, totaling 31,557,600 s. The Julian year is theaverage length of the year in the Julian calendar usedin Western societies in previous centuries, and forwhich the unit is named. It included a leap yearevery four years without exception. Now it does not cor-respond to any of the many other ways of defining a year.

- Gregorian year (average): year on which our presentcalendar is based, 365.2425 days = 52.1775 weeks= 8,765.82 hours = 525,949.2 minutes = 31,556,952seconds (mean solar, not SI), which approximates verywell a tropical year of 365.2422 days. The 400-year cycleof the Gregorian calendar has 146,097 days and henceexactly 20,871 weeks (see “leap year”).

- leap year (or intercalary or bissextile year): year con-taining one extra day (or, in the case of lunisolar cal-endars, one month) in order to keep the calendar year

18

synchronized with the astronomical or seasonal year. Be-cause seasons and astronomical events do not repeat ina whole number of days, a calendar that had the samenumber of days in each year would, over time, drift withrespect to the event it was supposed to track. By oc-casionally inserting an additional day or month into theyear, the drift can be approximately corrected. A yearthat is not a “leap year” is called a “common year”.

In the Gregorian calendar that we currently employ(a solar calendar), February in a leap year has 29 daysinstead of the usual 28, so the year lasts 366 days insteadof the usual 365. Similarly, in the Hebrew calendar (alunisolar calendar), a 13th lunar month is added 7 timesevery 19 years to the twelve lunar months in its commonyears, to keep its calendar year from drifting through theseasons too rapidly.

Since a tropical year lasts 365.2422 mean days, theGregorian calendar includes 97 leap years every 400years, which gives a mean duration per year of 365.2425days. The rule is the following:

(1) If a year number is evenly divisible by 4 and notby 100, then it is a leap year.

(2) Years that are evenly divisible by 100 are not leapyears, unless they are also evenly divisible by 400, inwhich case they are leap years (for example, 1700, 1800,and 1900 were not leap years, but 1600 and 2000 were).

- epact (Latin, epactae): age of the Moon in days(number of days from the last new Moon) on January 1.It mainly appears in connection with tabular methodsfor determining the date of Easter, and varies (usuallyby 11 days) from year to year, because of the differencebetween the solar year of 365 days and the lunar year of354 days.

DAY

- apparent solar day: the interval between two suc-cessive returns of the (true) Sun to the local meridian.It can be up to about 20 seconds shorter or 30 secondslonger than a mean day.

- mean solar day: 24 hours. In principle, in theInternational System, one day = 24 hours = 86,400 s.However, the Earth’s day has increased in length overtime, mainly due to tides raised by the Moon whichslow Earth’s rotation. Because of the way the secondhas been defined in the 20th century, the mean lengthof a solar day is now (beginning of the 21st century)fluctuating around values of 1-2 ms longer than 86,400 s.Averaging over short-time fluctuations, it is increasingby about 1.7 ms per century (an average over the last2,700 years). The length of one day should be about 21.9hours 620 million years ago, as recorded by rhythmites(alternating layers in sandstone). (Note that at anincreasing of 1.7 ms/century and assuming it constantover geological periods, we extrapolate backwards tofind a day of 21 hours 4 min, somewhat shorter than

TABLE I: Comparison between different years and days.

Mean days seconds

Tropical year 365.2422 31,556,926

Sidereal year 365.2564 31,558,153

Anomalistic year 365.2596 31,558,429

Mean day 1 86,400

Sidereal day 0.997270 86,164

the value given above. To coincide with the rhythmitesdata, we need an average increase of 1.2 ms/century).

- synodic day: period of time it takes for a planetto rotate once in relation to the body it is orbiting (asopposed to a sidereal day which is one complete rotationin relation to the stars). For Earth, synodic day is thesame as solar day, and is about 24 hours long.

- stellar day: an entire rotation of a planet withrespect to the distant (fixed) stars. For the Earth itamounts to 86164.098 s of mean solar time (UT1),i.e., it is 3 min 56 s shorter than a mean solar day.This difference between stellar and mean solar daycorresponds to an angle ∆ϕ = 360o/365.2425 days =0.9856o that the Earth moves in its orbit around the Sun(which converts into ∼ 4 min in the rotational motion).

- sidereal day: the time it takes the Earth to makeone rotation relative to the vernal equinox. A meansidereal day is 23 hours, 56 min, 4.091 s (23.93447 hoursor 0.99726957 mean solar days). Since the mean vernalequinox precesses slowly westward relative to the fixedstars, completing one revolution in about 25,770 years,the misnamed sidereal day is some 9 ms shorter thanthe Earth’s period of rotation relative to the fixed stars(stellar day). This comes from ∆ϕ = 360o/(25770×365)days = 3.8o × 10−5/day, i.e. δt = 4 × 3.8 × 10−5 min =9 ms.

Appendix C: Some mathematical expressions

Here we summarize some expressions employed alongthe notes.

(1) Some derivativesWe have r · r = rr, since

r =d

dt(r · r) 1

2 =1

2

2r · r(r · r) 1

2

=r · rr

(C1)

Also:

d

dt

(

1

r

)

= − 1

r2

dr

dt= −r · r

r3(C2)

19

(2) Vector triple product

a × (b × c) = (a · c)b − (a · b)c (C3)

(3) Scalar triplet product

a · (b × c) = c · (a × b) = b · (c × a) (C4)

(4) Derivative of the distance r

The equation of the conic in polar coordinates is:

r =p

1 + e cosϕ(C5)

Taking a time derivative:

r = pe ϕ sin ϕ

(1 + e cosϕ)2=

r2

pe ϕ sinϕ (C6)

and

r =e

pr2ϕ sin ϕ =

2 e C

psinϕ (C7)

since C = 12r2ϕ [Eq. (18)]

Appendix D: Motion in polar coordinates

For many purposes, it can be convenient to describe aplanar motion in polar coordinates, rather than Carte-sian coordinates. Then, one has the coordinates (r, ϕ),related to the Cartesian coordinates by: x = r cosϕ, andy = r sin ϕ. This gives for the components of the velocity:

x = r cosϕ − rϕ sinϕ (D1)

y = r sin ϕ + rϕ cosϕ

Then, in three-dimensional space we have r = (x, y, 0)and r = (x, y, 0), so that r × r = (0, 0, xy − yx), and thethird component of the cross product results to be forkeplerian motion 2C = r2ϕ. [Remember that the crossproduct is a constant of motion that we called: r × r =2C, see Eq. (16)].

For the velocity we have:

v2 = |r|2 = x2 + y2 = r2 + r2ϕ2 (D2)

A rather simple way of dealing with polar coordinatesfor planar motion is considering the position vector asa complex variable r ≡ r(cos ϕ + i sin ϕ) = r exp(iϕ).Taking into account that i exp(iϕ) = exp(i(ϕ+π/2)), wehave:

v =dr

dt≡ reiϕ + rϕ ei(ϕ+ π

2) (D3)

This means that r has components r and rϕ in the di-rections parallel and perpendicular to r, respectively.

r

ϕ

r

^^

ϕ

FIG. 19: Mobile unit vectors r and ϕ used to study two-dimensional motion.

Taking another time derivative, we find for the accel-eration:

a =dv

dt≡ (r − rϕ2) eiϕ + (2rϕ + rϕ) ei(ϕ+ π

2) (D4)

which yields for the radial and tangential components ofthe acceleration:

ar = r − rϕ2

at = 2rϕ + rϕ (D5)

For gravitational forces one has at = 0. For centralforces in general we have r‖r, and the vector C = 1

2r× r

is a constant of motion [see Eq. (16)]. In particular, itsmodule C is a constant given by C = 1

2r2ϕ [see Eq. (18)].Thus

dC

dt= rrϕ +

1

2r2ϕ = 0 , (D6)

and then (since r 6= 0): 2rϕ + rϕ = 0, so that our calcu-lation is consistent with at = 0.

Another interesting point is that the 1/r2 law for thegravitational force can be derived assuming that the orbitis elliptical (as in fact was done from Kepler’s observa-tions). We have:

r =p

1 + e cosϕ, r =

2 e C

psin ϕ , r =

2 e C

pcosϕ ϕ

(D7)Replacing these expressions into ar = r−rϕ2, and takinginto account that C = 1

2r2ϕ, we find

ar = −2Cϕ

p= −4C2

p

1

r2= −k2

r2= −GMT

r2(D8)

Another equivalent way of dealing with two-dimensional motion consists in using a mobile coordi-nate system with unit vectors r and ϕ (see Fig. 19). r

is parallel to the position vector, r = r/r, i.e., it hascomponents (cosϕ, sin ϕ), and the unit vector ϕ, perpen-dicular to r, has components (cos(ϕ+π/2), sin(ϕ+π/2))= (− sinϕ, cosϕ). Now note that the time derivative ofa unit vector is perpendicular to it:

|r| = 1 , r · r = 1 ,d

dt(r · r) = 2r · dr

dt= 0 ,

dr

dt⊥ r.

(D9)

20

In fact, for the derivatives of the unit vectors we have:

dr

dt= (− sin ϕ ϕ, cosϕ ϕ) = ϕ ϕ (D10)

dϕ

dt= (− cosϕ ϕ,− sinϕ ϕ) = −ϕ r (D11)

Then, the derivative of the position vector r = r r, is:

r = rr + rϕϕ (D12)

Taking a second derivative, we find:

r = (rr + rdr

dt) + (rϕϕ + rϕϕ + rϕ

dϕ

dt) (D13)

or

r = (r − rϕ2)r + (2rϕ + rϕ)ϕ , (D14)

where we recognize the components ar and at given abovein Eq. (D5).

Appendix E: Effect of the obliquity on the equation

of time

Here we give some details on the influence of the obliq-uity of the ecliptic on the equation of time. As inSec. III.B we assume a circular trajectory on a planeforming an angle ǫ with the equator. From the sphericaltriangle shown in Fig. (7), we have:

tan α = tanλ cos ǫ (E1)

To obtain variations of α as a function of λ, we calculatethe derivative

dα

dλ=

cos2 α

cos2 λcos ǫ =

1 + tan2 λ

1 + tan2 αcos ǫ (E2)

and for a one-day interval we can approximate:

∆α =cos2 α

cos2 λcos ǫ ∆λ (E3)

- At the equinoxes, cosα = cosλ = 1, and ∆α =0.9175 ∆λ.

- At the solstices, cosα = cosλ = 0, and the quotientin Eq. (E3) is singular. The limit at those points canbe easily found by replacing tanλ from Eq. (E1) intoEq. (E2), which yields

dα

dλ=

cos2 ǫ + tan2 α

1 + tan2 α

1

cos ǫ, (E4)

and from here:

limα→π/2

dα

dλ=

1

cos ǫ, (E5)

which gives for the summer solstice: ∆α = ∆λ/ cos ǫ, or∆α = 1.0899 ∆λ (and the same for the winter solstice).

P

E

P’

ϕ

a ry

xFO

FIG. 20: Given a point P on the ellipse, its eccentric anomaly

is the angle E, given by cos E = (x + c)/a. P ′ is the cor-responding point to P , on a circumference of radius a (thesemi-major axis of the ellipse).

Appendix F: Eccentric anomaly

For an ellipse with semi-axes a and b, we define an aux-iliary circumference of radius a that circumscribes the en-tire ellipse. Let us consider a point P on the ellipse, withpolar coordinates (r, ϕ), or Cartesian coordinates (x, y).The eccentric anomaly corresponding to point P is theangle E in Fig. 20. This angle is defined as follows. Drawthe straight line parallel to the (vertical) semi-minor axisand passing through P (there is one and only one parallel,according to Euclid). This line intersects the circumfer-ence in point P ′ (see Fig. 20; there is another intersectionpoint on the other side of the major axis, that we do notconsider). P ′ is called the corresponding point to P . Theradius of the auxiliary circle passing through P ′ makesan angle E with the major axis.

Taking the origin of coordinates on the focus F , wehave for the ellipse in Cartesian coordinates [see Eq. (27)]

(x + c)2

a2+

y2

b2= 1 (F1)

and for the angle E:

cosE =x + c

a(F2)

From these two equations, one has sin E = y/b.To find out the distance r as a function of E, we write

r2 = x2+y2 = (a cosE−ae)2+b2 sin2 E = a2(1−e cosE)2

(F3)so that

r = a(1 − e cosE) (F4)

which coincides with Eq. (47).

21

Appendix G: Area of the ellipse

Taking into account that the elementary area is

dS =1

2r.rdϕ =

1

2r2dϕ (G1)

we can calculate the area swept in a complete revolution(the area of the ellipse). Thus we have in polar coordi-nates:

S =

∫ 2π

0

dS =

∫ π

0

r2 dϕ (G2)

Using for r the expression in Eq. (25) for the conicaltrajectory, one has

S =

∫ π

0

p2

(1 + e cosϕ)2dϕ (G3)

To calculate this integral, we divide it into two parts:

S =p2

1 − e2

∫ π

0

dϕ

1 + e cosϕ+

ep2

e2 − 1

∫ π

0

e + cosϕ

(1 + e cosϕ)2dϕ

(G4)We calculate a primitive for the first part using thechange t = tan ϕ

2 , and find

∫

dϕ

1 + e cosϕ=

2

(1 − e2)3/2tan−1

[

√

1 − e

1 + etan

ϕ

2

]

(G5)For the second part, a primitive is:

∫

e + cosϕ

(1 + e cosϕ)2dϕ =

sin ϕ

1 + e cosϕ. (G6)

Taking the integration limits, the second part vanishes,so that only the first part remains, and

S =2p2

(1 − e2)3/2

π

2=

πp2

(1 − e2)3/2(G7)

Since

a b =p2

(1 − e2)3/2, (G8)

[see Eq. (28)] we finally find

S = πa b (G9)

Note that the area of an ellipse can be more easilycalculated in Cartesian coordinates:

x2

a2+

y2

b2= 1 , (G10)

and

S = 2

∫ a

−a

y+(x) dx = 2

∫ a

−a

b

√

1 − x2

a2dx . (G11)

Putting x = a sin θ, we find

S = 2ab

∫ π/2

−π/2

cos2 θ dθ = πa b (G12)

S

N

δ

Hn

s

Z

90 − h

A

90 − φ

φ

Horizon

Equatorw

FIG. 21: Celestial sphere including details of equatorial andhorizontal coordinate systems. φ: local latitude, h: altitude,A: azimuth (measured from the local south), H : local hourangle (H = θ − α; θ, local sidereal time), Z: zenith, N andS: North and South poles in the equatorial system. δ: decli-nation

Appendix H: Astronomical coordinate systems

1) Horizontal coordinates

Horizon: fundamental plane.

Primary direction: north or south point of horizon.

Zenith (Z): pole of the upper hemisphere.

Nadir: pole of the lower hemisphere.

Zenith distance (z): complement of altitude (i.e. z =90o − h).

Vertical circle: great circle on the celestial sphere thatpasses from the observer’s zenith through a given celestialbody. Vertical circles are perpendicular to the horizon.

Principal vertical (or Local Celestial Meridian): verti-cal circle which is on the north-south direction.

Prime vertical: Vertical circle on the east-west direc-tion.

Azimuth (A): angle between planes ofthe principal vertical and the vertical cir-cle of a celestial object. We measure it herefrom the south increasing towards the west. Nowa-days, it is more usual to measure it from the northincreasing towards the east.

Altitude (h) (also called elevation): angle between acelestial object and the observer’s local horizon. For ob-jects above the horizon, h ∈ (0, 900].

Almucantar (also spelled almucantarat or almacan-tara): a circle on the celestial sphere parallel to thehorizon. Celestial objects lying on the same almucantarhave the same altitude.

22

Q z = 90 − h

90 − δ

P

Z

90 − φH

N

180 − A

FIG. 22: Spherical triangle employed to pass from equato-rial to horizontal system (see Fig. 21). φ: local latitude, h:altitude, δ: declination, A: azimuth (measured from the lo-cal south), H : local hour angle (H = θ − α, θ local siderealtime), Q: parallactic angle, Z: zenith, N : North pole in theequatorial system. P : position of the celestial object.

2) Equatorial coordinates

Celestial Equator: fundamental plane.Primary direction: vernal point γCelestial Poles: North and South.Meridian: great circle passing through the celestial

poles and the zenith of a particular place on Earth. Itcontains the horizon’s north and south points and is per-pendicular to the celestial equator and the celestial hori-zon. The meridian is divided into the local meridian(which contains the zenith and is terminated by the celes-tial poles) and the antimeridian (opposite half containing

the nadir).Hour circle of a celestial object is the great circle

through the object and the celestial poles. It is perpen-dicular to the celestial equator.

Declination (δ): angular distance of a point on thecelestial sphere from the celestial equator, measured onthe great circle passing through the celestial poles and thepoint in question. Points north of the celestial equatorhave δ > 0, while those south have δ < 0. Declination isusually measured in sexagesimal degrees (o), minutes (’),and seconds (”).

Right ascension (α): angular distance on the celestialequator measured eastward from the vernal equinox tothe hour circle of a given point on the celestial sphere. Itis usually measured in hours, minutes and seconds (time).

Hour angle (H): angular distance on the celestialsphere measured westward along the celestial equatorfrom the local meridian to the hour circle passing througha given point or celestial object. The hour angle is relatedto the right ascension through the local sidereal time θ:H = θ−α. Note that, unlike right ascension, hour angleis always increasing with the rotation of the Earth.

Sidereal time (θ): at a given Earth’s place defined byits geographical longitude, and any moment, the siderealtime is the hour angle of the vernal equinox at that place.It has the same value as the right ascension of any ce-lestial object that crosses the local meridian at the sametime. At the moment when the vernal point γ crosses thelocal meridian, the local sidereal time is 00:00. Green-wich sidereal time is the hour angle of the vernal equinoxat the prime meridian at Greenwich.