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Home , Axial precession, Dynamical time scale, Equation of time, Position of the Sun

SOME NOTES ON THE EQUATION OF

by Carlos Herrero

Version v2.1, February 2014

I. INTRODUCTION N

Since ancient humans have taken the as a reference for measuring time. This seems to be a natu- ral election, for the strong influence of the Sun on our daily life, with a perpetual succession of days and nights. However, it has also been observed long time ago (e.g., ε ancient Babylonians) that our Sun is not a perfect time keeper, in the sense that it sometimes seems to go faster, λ δ and sometimes slower. In particular, it is known that the time interval between two successive transits of the γ α Sun by a given is not constant along the . 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 defined the to quantify deviations of the time directly measured from the Sun position respect to an assumed perfect time keeper. In fact, the equation of S time is the difference between apparent and mean solar time (as yielded by in modern times). FIG. 1: showing the on the ecliptic. α, ; δ, ; λ, ecliptic At any given instant, this difference will be the same for . γ indicates the vernal point, and ǫ is the obliquity every observer on the . of the ecliptic. Apparent (or true) solar time can be obtained for ex- ample by measuring the current position ( angle) of the Sun, as indicated (with limited accuracy) by a sun- time for that . (Another method used stellar ob- dial. Mean solar time, for the same place, would be the servations to give , in combination with the time indicated by a steady set so that over the year relation between sidereal and solar time.) Values of the its differences from apparent solar time average to zero equation of time for each of the year, compiled by as- (with zero net gain or loss over the year). Apparent time tronomical observatories, were widely listed in can be ahead (fast) by as much as 16 min 33 s (around and ephemerides. Now, it can be found in many places, 3 November), or behind (slow) by as much as 14 min 6 in particular in numerous web pages, e.g., the so-called s (around 12 February). The equation of time has zeros Procivel in Rodamedia.com. near 15 April, 13 June, 1 September, and 25 December. Note that the name “equation of time” can be mislead- It changes slightly from one year to the next. ing, as it does not refer to any equation in the modern The graph of the equation of time is closely approx- sense of this word (a mathematical statement that asserts imated by the sum of two curves, one with a pe- the equality of two expressions, often including quanti- riod of a year and another with a period of half a year. ties yet to be determined, the unknowns). Here, the These curves reflect two effects, each causing a different equation is employed in its Medieval sense, taken from non-uniformity in the apparent daily of the Sun the Latin term aequatio (which means equalization or relative to the stars: the obliquity of the ecliptic, which adjustment), and that was used for ’s difference is inclined by about 23.44o relative to the plane of the between mean and true solar time. Earth’s equator, and the eccentricity of the Earth’s orbit In the following we present some questions related to around the Sun, which is about 0.017. the equation of time. For convenience, we will consider The equation of time has been used in the past to set motion of the Earth around the Sun or motion of the clocks. Between the invention of rather accurate clocks Sun as seen from Earth, depending on the discussion around 1660 and the advent of commercial time distri- at hand. For example, when discussing orbital motion bution services around 1900, one of two common land- we have in mind the movement of the Earth. However, based ways to set clocks was by observing the passage of when displaying the celestial sphere it is the Sun that the Sun across the local meridian at . The moves on the ecliptic, as shown in Fig. 1. the Sun passed overhead, the clock was set to noon, off- set by the number of given by the equation of These notes are organized as follows: 2 N

b p r Ecliptic c ϕ a F S F S Equator T Λ y γ α S x M M

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

- In Sec. II we define the “artificial ” that are used S to (presumably) simplify the discussion on the motion of the true Sun and the definition of mean time. FIG. 3: Celestial sphere displaying the position of the (true) - In Sec. III we discuss the two major contributions Sun ST and the fictitious (dynamical mean) Sun SF on the to the equation of time: eccentricity of the orbit and ecliptic, as well as the mean Sun SM in the equator. P in- dicates perigee; γ, vernal point; αM , right ascension of the obliquity of the ecliptic. mean Sun; Λ, ecliptic longitude of SF . Note that αM = Λ. - In Sec. IV we present the mathematical details of our calculations, based only on Newton’s laws and elliptic orbits. and slower near the apogee (aphelion for the Earth). We - The position of the Gregorian on the Earth’s will call it S . Its position is given by the , orbit is discussed in Sec. V. T ϕ, which is the angle between S and the perigee; see - In Sec. VI we present a schematic way to calculate T Fig. 2. rather precisely the equation of time with some very basic assumptions and simplifications. 2 - Fictitious Sun (or dynamical mean Sun). This is - In Sec. VII we compare the results of our calculations only an intermediate tool between true Sun and mean with those found by using other approaches. Sun. We will call it SF . It moves on the celestial sphere - In Sec. VIII we present the . following the ecliptic with uniform motion. To be precise, - In Sec. IX we introduce a simple correction to the SF is an imaginary body that moves uniformly on the equation of time, due to the lunar perturbation. ecliptic with the mean of the true Sun, - Finally, we give some appendixes, including the no- and which coincides with ST at perigee and apogee. The tation we have employed, a glossary of the main terms, position of SF is given by its ecliptic longitude Λ; see and some mathematical formulas. Fig. 3. 3 - Mean Sun. We call it SM . It moves uniformly on the equator. Its position is measured by the mean II. HOW MANY SUNS ? anomaly M on the equator, in contrast with SF for which the position is measured on the ecliptic. To be concrete, the mean Sun is supposed to move on A useful tool to study the equation of time is the the equator in such a manner that it right ascension, so-called mean Sun, which is a mental artifact giving us α , is equal to the ecliptic longitude of the (fictitious) a reliable time keeper, that should coincide with our best M dynamical mean Sun, Λ. S is directly related to our clocks if the Earth had a constant speed (which M clocks, as it is used to define the solar mean time. unfortunately is not the case). This due to the variable angular velocity will not be considered here, as in a first approximation is not relevant for our Now we put the three objects in motion: calculations. Its influence on the different time scales 1 - ST does not need to start, it has been moving for presently used can be found in the glossary at the end many on the ecliptic. of the text. 2 - Now we humans wait until ST goes through the perigee, and then SF starts on the same position and the 1 - True Sun (or apparent Sun). Suns there is only one, same direction as ST . Since SF moves uniformly on the the others are mental artifacts to simplify the calculations ecliptic, at the beginning it will move slower than ST . and mainly to understand the whole thing. The true Sun 3 - We wait until SF arrives at the vernal point γ. At moves on the ecliptic with nonuniform velocity, i.e., it that moment, the mean Sun SM starts moving at γ with goes faster close to the perigee (perihelion for the Earth) the same velocity as SF , BUT ON THE EQUATOR. 3

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

Note that when we will simply speak about the Sun, 5 we will obviously mean “true” Sun.

We will define the equation of time ∆t at a certain 0 moment as the difference between the right ascension of the mean Sun and that of the true Sun:

∆t = α α (1) Difference M - phi -5 M − Equivalently, it is the difference between the hour angles of the mean Sun and true Sun: ∆t = H HM . We -10 − 0 100 200 300 note that sometimes ∆t is defined as α αM , but this definition will not be used here. With our− definition of , M (degrees) equation of time, ∆t > 0 means that the Sun crosses a given meridian before SM , and ∆t< 0 indicates that ST crosses it after SM . Thus, for ∆t > 0 the Sun is ahead FIG. 4: Contribution of the eccentricity of the Earth’s orbit to (fast), and for ∆t< 0 it is behind (slow). the equation of time (given in minutes of time), as a function of the mean anomaly M. It is assumed that the motion takes place on the equator plane. III. THE TWO MAJOR CONTRIBUTIONS TO THE EQUATION OF TIME

This is a qualitative explanation of the origin of the equation of time. For a more rigorous calculation one ∆ϕ should go to Sec. IV. Earth

A. Eccentricity of the Earth’s orbit ∆ϕ As seen from Earth, the Sun appears to revolve once around the Earth through the background stars in one year. If the Earth orbited the Sun with a constant speed, in a and on a plane perpendicular to the Sun Earth’s axis, then the Sun would culminate (would cross FIG. 5: Schematic representation of the Earth’s motion a given meridian) every day at exactly the same time. In around the Sun. The angle ∆ϕ swept by the Earth (as seen that hypothetical case, the Sun would be a rather good from the Sun) in one day is the same as the angle that the time keeper, similar to the UTC time given by modern Earth has to rotate to complete a solar day, in addition to the atomic clocks (except for the small effect of the slowing 360o corresponding to a sidereal day. rotation of the Earth). But the orbit of the Earth is an ellipse, and thus: (1) its orbital speed varies by about 3.4% between aphelion and perihelion (29.291 and 30.287 km/s), ac- 7.66 minutes and a period of one year to the equation of cording to Kepler’s laws of planetary motion; time. The zero points are reached at perihelion (at the (2) its angular velocity changes accordingly, being beginning of January) and aphelion (beginning of July), maximum at the perihelion and minimum at the aphe- while the maximum values are in early April (negative) lion, and and early October (positive). In Fig. 4 we plot this vari- (3) the Sun appears to move faster (in its annual mo- ation along the year, as calculated by the method de- tion relative to the background stars) at perihelion (cur- scribed in Sec. IV. The mean anomaly appearing in this rently around January 3) and slower at aphelion a half plot is defined as the angle from the periapsis to the dy- year later. namical mean Sun. At these extreme points, this causes the apparent solar A simple (although not rigorous) derivation of the day to increase or decrease by about 7.9 s from its mean maximum changes in the solar day length, caused by el- of 24 . This daily difference accumulates along the lipticity of the orbit, is the following (see Fig. 5). For an days. As a result, the eccentricity of the Earth’s orbit elliptical orbit, we know, from conservation of the angu- contributes a sine wave variation with an amplitude of lar momentum, that [this is explained with more detail 4 below, see Eq. (19)] 10 1 1 C = r2ϕ˙ rv (2) 2 ≈ 2 5 (r: distance Sun–Earth; v: velocity) since v rϕ˙ (be- cause the eccentricity e 1) and C is a constant.≈ Now ≪ we call ∆ϕ the change of ϕ in one solar day, i.e. between 0 two successive returns of the Sun to the local meridian. For small ∆ϕ (as happens for one day): -5 2C

∆ϕ ∆t (3) Difference alpha - alpha_M ≈ r2 -10 Then, putting “P” for perihelion and “A” for aphelion, we have: 0 100 200 300 Mean anomaly, M (degrees) (∆ϕ) r2 1+ e 2 P = A = =1.0691 (4) (∆ϕ) r2 1 e A P  −  FIG. 6: Contribution of the obliquity of the ecliptic to the (We have used e = 0.0167 for the Earth’s orbit). This equation of time (measured in minutes of time), as a function means that ∆ϕ varies in about 6.9% from its maximum of the mean anomaly M. It is assumed that the orbit is to its minimum value, i.e. 3.45% with respect to the ± circular. mean day (24 hours). For the mean day, (∆ϕ)M = 360o/365.2564 days = 0.9856o/day. This means that in average the Earth sweeps in its translational motion an and , while the extreme values appear at the be- angle of 0.9856o per day, which is exactly the angle that ginning of February and August (negative), and the be- the Earth has to rotate in addition to 360o to complete ginning of May and November (positive). In Fig. 6 we a whole solar day (see Fig. 5). This angle corresponds plot this variation along the year, as calculated by the to a delay of 3.94 min of the mean solar day respect the method described in Sec. IV. sidereal day. The actual delay will change along the year, As indicated above, the contribution of obliquity to the so that close to the perihelion it will be larger (the Earth change in duration of apparent solar days is maximum at moves faster), and near the aphelion it will be shorter. the solstices and equinoxes. For the latter, we now derive Thus, the difference with respect to the mean day will be in a simple way this contribution, assuming that the orbit at perihelion and aphelion: 0.0345 3.94 min 8 s. is circular. First note that the daily change in ecliptic ± × ≈± longitude is: ∆λ = 360o/365.2564 = 0.9856o/day. From the spherical triangle shown in Fig. 7, we have: B. Obliquity of the ecliptic tan α = tan λ cos ǫ (5) If the Earth’s orbit were circular, the motion of the Sun, as seen from the rotating Earth, would still not be and the daily change in right ascension at the uniform. This is a consequence of the tilt of the Earth’s will be: rotation axis with respect to its orbit, or equivalently, to − the obliquity of the ecliptic with respect to the equator. ∆α = tan 1[tan(∆λ) cos ǫ] , (6) The projection of this motion onto the , along which “clock time” is measured, is a maximum at which gives ∆α = 0.9042o. This translates into a time the solstices, when the yearly movement of the Sun is interval δt = 3.6168 min = 3 min 37 s, which is the differ- parallel to the equator and appears as a change in right ence between the corresponding solar day and a sidereal ascension (the time derivative of the solar declination δ is day. (This is similar to the discussion above in Sec. III.A, then zero, dδ/dt = 0 ). That projection takes a minimum see Fig. 5) Taking into account that for a mean day δt = at the equinoxes, when the Sun moves in a sloping direc- 3 min 56 s, we find that at the equinoxes the obliquity tion and dδ/dt is maximum, leaving less for the change contributes to shorten the solar day in about 20 s. In in right ascension,| | which is the only component that af- Appendix E we give some more details on the estimation fects the duration of the solar day. At the equinoxes, the of δt at different times. Sun is seen slowing down by up to 20.3 every day Note: As a rule of thumb, if a given day the change and at the solstices speeding up by a similar amount. of solar right ascension ∆α is larger than the mean value Concerning the equation of time, the obliquity of the (∆α)M , then the Sun takes more than 24 hours to return ecliptic contributes a sine wave variation with an ampli- to a meridian (Sun slower than clock). On the contrary, tude of 9.87 minutes and a period of half a year. The if ∆α< (∆α)M , the Sun turns to a meridian in less than zero points of this sine wave are reached at the equinoxes 24 hours (Sun faster than clock). 5

and that exerted by the planet on the Sun is m m F = m ¨r = G 0 r , (8) λ ps 0 s r3 δ r Equator ε where m0 and s are the mass and position of the Sun; m and rp those of the planet, and r = rp rs is the planet γ α position, as seen from the Sun. Also, −r = r , and dots indicate time derivatives. | | Note that Ecliptic Fsp + Fps =0 , (9) FIG. 7: Spherical triangle showing the Sun’s position on the as should be, or celestial sphere. γ, vernal point; ǫ, obliquity of the ecliptic; α, right ascension; δ, declination; λ, ecliptic longitude. m ¨rp + m0 ¨rs =0 , (10) from where IV. KEPLERIAN ELLIPTIC MOTION MT R = A t + B , (11) A. Basic assumptions (A and B are integration constants) with the center-of- mass position: The present calculations give a rather accurate value m r + m r for the equation of time. However, they are based on the R = p 0 s , (12) very simple assumption that Sun and Earth are isolated m0 + m in the Universe without any external interactions. This and the total mass MT = m0 + m. The center-of-mass means: has inertial motion. - One neglects gravitational (for some purposes impor- For the relative position r we have ¨r = ¨rp ¨rs, so that tant) interactions with the and other objects in the we find the following differential equation − Solar System (mainly Jupiter). A correction due to the r lunar perturbation is presented in Sec. IX. ¨r = k2 , (13) - Sun and Earth are considered as point masses, which − r3 would be precise for spherical bodies with uniform mass where we have defined distribution. This means that we neglect oblateness of 2 Sun and Earth, as well as Earth deformations. In partic- k = GMT = G(m0 + m) (14) ular, of the Earth axis is not taken into account in the calculations. - The gravitational interaction is assumed to be C. Equation of the trajectory Newtonian, i.e., we do not consider corrections due to General Relativity. We note first that r ¨r, so that r ¨r = 0, from where k × These effects give rise to changes in the Earth orbit, d(r r˙) × = r˙ r˙ + r ¨r = r ¨r = 0 (15) such as of the equinoxes and precession of the dt × × × periapsis, that can be considered in an effective way by Thus, the cross product r r˙ is constant along the tra- locating the orbit according to the known position of the jectory, and we will write: × vernal point and perihelion for a given date. r r˙ =2C (16) × B. Two-body problem This means: First, that the (parallel to C) is a constant of motion; , that the relative We consider a problem of two bodies interacting grav- motion of the two bodies takes place on a plane normal to the vector C (since r C = 0), and third, that the motion itationally. To study the trajectory and dynamics, the · basic ingredients are the gravitation law (attraction force verifies the so-called “law of equal areas” (second Kepler’s proportional to the product of masses and inverse to the law of planetary motion). In fact, if we describe the squared distance) and the second law of motion, both motion in planar polar coordinates (r, ϕ) (with origin on r postulated by Newton. the focus), the velocity ˙ has componentsr ˙ and rϕ˙ in the r The force exerted by the Sun on the planet is: directions parallel and perpendicular to , respectively. Then, we have for the velocity: m0m F r r 2 2 2 2 2 sp = m ¨p = G 3 , (7) v = r˙ =r ˙ + r ϕ˙ (17) − r | | 6 and from Eq. (16): - For e = 1, a parabola - For e> 1, a hyperbola 1 1 C = C = r r˙ = r2ϕ˙ (18) In the case of planetary motion we have elliptic orbits, | | 2| × | 2 i.e. the motion is bound with negative energy h< 0 (see Now note that the elementary area swept by the vector below). For nonnegative h the motion is neither bound radius r in the orbit plane is nor periodic (parabolic for h = 0 or hyperbolic for h> 0). We are interested here in the case 0

k2 k4 2h = (e2 1) = (e2 1) . (36) dr = ae sin E dE (48) p − 4C2 − so that Eq. (46) reduces to In particular, for elliptic motion, one has from Eq. (28) n dt = (1 e cos E) dE (49) p = a(1 e2) , (37) ± − − Here we use the eccentric anomaly E as an intermediate so that variable to connect the true anomaly ϕ with the mean 2k2 2 1 anomaly M (see below). For more details on the geo- v2 =2h + = k2 (38) metric meaning of E, see Appendix F and Fig. 20. r r − a   By integration of Eq. (49) we find Note that the constant C (which gives the “areal ve- E e sin E = n(t T ) (50) locity”) can be written as − −

Se πab where T is an integration constant which coincides with C = = , (39) P P the instant of transit by the periapsis, as for t = T we have E = 0 and r = a(1 e)= a c (see Fig. 2). Eq. (50) S being the area of the ellipse. Then (note that p = e is the so-called Kepler equation.− − The quantity b2/a): M = n(t T ) (51) − 4C2 4π2a2b2 a a3 2 2 is the mean anomaly, i.e., the angle from the periap- GMT = k = = 2 2 =4π 2 (40) p P b P sis to the dynamical fictitious Sun (along the ecliptic, see Sec. II), or equivalently the angle from the point α0 with MT = m0 + m. This is the third Kepler’s law. For an elliptic keplerian motion it is usual to use, in- to the dynamical mean Sun (along the equator). Thus, stead of the P , the average angular velocity this equation gives us a relation between the eccentric (called average motion) n: anomaly E and the mean anomaly M. Note that M is not the total mass of the system, which is called MT . 2π Going back to Eqs. (25) and (47), we have for elliptic n = (41) P motion: Then, Eq. (40) can be rewritten as a(1 e2) r = − = a(1 e cos E) (52) 1+ e cos ϕ − k2 = n2a3 (42) where we have a relation between the eccentric anomaly and we have E and the true anomaly ϕ. From the last two expressions 4C2 pk2 n2a4(1 e2) in Eq. (52), we can find out cos ϕ by elementary algebra: r2ϕ˙ 2 = = = − , (43) r2 r2 r2 cos E e cos ϕ = − (53) where we have employed Eqs. (18), (26), (37), and (42). 1 e cos E − Now, using Eq. (38) for the velocity, and taking into and from here: account that v2 =r ˙2 + r2ϕ˙ 2, we have: ϕ 1 cos ϕ (1 + e)(1 cos E) 1+ e E n2a4(1 e2) 2 1 tan2 = − = − = tan2 r˙2 + − = n2a3 , (44) 2 1+cos ϕ (1 e)(1 + cos E) 1 e 2 r2 r − a − −   (54) and finally from where one has n2a2 ϕ 1+ e E r˙2 = [a2e2 (a r)2] (45) tan = tan (55) r2 − − 2 1 e 2 r − 8

20

λp 10 P A P'

0

γ' γ -10 Equation of time (min) 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 a -20 0 100 200 300 future position of the vernal point due to equinox precession. The perihelion precesses from P to P’. The latter motion is Mean anomaly, M (degrees) much slower than that of γ. Note that the true Earth’s orbit is much less eccentric than that plotted here.

FIG. 8: Equation of time as a function of the mean anomaly M, as derived from the method given in Sec. IV. is our reference for angles on the , it has to be accurately located. Its position changes along the years, and is referred to the vernal equinox by means of or o its ecliptic longitude λp. In 2011, λp(2011) = 283.084 , o − 1+ e E and it increases at a rate of 0.017 per year: ϕ = 2 tan 1 tan (56) 1 e 2 "r − # λp = λp(2011) + 0.0170 ∆n (59) which gives us the true anomaly ϕ as a function of the where ∆n = ny 2011, and ny is the year. This shift eccentric anomaly E. is due to the combination− of the equinox precession and Knowing ϕ we calculate the Sun’s longitude λ on the the precession of the Earth’s perihelion itself (see Fig. 9). ecliptic as It causes a change (advance) of the mean perihelion in 24.83 min per year, which corresponds to the 0.017o shift λ = λ + ϕ (57) p mentioned above. where λp is the ecliptic longitude of the periapsis (λp = The instant of the mean perihelion in our calendar 283.084o on 1.1.2011). Note that the mean anomaly M is changes also due to the non-commensurability of the cal- related to λp through the ecliptic longitude of the mean endar years with the anomalistic year. Then, for years Sun, Λ, i.e., M =Λ λp. following a common (not leap) year there appears a shift Knowing λ, we calculate− the right ascension α by relat- (ahead) of 6.233 hours (= 6 hours 14 min), and for years ing it to the angles ǫ and λ in the right spherical triangle following a a retard of 24 - 6.233 hours = 17 shown in Fig. 7: hours 46 min. (This is because inserting the 29th Febru- ary affects the following perihelion in the next January. tan α = cos ǫ tan λ (58) Also note that these 6 hours 14 min include the 24.83 min of the perihelion shift). Thus, in 2011 the mean per- Finally, ∆t can be calculated from the difference ∆t = ihelion was on 3 Jan at 19:54 UTC, in 2012 it occurred α α, where the right ascension of the mean Sun is M − on 4 Jan at 2:08 UTC, in 2013 it took place on 3 Jan at given by αM = M + λp. This gives the curve shown in 8:22 UTC, and so on. Fig. 8. A practical rule: Given a year ny > 2011, the instant of the mean peri- V. POSITION OF THE CALENDAR ON THE helion, T , is given (in hours) by ORBIT T = T +6.233∆n 24 i (60) 2011 − x With the above calculations the problem of calculating where ∆n = n 2011, and i is the number of leap y − x the equation of time is formally solved, but now we have years from 2011, given by the expression: ix = int[(ny to translate the mean anomaly M to a precise date in 2009)/4)]. Here “int” means the integer part of the quo-− our . This can be done in two steps: tient, and thus one has ix = 0 for 2011 and 2012, 1 for (1) Locate the mean perihelion at its precise date of 2013–2016, 2 for 2017–2020, and so on. For complete- the year we are interested in. Since the mean perihelion ness, we mention that this rule is valid also for 2009 and 9

2010. For earlier years one can use a similar rule, taking Better approximations can be obtained in a recursive the same reference of year 2011: way, by inserting the obtained value again in the equation: E1 = M + e sin M, E2 = M + e sin E1, T = T2011 +6.233∆n + 24 kx (61) E3 = M + e sin E2, ..., until obtaining the required precision. This works well because e 1. where kx = int[(2008 ny)/4)] + 1, so that kx = 1 for ≪ 2005–2008, 2 for 2001–2004,− and so on. Note that the actual perihelion may differ from the 3) Calculate the true anomaly ϕ from the eccentric mean perihelion by up to about 30 hours, mainly due anomaly E [see Eq. (56)]: to the influence of the Moon, but we will not discuss − 1+ e E this here. The important point to be emphasized here ϕ = 2 tan 1 tan (66) is that to calculate the equation of time we have to 1 e 2 "r − # “place” the whole orbit in the calendar as accurately as possible (associate a precise date to each angle M), and 4) Find the ecliptic longitude of the (true) Sun: this can be only done with respect to the mean perihelion. λ = λp + ϕ (67) (2) The mean anomaly M is an angle that increases where λ is the ecliptic longitude of the periapsis (in year uniformly (clock) from 0o at the mean perihelion to 360o p 2011, λ = 283.084o; for the following years see Eq. (59)). at the next perihelion. In fact, at the next perihelion p this angle is slightly larger than 360o due to perihelion 5) Calculate the right ascension α by relating it to the precession, and 360o corresponds to a (a angles ǫ and λ in the right spherical triangle shown in little shorter than the anomalistic year). Then, given a Fig. 7: date, we translate it to an angle M by calculating the − number of days N (not necessarily an integer) from the α = tan 1(cos ǫ tan λ) (68) mean perihelion, so that 6) Obtain the right ascension α of the mean Sun as: N M M = 360o (62) 365.2425 × αM = λp + M (69) where 365.2425 is the number of days in a tropical year 7) Finally, calculate the equation of time as: (in fact, mean Gregorian year). ∆t = α α (70) M − and we obtain ∆t as a function of the mean anomaly M. VI. A RECIPE TO CALCULATE THE EQUATION OF TIME 8) If we are interested in converting M into an actual date in our Gregorian calendar, the number of days N has A schematic and simple procedure to obtain an ap- to be added to the date of the perihelion in the year under proximate value for the equation of time is the following. consideration. A practical rule to find rather precisely It is based on Newton equations of motion, which give the instant T of the mean perihelion is given in Sec. V. rise to conical (elliptical for planets) trajectories in the two-body problem (Sun and planet). In fact, this is a summary of the procedure presented in Sec. IV. VII. COMPARISON WITH OTHER METHODS 1) Give the time, i.e., the Sun’s mean anomaly M TO CALCULATE ∆t N M = 360o , (63) A. Precision of our method 365.2425 × where N is the number of days from the periapsis (in We remember that the calculations presented above general, a non-integer number), and 365.2425 is the are based on the assumption that Earth and Sun are ce- number of days in a mean Gregorian year [see Eq. (62)]. lestial bodies isolated in the Universe, and we solved the two-body problem with Newtonian gravitational interac- 2) From M we calculate the eccentric anomaly E tion. We pointed out, however, the presence in the real through Kepler’s equation [see Eq. (50)]: world of interactions with other celestial bodies (mainly the Moon and other planets) which cause perturbations M = E e sin E . (64) − on the Earth’s orbit, such as precession of the equinoxes This transcendental equation cannot be solved analyti- and precession of the perihelion. Also orbital parameters cally in E. A first and rather accurate approximation as the eccentricity e and the obliquity ǫ change with time, consists in taking but this change is very slow for our present purpose, so that they may be considered constant. For our calcula- E M + e sin M . (65) tions on the equation of time, an important point is the ≈ 10

2 2

1 1

0 0

-1 -1 Error (difference C-P) (s) Error (difference C - P) (s)

-2 -2 0 100 200 300 0 500 1000 Days (year 2011) Days (since 1 Jan 2011)

FIG. 10: Difference EC = (∆t)C −(∆t)P along the year 2011. FIG. 11: Difference EC = (∆t)C − (∆t)P from January 2011 Points are plotted at five-day intervals. to December 2013. difference between the actual Earth’s perihelion (the in- discussed in more detail in Sec. IX. stant when the distance Sun-Earth is a minimum) and Concerning contribution (2) of the error EC , we cannot the mean perihelion, which is used in our calculations as give at this moment a precise reason for it, as it could if the Moon were not present at all. be due to a sum of different contributions. An important To assess the precision and reliability of our calcu- point is that we do not find at first sight any apparent lations based on a simple two-body problem, we will periodicity in this contribution along the years, as can be compare our results with precise values of the equation observed in Fig. 11, where we have plotted EC for three of time, as those given in the Procivel tables (see Ro- consecutive years, from January 2011 to December 2013. damdia.com). Values presented in these tables are based on the actual position of the Sun at each moment, as derived from reliable astronomical calculations. B. Sum of two independent contributions Given an instant of time (date), we call error EC the difference between the equation of time calculated from In Sec. III we discussed the two main contributions our approximation, (∆t)C , and that given in the Procivel to the equation of time: eccentricity of the orbit and tables, (∆t)P : obliquity of the ecliptic. A simplification to calculate approximately ∆t consists in adding both contributions, EC = (∆t)C (∆t)P (71) − as if they could be considered totally independent. This This difference is shown in Fig. 10 for year 2011, where means that one calculates first the contribution of ec- data points are presented every five days. It is remarkable centricity, assuming that the Sun moves on the equator that the absolute error of the procedure described here is (ǫ 0), and then the contribution of obliquity assuming → less than 2 s along the whole year. It is also remarkable that the orbit is circular (e 0). → that the evolution of EC along the year can be separated For this estimation, we have carried out the same cal- into two main contributions: (1) a high-frequency oscil- culations presented in Sec. IV, but with the correspond- lation with a period Th of about 30 days and amplitude ing simplifications: less than one second, and (2) a low-frequency background (1) We neglect the obliquity of the ecliptic putting ǫ = of larger amplitude. 0, and find the contribution of the eccentricity, (∆t)exc. Contribution (1) is related to the influence of the The resulting contribution to the equation of time (that Moon, since a careful analysis indicates that the period can be written in this case as αM α = M ϕ) is shown − − Th coincides with the synodic period of the Moon. Taking in Fig. 4 as a function of the mean anomaly M along one into account that the influence of the Moon has not been year. As expected, one finds a periodic behavior with a considered in our calculations for the two-body problem, period of 2π (one year). its gravitational interaction with the Earth should ap- (2) We neglect the eccentricity of the orbit putting e = pear as a residual source of error in our results. We note 0, and obtain the contribution of the obliquity, (∆t)obl. in passing that the Moon interaction causes maxima and This contribution is presented in Fig. 6 as a function of minima in the sine-shape oscillation displayed in Fig. 10 the mean anomaly M. One finds a periodic function, for first and third quarter Moon, respectively. This con- with a period of π (half a year). tribution vanishes for full and new Moon. This will be The sum of both contributions has a shape similar to 11

Sum of two contributions - "exact" value We now look for a simplified relation between the true 0.8 anomaly ϕ and the mean anomaly M. To this end, we start from a relation between ϕ and the eccentric 0.6 anomaly E. From Eq. (55) we have

0.4 ϕ 1+ e E tan = tan . (77) 2 1 e 2 0.2 r − First, we note that e 1, and an expansion of the func- 0 tion containing e gives:≪

-0.2 1 1+ e 2 =1+ e + O(e2) (78) Error (minutes time) -0.4 1 e  −  -0.6 and we will retain only terms up to first order in e. Then, we have: -0.8 0 100 200 300 ϕ E E tan tan + e tan (79) Mean anomaly, M (degrees) 2 ≈ 2 2 Since the difference ϕ E is small (vs. π), we put ϕ/2= FIG. 12: Difference between our ”precise” calculation for the − equation of time, (∆t)C , and ∆t calculated as a sum of two E/2 + ∆, with ∆ a small parameter. Taylor expanding independent contributions. tan(ϕ/2) with the parameter ∆ we find ϕ E 1 tan = tan + ∆+ O(∆2) (80) 2 E the more precise ∆t displayed in Fig. 8, but it differs from 2 2 cos 2 the latter by an amount that changes along the year, and is always less than 40 s (see Fig. 12). Now, comparing Eqs. (79) and (80), and identifying lin- ear terms in e and ∆, we have e ∆= sin E (81) C. Milne’s formula 2 and finally A simple analytical formula, presented by R. M. Milne in 1921 (“Note on the Equation of Time”, The Mathe- ϕ E + e sin E . (82) matical Gazette 10, 372-375), gives an approximation for ≈ the equation of time as a function of the mean anomaly With this result, and taking into account that E M + ≈ M. To derive this approximation, we consider Eq. (58), e sin M [see Eq. (65)], we find that the true anomaly ϕ which relates the right ascension α to the ecliptic longi- can be written as a function of M: tude λ: ϕ M +2 e sin M . (83) tan α = cos ǫ tan λ (72) ≈ Since λ = λp + ϕ, Eq. (76) transforms into Remembering that 2 ǫ 2 ǫ α λp + M +2 e sin M tan sin(2M +2λp) . (84) 1 tan 2 ≈ − 2 cos ǫ = − 2 ǫ (73) 1 + tan 2 The right ascension of the mean Sun is αM = λp + M, so that we have for the difference α α: we have M −

2 ǫ 2 ǫ 2 ǫ 1 + tan sin α cos λ = 1 tan cos α sin λ (∆t)Milne = 2 e sin M + tan sin(2M +2λp) (85) 2 − 2 − 2     (74) or This equation gives approximately the equation of time as a simple analytical function of the mean anomaly M, 2 ǫ sin(α λ)= tan sin(α + λ) . (75) with the orbital parameters e, ǫ, and λ . It has a clear − − 2 p physical explanation as the sum of two terms, the first Now, since the right ascension α and the ecliptic longi- one due to eccentricity of the orbit, and the second one tude λ take similar values, the difference α λ is small due to obliquity of the ecliptic. Introducing the values: − o (compared to π; in fact, α λ is always less than 3 ), e = 0.0167, ǫ = 23.44o, and ecliptic longitude of the | − | o and Taylor expanding the sine function to first order, we periapsis on 2011-1-1: λp = 283.084 , one finds have: ǫ (∆t)Milne = 7.655 sin M +9.863 sin(2M + 206.168) α λ sin(2λ) tan2 . (76) − ≈ − 2 (86) 12 where ∆t is given in minutes of time and M in degrees. 90 o This equation was first derived by Milne, who wrote it in Year 2011 - 12 UTC - 40 terms of the ecliptic longitude of the (fictitious) dynam- 80 1 June ical mean Sun, Λ = λp + M. 70 The absolute error of this formula is less than 45 s throughout the year, with its largest value 44.8 s at the 60 beginning of October. 50 1 Oct 1 Mar D. A simple and rather accurate approximation 40 Altitutde (degrees) 30 A very simple approximation, that is however more 1 Jan accurate than Milne’s method is the following. Given 20 the mean anomaly M, the Sun’s ecliptic longitude can 10 be approximated as: -6 -4 -2 0 2 4 6 (degrees) λ = λ + ϕ λ + M +2 e sin M , (87) p ≈ p FIG. 13: Analemma for year 2011 at 12 UTC as seen from a where we have used Eq. (83). From λ, one calculates the place with longitude 0o and latitude 40o N. right ascension of the true Sun, α, as [see Eq. (58)]:

tan α = cos ǫ tan λ . (88) and therefore the analemma would be a simple dot, as happens on Earth for the mean Sun described above. The right ascension of the mean Sun is α = λ + M, M p For a celestial body with a circular orbit but appreciable from where one finds the equation of time ∆t = α α. M , the analemma would be a figure like “8” with The largest error of this approximation for ∆t is about− northern and southern lobes equal in size. For an object 6 s. This simple approximation is much more precise than with an eccentric (i.e., non-circular) orbit but no axial Milne’s formula. The main source of error in Milne’s for- tilt, the analemma would be a straight line (in fact, a mula is the approximation for the right ascension α given segment). At noon this line would appear in the east- in Eq. (76), which is obtained here directly from Eq. (88) direction at an altitude h = 90o φ (φ: latitude). without any approximation. The main approximation In Fig. 13 we present an analemma− as seen from the introduced here is that given in Eq. (87) for λ, which Earth’s northern hemisphere. It is a plot of the position appears also in the derivation of Milne’s formula. of the Sun at 12:00 UTC (noon) as seen from a point on the Greenwich meridian (longitude 0o) and latitude o VIII. ANALEMMA φ = 40 N during year 2011. The horizontal axis is the azimuth angle A in degrees (0o is facing south), and the vertical axis is the altitude h measured in degrees above An analemma is a curve showing the angular offset of the horizon. The first day of each is shown as a celestial body (usually the Sun) from its mean position a circle, and the solstices and equinoxes are shown as on the celestial sphere, as viewed from another celestial squares. It can be seen that the equinoxes occur at al- body (usually the Earth). This name is commonly ap- titude h = 90o φ = 50o, and the solstices appear at plied to the figure traced in the when the position of altitudes h = 90−o φ ǫ, where ǫ is the axial tilt of the Sun is plotted at the same clock time each day over a the Earth (obliquity− of± the ecliptic, ǫ = 23o 26’). Note calendar year from a fixed position on Earth. The result- that the analemma is plotted with its width exaggerated, ing curve resembles a lemniscate of Bernoulli. The word to permit observing that it is asymmetrical. A diamond analemma comes from Greek and means “pedestal of a in Fig. 13 indicates the fixed position of the mean Sun ”, since it was originally employed to describe the throughout the year at the selected clock time. line traced by the shadow of the sundial’s gnomon along We calculate the analemma by using the Sun equatorial the year. coordinates (α, δ), and transforming them to horizontal The actual shape of the Sun’s analemma depends on: coordinates (A, h) for a given place on Earth at a given (1) general parameters for all observers on Earth, such as time. The Sun’s right ascension α is obtained from the the eccentricity e of the Earth’s orbit and the obliquity expressions given above in Secs. IV and VI, i.e., of the ecliptic, ǫ, and (2) particular parameters of the − observer, as his/her geographic latitude and the actual α = tan 1(cos ǫ tan λ) (89) observation time. Viewed from an imaginary planet with a circular orbit and the declination δ is obtained from the expression (e = 0) and no axial tilt (equator parallel to ecliptic, sin δ = sin λ sin ǫ (90) ǫ = 0), the Sun would always appear at the same point in the sky at the same time of day throughout the year, obtained from the spherical triangle shown in Fig. 7. 13

60 o o 30 Year 2011 - 12 UTC - Latitude 90 Year 2011 - 9 UTC - Latitude 40

50 1 June 20 1 Aug 1 May 10 1 Aug 40 HORIZON 0 1 Apr 1 Oct 1 Oct 1 Mar 30 -10

Altitude (degrees) Altitude (degrees) 1 Mar

-20 20 1 Jan -30 1 Jan 10 -4 -2 0 2 4 -80 -70 -60 -50 -40 Azimuth (degrees) Azimuth (degrees)

FIG. 14: Analemma for year 2011 at 12 UTC as seen from the FIG. 15: Analemma for year 2011 at 9 UTC as seen from a Pole. (Note that on the Pole the actual time is rather place with longitude 0o, latitude 40o N. irrelevant for the analemma’s shape).

Now, to convert to horizontal coordinates we use the 60 o spherical triangle displayed in Fig. 22. In this triangle Year 2011 - 15 UTC - Latitude 40

cos z = sin δ sin φ + cos δ cos φ cos H (91) 50 1 Aug from where we find z [0, 180o]. Here H is the local 1 June ∈ , given by H = θ α, with θ the local sidereal 40 time (i.e., the hour angle of− the vernal point γ). Once z is known, we use the expression: 30 1 Mar sin H sin A 1 Oct

= (92) Altitude (degrees) sin z cos δ 20 to obtain the azimuth A [0, 360o). ∈ The analemma is oriented with the smaller loop ap- 1 Dec pearing north of the larger loop (see Fig. 13). At the 10 , the analemma is totally upright (an 8 with 40 50 60 70 80 the small loop at the top), and one is able to see only Azimuth (degrees) the top half of it (the trajectory of the Sun in half a year). This is displayed in Fig. 14, where the symbols FIG. 16: Analemma for year 2011 at 15 UTC as seen from a place with longitude 0o, latitude 40o N. have the same meaning as in Fig. 13. We now move south and cross the , then we can see the whole analemma. If we look at it at noon, it is still up- right, and moves higher from the horizon as we go south. so that the small loop is beneath the large loop in the When we are on the equator, the analemma is overhead. sky. When we cross the Antarctic Circle, the analemma If we continue going further south, it moves toward the appears almost completely inverted, and it begins to dis- northern horizon, and is now seen with the larger loop at appear below the horizon, and finally only a part of the the top. larger loop is visible when we are on the . At noon the analemma appears rather “vertical” on If we look at the sky at an earlier (or later) time from the sky. However, it appears inclined when observed at a point with latitude 40o N, then we see only a part of other day times. Imagine now that we are looking at the analemma, as the rest lies below the horizon. This is the analemma in the early morning or evening. Then it shown in Fig. 17 at 6 UTC. starts to tilt to one side as we move southward from the The analemma can be used to find the dates of the North Pole. In Figs. 15 and 16 we show the analemma earliest and latest and of the year, which as seen from a place with longitude 0o and latitude 40o do not occur on the dates of the solstices. An analemma N at 9 UTC and 15 UTC, respectively. When we arrive in the eastern sky with its lowest point just above the at the equator, the analemma appears totally horizon- horizon corresponds to the latest of the year, since tal. If we continue going south, it still continues rotating for all other points (dates) on the analemma, the sunrise 14

2 o 20 Year 2011 - 6 UTC - Latitude 40

1 June 1 10 1 Aug

1 Oct 0 0 HORIZON

Altitude (degrees) 1 Mar -1 -10 Error (difference C - P) (s)

1 Jan -20 -2 -120 -100 -80 -60 0 500 1000 1500 Azimuth (degrees) Days (since 1 Jan 2011)

FIG. 17: Analemma for year 2011 at 6 UTC as seen from a FIG. 18: Difference EC = (∆t)C − (∆t)P from January 2011 place with longitude 0o, latitude 40o N. to December 2014, after correction of the Lunar perturbation, as described in the text.. occurs earlier. Therefore, the date when the Sun is at this lowest point is the date of the latest sunrise. Likewise, - The Moon moves on the ecliptic plane. when the Sun is at the highest point on the analemma, These approximations, although very crude, are near its top-left end, the earliest sunrise of the year will enough for the precision required here. occur. Similarly, the earliest will occur when the We then assume that the true anomaly ϕ calculated Sun is at its lowest point on the analemma when it is in the two-body problem above refers to the close to the western horizon, and the latest sunset when B. We will call ϕ′ the true anomaly of the Earth. For it is at the highest point. motion on a plane, we write (x′,y′) for the position of Note that in many places in the web, one reads that the Earth and (x, y) for the position of B. We have: the east-west component of the analemma is the equation of time, but this is not right. The equation of time is the ′ x = x + xT = r cos ϕ + d cos θ (93) difference between solar time and mean time (or between ′ true and mean right ascension), but in the analemma we y = y + yT = r sin ϕ + d sin θ (94) have the local azimuth of the Sun, as given by the local coordinates. where (xT ,yT ) is the position of the Earth respect B. r is the distance Sun-B (assumed to be constant), d is the distance Earth-B (also assumed to be constant), and θ IX. LUNAR PERTURBATION ON THE changes from 0 to 2π in a sidereal month (period Ts = EARTH’S ORBIT 27.321662 mean days). To give an initial position for the three-body system, we put θ = ϕ + π for the , where r′ = r d As shown above, there appears in our calculated equa- − tion of time a modulation apparently due to the Lunar (Earth closer to the Sun). Thus perturbation on the Earth’s orbit, as its period coincides with the synodic period of the Moon. 2π θ = θ0 + (t t0) (95) To take into account this perturbation, we assume that Ts − the two-body calculations presented above refer to the motion of the system Earth-Moon around the Sun, i.e. with t0 the instant of a full Moon, and θ0 = ϕ0 + π. they give the trajectory of the barycenter B of the Earth- In the calculations, we use the distance ratio R = r/d. Moon system. Now we have to calculate the position Putting r = 149,598,000 km and d = 4678 km, we have of the Earth respect the barycenter B at each moment. R = 31979.05 (note that B is inside the Earth). We approach this problem by making some very basic Introducing the modified true anomaly ϕ′ instead of ϕ approximations: to calculate the equation of time ∆t, we find data that - The actual three-body problem is replaced by a two- compared with (∆t)P from Procivel tables give the resid- body problem (Sun plus barycenter B), and then we ual error shown in Fig. 18. Now the oscillations shown add a correction due to the relative motion of Earth and in Figs. 10 and 11 have disappeared almost completely. Moon. Taking into account the simplicity of the model, this can - The orbits of Earth and Moon around B are circular be considered a good accomplishment. 15

Appendix A: Notation by about 1.5o.

Ellipse: - obliquity of the ecliptic: angle between the celestial - a : semi-major axis of the ellipse equator and the ecliptic. It is usually denoted as ǫ, and o - b : semi-minor axis of the ellipse presently it amounts to about 23 26’. - e : eccentricity of the conical orbit. For the Earth’s orbit, e =0.0167 - vernal point ( point): point on the celestial - c : linear eccentricity of the ellipse (semi-focal distance, sphere where the ecliptic intersects with the celestial c = ae) equator, and the Sun crosses the latter from south to - p : parameter of the conic, semi-latus rectum north, i.e., its declination changes from negative to [p =4C2/k2 = a(1 e2)] positive. This corresponds to the equinox in the − Northern hemisphere, and happens around the 20th Orbit: - 21st of March. Due to , the vernal - t : dynamical time, the independent variable in the equinox moves slowly westward relative to the fixed theory (Newtonian, ideal) stars, completing one revolution in about 25,770 years. - P : orbital period o - λp : ecliptic longitude of the periapsis (= 283.084 on - axial precession: -induced, slow and continu- 2011-1-1) ous change in the orientation of an astronomical body’s -Λ = λp + M, ecliptic longitude of the (fictitious) rotational axis. For our planet, it refers to the gradual dynamical mean Sun shift in the orientation of Earth’s axis of rotation, which - λ = λp + ϕ, Sun’s true longitude on the ecliptic traces out a pair of cones joined at their apices, in a - T : instant of transit by the periapsis cycle of approximately 25,770 years. Earth’s precession - ǫ: obliquity of the ecliptic (angle between equatorial has been historically called precession of the equinoxes and ecliptic planes). Now ǫ = 23o 26’ because they move westward along the ecliptic relative to - v : velocity the fixed stars, opposite to the motion of the Sun along o - n : average motion (average angular velocity) the ecliptic. This precession amounts to 0.0140 /year or - α : right ascension of the (true) Sun 50.29”/year. - αM : right ascension of the mean Sun, αM = λp + M - nutation: a rocking, swaying, or nodding motion in Anomalies: the axis of rotation of a largely axially symmetric object, - ϕ : Sun’s true anomaly such as a , planet, or bullet in flight, or as an - M : Sun’s mean anomaly intended behavior of a mechanism. - E : Sun’s eccentric anomaly - Earth’s nutation: nutation in Earth’s axis mainly Constants: due to tidal forces of the Sun and the Moon, which - G : Newton’s gravitational constant continuously change location relative to each other. It can be decomposed in several components, the largest - m0 : Sun mass - m : planet mass one having a period of 18.6 years, the same as the precession of the Moon’s orbital nodes. It reaches plus - MT = m0 + m, total mass or minus 17” in longitude and 9” in obliquity. All - k = √GMT , constant for the calculations - h : constant proportional to the mechanical energy other terms are much smaller. The next-largest, with a period of 183 days ( half a year), has amplitudes (kinetic plus potential) ∼ - C : constant proportional to the angular momentum 1.3” and 0.6”, respectively. The periods of all terms larger than 0.0001” lie between 5.5 and 6798 days. To explain accurately Earth’s nutation, one has to account for deformations of the solid Earth.

Appendix B: Glossary - periapsis: closest position of a planet to the focus (Sun). For celestial bodies orbiting around the Sun, it is ORBIT called perihelion.

- ecliptic: plane of the Earth’s orbit around the Sun. - apoapsis: farthest position of a planet from the focus In astronomical terms, it is the intersection of the celes- (Sun). For celestial bodies orbiting around the Sun, it is tial sphere with the ecliptic plane. This plane is different called aphelion. from the of the solar system, which is perpendicular to the vector sum of the angular momenta - perihelion precession (or absidal precession): motion of all planets, Jupiter being the main contributor. The of the perihelion and the major axis of a planet’s elliptical present ecliptic plane is inclined to the invariable plane orbit within its orbital plane, partly in response to per- 16 turbations due to changing gravitational forces exerted which serves as a modern replacement for Greenwich by other planets. This causes the orbit to be not really Mean Sidereal Time). an ellipse but a flower-petal shape. Because of the ecliptic longitude of the Earth’s perihelion - UTC (Coordinated ): atomic time slowly increases. It takes about 112,000 years for the scale that approximates UT1, introduced on 1 January ellipse to revolve once relative to the fixed stars. 1972 . It is the international standard on which civil As a consequence of this, the anomalistic year is time is based. It ticks SI seconds, in step with TAI slightly longer than the , while the tropical (International Atomic Time). It usually has 86,400 year (which attempt to track) is shorter due to SI seconds per day, but is kept within 0.9 seconds of the precession of Earth’s rotational axis, so that the two UT1 by the introduction of occasional intercalary leap forms of precession add (see Fig. 9). Thus, it takes about seconds. Until now (January 2014) these leaps have 21,000 years for the ellipse to revolve once relative to the always been positive, with a day of 86401 seconds. When vernal equinox, that is, for the perihelion to return to an accuracy better than one second is not required, the same date (given a calendar that tracks the UTC can be used as an approximation of UT1. The perfectly). The dates of perihelion and aphelion advance difference between UT1 and UTC is known as DUT1 (= on this cycle an average of one day every 58 years. (Note UT1 - UTC). Weekly updated values of DUT1 with 0.1 that all this refers to the mean perihelion, and does not s precision are broadcast by several services. take into account short-period changes, as those caused by the presence of the Moon, and that may change the - : positive or negative one-second ad- instant of the actual perihelion in about 1 day). ± justment to the Coordinated Universal Time (UTC) scale that keeps it close to mean solar time. UTC is TIME maintained using atomic clocks, and is the basis for official time-of-day radio broadcasts for . To - apparent solar time or true solar time: time given keep the UTC time scale close to mean solar time, it by the daily apparent motion of the (true) Sun is occasionally corrected by an intercalary adjustment of one second (“leap” second), so that UTC remains - mean solar time: hour angle of the imaginary within the range -0.9 s < DUT1 < +0.9 s. There are mean Sun. It is realized with the UT1 time scale. two reasons for this correction: (1) the rate of rotation Due to the nonuniformity of the Earth’s angular ve- of the Earth is not constant, due to tidal braking and locity, other more regular procedures are now used to redistribution of mass within the Earth, including oceans measure time, mainly based on atomic clocks (i.e., UTC). and atmosphere, and (2) the SI second (used for UTC) was already, when adopted, shorter than the second of - second: International System (SI) , de- mean solar time. The timing of leap seconds is now fined as the duration of 9,192,631,770 periods of the ra- determined by the International Earth Rotation and diation corresponding to the transition between the two Reference Systems Service (IERS). Since June 1972, hyperfine levels of the ground state of 133Cs (caesium). there have been 25 leap seconds (all positive) until To be precise, this definition refers to a Cs atom at rest at January 2014. The most recent one happened on June a temperature of 0 K (absolute zero), and with zero exter- 30, 2012 at 23:59:60 UTC. nal radiation effects (i.e., zero local electric and magnetic fields). - TAI (International Atomic Time): high-precision 133 7 The nuclear spin of Cs is I = 2 , and the electronic atomic coordinate- based on 2 spin in the ground state is S1/2, so that the total spin on Earth’s geoid. It is the basis for Coordinated Uni- of the two lowest levels is F = 3 and 4. The ground versal Time (UTC), which is used for civil timekeeping state corresponds to F = 3, which under a magnetic all over the Earth, and for , used in field splits into its seven mF sublevels (mF = 3, ..., 3). . Since 2012-6-30 when the last leap second − was added, TAI has been 35 seconds ahead of UTC. - UT1: principal form of Universal Time. Concep- This is still true as of 2014-1-1. These 35 seconds result tually it is mean solar time at 0o longitude. However, from the initial difference of 10 seconds at the start of precise measurements of the Sun are difficult, and UT1 1972, plus 25 leap seconds introduced in UTC since is computed in fact from observations of distant quasars 1972. Thus, in January 2014, UTC = TAI – 35 s. using long baseline interferometry, laser ranging of the Moon and artificial satellites. UT1 is the same every- - TT (Terrestrial Time): modern astronomical time where on Earth, and is proportional to the rotation angle standard defined by the International Astronomical of the Earth with respect to distant quasars, specifically, Union, primarily for time-measurements of astronomical the International Celestial Reference Frame (ICRF), observations made from the surface of the Earth. neglecting some small adjustments. The observations Presently, the Astronomical uses TT for its allow the determination of the Earth’s angle with respect tables of positions of the Sun, Moon and planets as seen to the ICRF, called the Earth Rotation Angle (ERA, from the Earth. TT continues Terrestrial Dynamical 17

Time (TDT), which in turn succeeded time s) (at the J2000.0 = 2000 January 1 12:00:00 TT). (ET). The unit of TT is the SI second, but it is not actually defined by atomic clocks. It is a theoretical - tropical year: period of time for the ecliptic longitude ideal, which real clocks can only approximate. TT is of the Sun to increase by 360o, i.e., length of time that distinct from UTC, the time scale currently used as a the Sun takes to return to the same position in the cycle basis for civil purposes, and indirectly underlies UTC of seasons. The mean tropical year is 365.24219 mean via International Atomic Time (TAI). To days (365 days, 5 hours, 48 minutes, 45 seconds). In the accuracy, TT runs parallel to TAI, and can be approxi- Gregorian calendar, one (mean) year = 265.2425 mean mated as TT = TAI + 32.184 s. (This offset arises from days. Because of the Earth’s axial precession and the historical reasons.) Since TAI = UTC + 35 s, we have associated shift of vernal equinox, this year is about 20 TT = UTC + 67.2 s. minutes shorter than the sidereal year. In fact, the ver- nal point precesses by ∆ω = 50.29”/year in the opposite - (GMT): a term originally direction to the apparent Sun motion (∆ω = 360o/25770 referring to mean solar time at the Royal Observatory in years), which converts into δt = 365.2422 ∆ω/360o = Greenwich, London. It is arguably the same as Coordi- 0.01417 days = 20.4 min. nated Universal Time (UTC) and when this is viewed as a the name Greenwich Mean Time is especially - anomalistic year: time taken for the Earth to used by bodies connected with the United Kingdom, complete one revolution with respect to the apsides of such as the BBC World Service, the Royal Navy, the its orbit (perihelion and aphelion). It is usually defined Met Office, and others. Before the introduction of UTC as the time between perihelion passages. Its average du- on 1 January 1972 Greenwich Mean Time was the same ration is 365.25964 days (365 d 6 h 13 min 52.6 s) at the as Universal Time (UT), which is a standard astronom- epoch J2011.0. The perihelion moves ∆ω = 360o/112000 ical concept used in many technical fields. The term years = 0.00321o/year in the direction of the apparent “Greenwich Mean Time” is no longer used in Astronomy. Sun motion, i.e., it approaches the vernal point. Then, an anomalistic year is longer than a sidereal year by - (ET): refers to time in connection δt = 365.2564/112000 = 0.00326 days = 4 min 42 s. with any astronomical ephemeris. A practical definition The interaction between the anomalistic and tropical was proposed in 1948, to overcome the drawbacks of ir- cycle seems to be important in the long-term climate regular fluctuations in mean solar time, and thus define a variations on Earth, called the . uniform time based on Newtonian theory. ET is a (clas- sical) , defined implicitly from the - Besselian year: a year that begins at the moment observed positions of astronomical objects through the when the of the Sun is exactly 280o. This dynamical theory of their motion. moment falls near the beginning of the corresponding The unit and origin of ET are usually defined by adopt- Gregorian year. The definition depends on a particular ing a numerical expression for the geometric mean longi- theory of the orbit of the Earth around the Sun, that tude of the Sun. For this purpose, it has been tradition- of Newcomb (1895), which is now obsolete. For that ally employed the Newcomb formula: reason among others, the use of Besselian years has also o ′ 2 become obsolete. λm = 279 41 48.04”+ 129, 602, 768.13” T +1.089” T where T is counted in Julian of 36525 ephemeris - (average): 365.25 days of 86,400 s days. The origin of T is at the beginning of year 1900, each, totaling 31,557,600 s. The Julian year is the o average length of the year in the used when λm took the value 279 41’ 48.04”. This instant of time is dated 1900 January 0, 12 h ET exactly (as in Western societies in previous centuries, and for decided in 1958). Some corrections have been proposed which the unit is named. It included a leap year to obtain more precisely the solar longitude. However, it every four years without exception. Now it does not cor- was agreed in 1961 that “The origin and rate of ephemeris respond to any of the many other ways of defining a year. time are defined to make the Sun’s mean longitude agree with Newcomb’s expression”. - Gregorian year (average): year on which our present We note that ET can also refer to relativistic coor- calendar is based, 365.2425 days = 52.1775 dinate time scales, as that implemented by the JPL = 8,765.82 hours = 525,949.2 minutes = 31,556,952 seconds (mean solar, not SI), which approximates very ephemeris time argument Teph. well a tropical year of 365.2422 days. The 400-year cycle of the Gregorian calendar has 146,097 days and hence YEAR exactly 20,871 weeks (see “leap year”).

- sidereal year: time taken by a planet to orbit the - leap year (or intercalary or bissextile year): year con- Sun once with respect to the fixed stars. For the Earth, taining one extra day (or, in the case of lunisolar cal- it amounts to 365.2564 mean days (365 d 6 h 9 min 9.76 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 in TABLE I: Comparison between different years and days. a whole number of days, a calendar that had the same number of days in each year would, over time, drift with Mean days seconds respect to the event it was supposed to track. By oc- Tropical year 365.2422 31,556,926 casionally inserting an additional day or month into the Sidereal year 365.2564 31,558,153 year, the drift can be approximately corrected. A year Anomalistic year 365.2596 31,558,429 that is not a “leap year” is called a “common year”. Mean day 1 86,400 In the Gregorian calendar that we currently employ Sidereal day 0.997270 86,164 (a ), February in a leap year has 29 days instead of the usual 28, so the year lasts 366 days instead of the usual 365. Similarly, in the (a the value given above. To coincide with the rhythmites ), a 13th is added 7 times data, we need an average increase of 1.2 ms/). every 19 years to the twelve lunar in its common years, to keep its calendar year from drifting through the seasons too rapidly. - synodic day: period of time it takes for a planet Since a tropical year lasts 365.2422 mean days, the to rotate once in relation to the body it is orbiting (as Gregorian calendar includes 97 leap years every 400 opposed to a sidereal day which is one complete rotation years, which gives a mean duration per year of 365.2425 in relation to the stars). For Earth, synodic day is the days. The rule is the following: same as solar day, and is about 24 hours long. (1) If a year number is evenly divisible by 4 and not by 100, then it is a leap year. - stellar day: an entire rotation of a planet with (2) Years that are evenly divisible by 100 are not leap respect to the distant (fixed) stars. For the Earth it years, unless they are also evenly divisible by 400, in amounts to 86164.098 s of mean solar time (UT1), which case they are leap years (for example, 1700, 1800, i.e., it is 3 min 56 s shorter than a mean solar day. and 1900 were not leap years, but 1600 and 2000 were). This difference between stellar and mean solar day corresponds to an angle ∆ϕ = 360o/365.2425 days = o - (Latin, epactae): age of the Moon in days 0.9856 that the Earth moves in its orbit around the Sun (which converts into 4 min in the rotational motion). (number of days from the last new Moon) on January 1. ∼ It mainly appears in connection with tabular methods for determining the date of Easter, and varies (usually - sidereal day: the time it takes the Earth to make by 11 days) from year to year, because of the difference one rotation relative to the vernal equinox. A mean between the solar year of 365 days and the lunar year of sidereal day is 23 hours, 56 min, 4.091 s (23.93447 hours 354 days. or 0.99726957 mean solar days). Since the mean vernal equinox precesses slowly westward relative to the fixed stars, completing one revolution in about 25,770 years, DAY the misnamed sidereal day is some 9 ms shorter than the Earth’s period of rotation relative to the fixed stars - apparent solar day: the interval between two suc- (stellar day). This comes from ∆ϕ = 360o/(25770 365) cessive returns of the (true) Sun to the local meridian. days = 3.8o 10−5/day, i.e. δt = 4 3.8 10−5 min× = It can be up to about 20 seconds shorter or 30 seconds 9 ms. × × × longer than a mean day.

- mean solar day: 24 hours. In principle, in the International System, one day = 24 hours = 86,400 s. Appendix C: Some mathematical expressions However, the Earth’s day has increased in length over time, mainly due to raised by the Moon which slow Earth’s rotation. Because of the way the second Here we summarize some expressions employed along has been defined in the 20th century, the mean length the notes. of a solar day is now (beginning of the 21st century) (1) Some derivatives fluctuating around values of 1-2 ms longer than 86,400 s. We have r ˙r = rr˙, since Averaging over short-time fluctuations, it is increasing · d 1 1 2r ˙r r ˙r by about 1.7 ms per century (an average over the last 2 r˙ = (r r) = · 1 = · (C1) 2,700 years). The length of one day should be about 21.9 dt · 2 (r r) 2 r hours 620 million years ago, as recorded by rhythmites · (alternating layers in sandstone). (Note that at an Also: increasing of 1.7 ms/century and assuming it constant over geological periods, we extrapolate backwards to d 1 1 dr r ˙r = = · (C2) find a day of 21 hours 4 min, somewhat shorter than dt r −r2 dt − r3   19

(2) Vector triple product ϕ^ ^r a (b c) = (a c)b (a b)c (C3) × × · − · (3) Scalar triplet product r a (b c)= c (a b)= b (c a) (C4) · × · × · × ϕ (4) Derivative of the distance r FIG. 19: Mobile unit vectors ˆr andϕ ˆ used to study two- The equation of the conic in polar coordinates is: dimensional motion. p r = (C5) 1+ e cos ϕ Taking another time derivative, we find for the accel- eration: Taking a time derivative: dv π a = (¨r rϕ˙2)eiϕ + (2r ˙ϕ˙ + rϕ¨)ei(ϕ+ 2 ) (D4) e ϕ˙ sin ϕ r2 dt ≡ − r˙ = p 2 = e ϕ˙ sin ϕ (C6) (1 + e cos ϕ) p which yields for the radial and tangential components of the acceleration: and 2 e 2 e C ar =r ¨ rϕ˙ r˙ = r2ϕ˙ sin ϕ = sin ϕ (C7) − p p at = 2r ˙ϕ˙ + rϕ¨ (D5)

1 2 For gravitational forces one has at = 0. For central since C = 2 r ϕ˙ [Eq. (18)] r r C 1 r r forces in general we have ¨ , and the vector = 2 ˙ is a constant of motion [seek Eq. (16)]. In particular,× its module C is a constant given by C = 1 r2ϕ˙ [see Eq. (18)]. Appendix D: Motion in polar coordinates 2 Thus dC 1 For many purposes, it can be convenient to describe a = rr˙ϕ˙ + r2ϕ¨ =0 , (D6) planar motion in polar coordinates, rather than Carte- dt 2 sian coordinates. Then, one has the coordinates (r, ϕ), and then (since r = 0): 2r ˙ϕ˙ + rϕ¨ = 0, so that our calcu- related to the Cartesian coordinates by: x = r cos ϕ, and 6 lation is consistent with at = 0. y = r sin ϕ. This gives for the components of the velocity: Another interesting point is that the 1/r2 law for the gravitational force can be derived assuming that the orbit x˙ =r ˙ cos ϕ rϕ˙ sin ϕ (D1) − is elliptical (as in fact was done from Kepler’s observa- y˙ =r ˙ sin ϕ + rϕ˙ cos ϕ tions). We have: Then, in three-dimensional space we have r = (x, y, 0) p 2 e C 2 e C r = , r˙ = sin ϕ , r¨ = cos ϕ ϕ˙ and r˙ = (x, ˙ y,˙ 0), so that r r˙ = (0, 0, xy˙ yx˙), and the 1+ e cos ϕ p p × − third component of the cross product results to be for (D7) 2 2 keplerian motion 2C = r ϕ˙. [Remember that the cross Replacing these expressions into ar =r ¨ rϕ˙ , and taking r r 1 2 − product is a constant of motion that we called: ˙ = into account that C = 2 r ϕ˙, we find 2C, see Eq. (16)]. × 2 2 For the velocity we have: 2Cϕ˙ 4C 1 k GMT a = = = = (D8) r − p − p r2 − r2 − r2 v2 = r˙ 2 =x ˙ 2 +y ˙2 =r ˙2 + r2ϕ˙ 2 (D2) | | Another equivalent way of dealing with two- A rather simple way of dealing with polar coordinates dimensional motion consists in using a mobile coordi- for planar motion is considering the position vector as nate system with unit vectors ˆr andϕ ˆ (see Fig. 19). ˆr a complex variable r r(cos ϕ + i sin ϕ) = r exp(iϕ). is parallel to the position vector, ˆr = r/r, i.e., it has Taking into account that≡ i exp(iϕ) = exp(i(ϕ+π/2)), we components (cos ϕ, sin ϕ), and the unit vectorϕ ˆ, perpen- have: dicular to ˆr, has components (cos(ϕ+π/2), sin(ϕ+π/2)) = ( sin ϕ, cos ϕ). Now note that the time derivative of dr π − v = r˙eiϕ + rϕ˙ ei(ϕ+ 2 ) (D3) a unit vector is perpendicular to it: dt ≡ d dˆr dˆr r ˆr =1 , ˆr ˆr =1 , (ˆr ˆr)=2ˆr =0 , ˆr. This means that ˙ has componentsr ˙ and rϕ˙ in the di- | | · dt · · dt dt ⊥ rections parallel and perpendicular to r, respectively. (D9) 20

In fact, for the derivatives of the unit vectors we have: P' dˆr = ( sin ϕ ϕ,˙ cos ϕ ϕ˙) =ϕ ˙ ϕˆ (D10) dt − dϕˆ P = ( cos ϕ ϕ,˙ sin ϕ ϕ˙)= ϕ˙ ˆr (D11) a dt − − − r y Then, the derivative of the position vector r = r ˆr, is: E ϕ O F x r˙ =r ˙ˆr + rϕ˙ϕˆ (D12)

Taking a second derivative, we find: dˆr dϕˆ ¨r = (¨rˆr +r ˙ )+(˙rϕ˙ϕˆ + rϕ¨ϕˆ + rϕ˙ ) (D13) dt dt or

¨r = (¨r rϕ˙ 2)ˆr + (2r ˙ϕ˙ + rϕ¨)ˆϕ , (D14) 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- where we recognize the components ar and at given above responding point to P , on a circumference of radius a (the in Eq. (D5). semi-major axis of the ellipse).

Appendix E: Effect of the obliquity on the equation Appendix F: Eccentric anomaly of time For an ellipse with semi-axes a and b, we define an aux- Here we give some details on the influence of the obliq- iliary circumference of radius a that circumscribes the en- uity of the ecliptic on the equation of time. As in tire ellipse. Let us consider a point P on the ellipse, with Sec. III.B we assume a circular trajectory on a plane polar coordinates (r, ϕ), or Cartesian coordinates (x, y). forming an angle ǫ with the equator. From the spherical The eccentric anomaly corresponding to point P is the triangle shown in Fig. (7), we have: angle E in Fig. 20. This angle is defined as follows. Draw the straight line parallel to the (vertical) semi-minor axis tan α = tan λ cos ǫ (E1) and passing through P (there is one and only one parallel, according to ). This line intersects the circumfer- To obtain variations of α as a function of λ, we calculate ence in point P ′ (see Fig. 20; there is another intersection the derivative point on the other side of the major axis, that we do not ′ dα cos2 α 1 + tan2 λ consider). P is called the corresponding point to P . The = cos ǫ = cos ǫ (E2) ′ dλ cos2 λ 1 + tan2 α radius of the auxiliary circle passing through P makes an angle E with the major axis. and for a one-day interval we can approximate: Taking the origin of coordinates on the focus F , we have for the ellipse in Cartesian coordinates [see Eq. (27)] cos2 α ∆α = cos ǫ ∆λ (E3) cos2 λ (x + c)2 y2 + = 1 (F1) - At the equinoxes, cos α = cos λ = 1, and ∆α = a2 b2 0.9175 ∆λ. and for the angle E: - At the solstices, cos α = cos λ = 0, and the quotient in Eq. (E3) is singular. The limit at those points can x + c cos E = (F2) be easily found by replacing tan λ from Eq. (E1) into a Eq. (E2), which yields From these two equations, one has sin E = y/b. dα cos2 ǫ + tan2 α 1 = , (E4) To find out the distance r as a function of E, we write dλ 1 + tan2 α cos ǫ r2 = x2+y2 = (a cos E ae)2+b2 sin2 E = a2(1 e cos E)2 and from here: − − (F3) dα 1 so that lim = , (E5) α→π/2 dλ cos ǫ r = a(1 e cos E) (F4) which gives for the summer : ∆α = ∆λ/ cos ǫ, or − ∆α =1.0899 ∆λ (and the same for the winter solstice). which coincides with Eq. (47). 21

Appendix G: Area of the ellipse N 90 − φ Taking into account that the elementary area is H n 1 1 Z dS = r.rdϕ = r2dϕ (G1) 2 2 A we can calculate the area swept in a complete revolution 90 − h φ (the area of the ellipse). Thus we have in polar coordi- nates: δ 2π π 2 w S = dS = r dϕ (G2) Equator Z0 Z0 Using for r the expression in Eq. (25) for the conical Horizon trajectory, one has π p2 s S = dϕ (G3) (1 + e cos ϕ)2 Z0 To calculate this integral, we divide it into two parts: S p2 π dϕ ep2 π e + cos ϕ FIG. 21: Celestial sphere including details of equatorial and S = + dϕ horizontal coordinate systems. φ: local latitude, h: altitude, 1 e2 1+ e cos ϕ e2 1 (1 + e cos ϕ)2 − Z0 − Z0 A: azimuth (measured from the local south), H: local hour (G4) angle (H = θ − α; θ, local sidereal time), Z: , N and We calculate a primitive for the first part using the S: North and South poles in the equatorial system. δ: decli- ϕ change t = tan 2 , and find nation

dϕ 2 − 1 e ϕ = tan 1 − tan 1+ e cos ϕ (1 e2)3/2 1+ e 2 Appendix H: Astronomical coordinate systems Z − "r # (G5) For the second part, a primitive is: 1) Horizontal coordinates e + cos ϕ sin ϕ dϕ = . (G6) Horizon: fundamental plane. (1 + e cos ϕ)2 1+ e cos ϕ Z Primary direction: north or south point of horizon. Taking the integration limits, the second part vanishes, Zenith (Z): pole of the upper hemisphere. so that only the first part remains, and Nadir: pole of the lower hemisphere. 2p2 π πp2 Zenith distance (z): complement of altitude (i.e. z = o S = 2 3/2 = 2 3/2 (G7) 90 h). (1 e ) 2 (1 e ) − − − Vertical circle: great circle on the celestial sphere that Since passes from the observer’s zenith through a given celestial p2 body. Vertical circles are perpendicular to the horizon. ab = , (G8) (1 e2)3/2 Principal vertical (or Local Celestial Meridian): verti- − cal circle which is on the north-south direction. [see Eq. (28)] we finally find Prime vertical: Vertical circle on the east-west direc- S = πab (G9) tion. Azimuth (A): angle between planes of Note that the area of an ellipse can be more easily the principal vertical and the vertical cir- calculated in Cartesian coordinates: cle of a celestial object. We measure it here x2 y2 from the south increasing towards the west. Nowa- + =1 , (G10) a2 b2 days, it is more usual to measure it from the north increasing towards the east. and Altitude (h) (also called elevation): angle between a a a x2 celestial object and the observer’s local horizon. For ob- 0 S =2 y+(x) dx =2 b 1 2 dx . (G11) jects above the horizon, h (0, 90 ]. −a −a − a ∈ Z Z r Almucantar (also spelled almucantarat or almacan- Putting x = a sin θ, we find tara): a circle on the celestial sphere parallel to the π/2 horizon. Celestial objects lying on the same almucantar S =2ab cos2 θ dθ = πab (G12) have the same altitude. − Z π/2 22

N 90 − φ the nadir). H 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. Z 90 − δ 180 − A Declination (δ): angular distance of a point on the celestial sphere from the celestial equator, measured on the great circle passing through the celestial poles and the point in question. Points north of the celestial equator Q z = 90 − h have δ > 0, while those south have δ < 0. Declination is usually measured in sexagesimal degrees (o), minutes (’), P and seconds (”). Right ascension (α): angular distance on the celestial FIG. 22: Spherical triangle employed to pass from equato- equator measured eastward from the vernal equinox to rial to horizontal system (see Fig. 21). φ: local latitude, h: the hour circle of a given point on the celestial sphere. It altitude, δ: declination, A: azimuth (measured from the lo- cal south), H: local hour angle (H = θ − α, θ local sidereal is usually measured in hours, minutes and seconds (time). time), Q: parallactic angle, Z: zenith, N: North pole in the Hour angle (H): angular distance on the celestial equatorial system. P : position of the celestial object. sphere measured westward along the celestial equator from the local meridian to the hour circle passing through a given point or celestial object. The hour angle is related 2) Equatorial coordinates to the right ascension through the local sidereal time θ: H = θ α. Note that, unlike right ascension, hour angle − Celestial Equator: fundamental plane. is always increasing with the rotation of the Earth. Primary direction: vernal point γ Sidereal time (θ): at a given Earth’s place defined by Celestial Poles: North and South. its geographical longitude, and any moment, the sidereal Meridian: great circle passing through the celestial time is the hour angle of the vernal equinox at that place. poles and the zenith of a particular place on Earth. It It has the same value as the right ascension of any ce- contains the horizon’s north and south points and is per- lestial object that crosses the local meridian at the same pendicular to the celestial equator and the celestial hori- time. At the moment when the vernal point γ crosses the zon. The meridian is divided into the local meridian local meridian, the local sidereal time is 00:00. Green- (which contains the zenith and is terminated by the celes- wich sidereal time is the hour angle of the vernal equinox tial poles) and the antimeridian (opposite half containing at the at Greenwich.