This paper was awarded in the II International Competition (1993/94) ”First Step to Nobel Prize in Physics” and published in the competition proceedings (Acta Phys. Pol. A 88 Supplement, S-49 (1995)). The paper is reproduced here due to kind agreement of the Editorial Board of ”Acta Physica Polonica A”. EQUATION OF TIME — PROBLEM IN ASTRONOMY M. M ¨ uller Gymnasium M¨ unchenstein, Grellingerstrasse 5, 4142 M¨ unchenstein, Switzerland Abstract The apparent solar motion is not uniform and the length of a solar day is not constant throughout a year. The difference between apparent solar time and mean (regular) solar time is called the equation of time. Two well-known features of our solar system lie at the basis of the periodic irregularities in the solar motion. The angular velocity of the earth relative to the sun varies periodically in the course of a year. The plane of the orbit of the earth is inclined with respect to the equatorial plane. Therefore, the angular velocity of the relative motion has to be projected from the ecliptic onto the equatorial plane before incorporating it into the measurement of time. The math- ematical expression of the projection factor for ecliptic angular velocities yields an oscillating function with two periods per year. The difference between the extreme values of the equation of time is about half an hour. The response of the equation of time to a variation of its key parameters is analyzed. In order to visualize factors contributing to the equation of time a model has been constructed which accounts for the elliptical orbit of the earth, the periodically changing angular velocity, and the inclined axis of the earth. PACS numbers: 95.10.Ce 1. Introduction 1.1. The measurement of time This paper deals with a problem of the astronomical measurement of time. Let us first introduce some basic definitions. The natural unit of time is the rotation of the earth, that is the apparent daily course of the sun. The length of time between two culminations of the sun is called a solar day. The time-system based on this unit is called apparent solar time. By this system, localities on a given meridian always have the same time-readings. A comparison of a sundial with a mechanical clock shows that the solar day has a variable length. Therefore, so-called mean solar time is commonly used. This is based on a unit which is defined as the average of a solar day. The mean solar time has been
Transcript
This paper was awarded in the II International Competition (1993/94) ”First Step to Nobel Prizein Physics” and published in the competition proceedings (Acta Phys. Pol. A 88 Supplement,S-49 (1995)). The paper is reproduced here due to kind agreement of the Editorial Board of”Acta Physica Polonica A”.
The apparent solar motion is not uniform and the length of a solar day is not constantthroughout a year. The difference between apparent solar time and mean (regular)solar time is called the equation of time. Two well-known features of our solar systemlie at the basis of the periodic irregularities in the solar motion. The angular velocityof the earth relative to the sun varies periodically in the course of a year. The planeof the orbit of the earth is inclined with respect to the equatorial plane. Therefore,the angular velocity of the relative motion has to be projected from the ecliptic ontothe equatorial plane before incorporating it into the measurement of time. The math-ematical expression of the projection factor for ecliptic angular velocities yields anoscillating function with two periods per year. The difference between the extremevalues of the equation of time is about half an hour. The response of the equationof time to a variation of its key parameters is analyzed. In order to visualize factorscontributing to the equation of time a model has been constructed which accounts forthe elliptical orbit of the earth, the periodically changing angular velocity, and theinclined axis of the earth.
PACS numbers: 95.10.Ce
1. Introduction
1.1. The measurement of time
This paper deals with a problem of the astronomical measurement of time. Let us firstintroduce some basic definitions. The natural unit of time is the rotation of the earth,that is the apparent daily course of the sun. The length of time between two culminationsof the sun is called a solar day. The time-system based on this unit is called apparent solartime. By this system, localities on a given meridian always have the same time-readings.
A comparison of a sundial with a mechanical clock shows that the solar day has avariable length. Therefore, so-called mean solar time is commonly used. This is basedon a unit which is defined as the average of a solar day. The mean solar time has been
1
2 M. Muller
fixed in such a way that it does not deviate too much from the apparent solar time. Thedeviations between apparent solar time and mean solar time are described by the equationof time
equation of time = (apparent solar time)− (mean solar time).
The derivation, suitable approximations and relevant aspects of the equation of time arediscussed in this paper. The derivation does not account for minor effects due to thegravitational fields of the moon and the planets. In principle, therefore, a comparison ofthe results of such an idealized equation of time with the actual observations can be usedto estimate the magnitudes of these effects. Furthermore, parameters of the orbit of theearth, such as its eccentricity, can be verified or calculated. It should be mentioned thatthe equation of time was very important for navigation in earlier times.
1.2. The periodicity in the solar motion
Two well-known features of our solar system are at the basis of the variations in theapparent motion of the sun:
1. According to Kepler’s second law, the angular velocity of the earth relative to thesun varies throughout a year.
2. Equal angles which the sun in its apparent movement goes through in the eclipticdo not correspond to equal angles we measure on the equatorial plane. However,it is these latter angles which are relevant for the measurement of time, since thedaily movement of the sun is parallel to the equatorial plane (see Fig. 1).
Fig. 1. Apparent path of the sun in a geocentric view. At the perihelion, the sunruns faster than at the aphelion. Equal angles β on the plane of the ecliptic do notcorrespond to equal angles γ on the equatorial plane. In this figure there is β1 = β2
but γ1 < γ2.
THE EQUATION OF TIME — A PROBLEM IN ASTRONOMY 3
2. The variable angular velocity of the earth
2.1. Kepler’s laws
Kepler’s first law tells us that all planets are moving in elliptical orbits around thesun, whereby the latter is positioned at one of the two focal points. Kepler’s second law— the so-called law of areas — describes the velocity of the planets. The area swept outper time interval is constant (= dA/dt).
Hence, during the time period t the radius vector from the sun to the earth sweepsout an area of
tdA
dt= t
πab
T=
t
Tπab (1)
(a, b — axes of the ellipse, T — duration of a revolution).Let us now derive the angular velocity of the earth as a function of time. The angle
covered by the earth after leaving the perihelion is called “true anomaly”, denoted herewith R (see Fig. 2). Let us imagine a “mean earth” which has also a revolution time T and
Fig. 2. Angles R, M and E at a specific time point. The affinity factor between theelliptical orbit and the circle going through the perihelion and the aphelion is givenby b/a. As the angular velocity of the “real earth” and the“mean earth” are bothconstant, the ratio between the two hatched areas is the same as between the areas ofthe circle and the ellipse, viz., 1 : b/a.
4 M. Muller
is running at a constant speed on a circular orbit with the sun at its centre. This “meanearth” would cover an angle, called “mean anomaly” (M), in the same period of time asthe true earth covers the angle R. In Fig. 2, M is drawn from the centre of the ellipse.The orbit of the “mean earth” is the circle through the perihelion and the aphelion. The“mean earth” starts from the perihelion at the same time as the true earth. Since theangular velocity of the “mean earth” is constant and its revolution lasts one year (T ), Msatisfies the simple equation
M = 2πt
T, (2)
where t is the time span after passage through the perihelion. It is very useful to definea third angle as a link between M and R (see Fig. 2). The perpendicular drawn from theposition of the true earth (E) onto the major axis intersects the circle, that is the orbitof the “mean earth”, at the point A. The angle PZA is called eccentric anomaly (E) andwas introduced by Johannes Kepler [1]. It can be used to calculate the area of ellipticsectors.
We get the following relations from Fig. 2
tanR =EB
SB=
baAB
ZB − ZS=
baa sinE
a cosE − ea=b
a
sinE
cosE − e=
√1− e2 sinE
cosE − e, (3)
cosR =
√1
1 + tan2 R=
√√√√ (cosE − e)2
(cosE − e)2 + (1− e2) sin2 E=
cosE − e1− e cosE
, (4)
tanR
2=
√1− cosR
1 + cosR=
√1− e cosE − cosE + e
1− e cosE + cosE − e=
√√√√(1 + e)(1− cosE)
(1− e)(1 + cosE)
=
√1 + e
1− e
√1− cosE
1 + cosE=
√1 + e
1− etan
E
2. (5)
Let us write Eq. (5) with√1− e1 + e
≡ cosα,R
2≡ y,
E
2≡ x,
tan y =tanx
cosα⇔ y = arctan
(tanx
cosα
). (6)
The differentiation of Eq. (6) yields
dy
dx=
cosα
1− sin2 α cos2 x=
cosα
1− sin2 α (1+cos 2x)2
=2 cosα
1 + cos2 α− sin2 α cos 2x= f(x). (7)
Since this expression is a periodic function of x it can be expanded into a Fourier series
f(x) =dy
dx=
cosα
1− sin2 α cos2 x
= a0 + a1 cosx+ b1 sin x+ . . .+ an cosnx+ bn sinnx+ . . . , (8)
where
a0 =1
2π
∫ 2π
0f(x)dx =
1
2π
∫ 2π
0
dy
dxdx =
1
2π
∫ 2π
0dy = 1 (9)
THE EQUATION OF TIME — A PROBLEM IN ASTRONOMY 5
and
an =1
π
∫ 2π
0f(x) cosnxdx, bn =
1
π
∫ 2π
0f(x) sinnxdx for n > 0. (10)
Replacing sine and cosine by the complex exponential function yields
an =1
π
∫ 2π
0
cosα(einx + e−inx
)1 + cos2 α− sin2 α
(e2ix+e−2ix
2
)dx
=i
π
∮ cosα(zn/2 + z−n/2
)−2z (1 + cos2 α) + sin2 α (z2 + 1)
dz
=i
π
∮ cosα(zn/2 + z−n/2
)sin2 α
(z2 − 2 (1+cos2 α)
sin2 αz + 1
)dz with z = e2ix. (11)
The contour of integration is twice the unit circle. The denominator has the roots
N1 =(
1− cosα
sinα
)2
= tan2 α
2, N2 =
(1 + cosα
sinα
)2
= cot2 α
2. (12)
Now, Eq. (11) can be written in the form
an =i
π
∮ cosα
sin2 α
(zn/2 + z−n/2
)(z − tan2 α
2
) (z − cot2 α
2
)dz
=i
π
∮ cosα
sin2 α
(zn/2 + z−n/2
)tan2 α
2− cot2 α
2
1(z − tan2 α
2
) − 1(z − cot2 α
2
) dz
= − i
4π
∮ (zn/2 + z−n/2
) 1(z − tan2 α
2
) − 1(z − cot2 α
2
) dz. (13)
As commonly known, it is∮zpdz =
∫ e4iπ
e0zpdz =
{0 for p 6= −1,
4iπ for p = −1.(14)
Therefore, we can develop the fractions of the integrand (13) into convergent series andgo on calculating only with terms of the form g(α)/z, because all other terms contributezero to the integral.
1
z − tan2 α2
=1
z
∞∑i=0
(tan2 α
2
z
)i, (15)
− 1
z − cot2 α2
= tan2 α
2
∞∑i=0
(z tan2 α
2
)i. (16)
The integrand (13) becomes
− i
4π
(zn/2 + z−n/2
) 1
z
∞∑i=0
(tan2 α
2
z
)i+ tan2 α
2
∞∑i=0
(z tan2 α
2
)i . (17)
Expressions in 1z
= z−1 only result if the exponent n2
is an integer. Consequently, theintegral is zero if n is odd, that is, an vanishes if n is odd. an = 0 for odd n. If n is even
6 M. Muller
we get the coefficient of 1/z in the integrand as the sum of two expressions, each fromone of the two series
− i
4π
1
z
(tann
α
2+ tan2 α
2tan−2+n α
2
)= − i
2π
1
ztann
α
2. (18)
The integration of this term yields
an = − i
2πtann
α
2
∫ e4iπ
e0
1
zdz = − i
2πtann
α
24iπ = 2 tann
α
2(19)
⇒ an = 2 tannα
2for even n > 0. (20)
The coefficients bn can be obtained similarly. If n is odd, bn disappears. If n is even, onehas to write instead of Eq. (18)
− 1
4π
(1
ztann
α
2− 1
ztann
α
2
)= 0. (21)
Consequently, the Fourier series only consists of cosine terms with an even coefficient inthe argument
dy
dx= 1 + 2
∞∑i=1
tan2i α
2cos 2ix, (22)
y = x+∞∑i=1
tan2i α
2
sin 2ix
i+ const. (23)
In Eq. (5), we have to set
cosα ≡√
1− e1 + e
→ tan2 α
2=
1− cosα
1 + cosα=
1−√
1−e1+e
1 +√
1−e1+e
=1− 2
√1−e1+e
+ 1−e1+e
1− 1−e1+e
=1 + e+ 1− e− 2
√1− e2
1 + e− 1 + e=
1−√
1− e2
e
≈1−
(1− e2
2− e4
8
)e
=e
2+e3
8. (24)
With R(E = 0) = 0 we get
R = E + 2∞∑i=1
(e
2+e3
8
)4sin iE
i
≈ E +
(e+
e3
4
)sinE +
e2
4sin 2E +
e3
12sin 3E. (25)
The angle E is so useful because the area swept out by the radius vector from the sunto the earth in the time span t can be calculated easily by using the affinity between thecircle and the ellipse
F = elliptic sector SEP =b
a(sector ZAP − triangle ZAS),
F =b
a
(E
2πareacircle −
ZS AB
2
)=b
a
(E
2a2 − aea sinE
2
)
THE EQUATION OF TIME — A PROBLEM IN ASTRONOMY 7
=b
a
(E
2a2 − a2 e sinE
2
)=ab
2(E − e sinE). (26)
On the other hand, the law of areas implies
F =t
Tπab, or with M =
t
T2π one gets F =
Mab
2. (27)
The comparison with Eq. (26) yields
E = M + e sinE (28)
Equation (28) is called Kepler’s equation. It is not possible to solve this equation forE in closed form. Therefore, the position of the earth — given by E — can only beapproximated as a function of time. The “reversion theorem” of Lagrange allows us toexpand any function in E into a series in e and M . Its basis is an equation in the form
z = y + xf(z). (29)
According to Lagrange, any function g(z) can be expanded into a series depending on xand y:
g(z) = g(y) + xg′(y)f(y) +x2
2!
∂
∂y
{g′(y) [f(y)]2
}
+x3
3!
∂2
∂y2
{g′(y)[f(y)]3
}+ . . . (30)
In the case of Kepler’s equation (28), we set
z = E, y = M, x = e, f(z) = sinE, f(y) = sinM, g(z) = z. (31)
E turns out as
E = M + e sinM +e2
2
∂
∂M(sin2 M) +
e3
6
∂2
∂M2(sin3 M) + . . .
= M + e sinM +e2
2sin 2M +
e3
8(3 sin 3M − sinM) + . . . (32)
To be able to express R in Eq. (25) as a function of M , we also have to expand the sinefunctions of E, 2E and 3E as far as needed
sinE =E −M
e= sinM +
e
2sin 2M +
e2
8(3 sin 3M − sinM) + . . . (33)
sin 2E = sin 2M + e sinM 2 cos 2M + . . .
= sin 2M + e(sin 3M − sinM) + . . . (34)
sin 3E = sin 3M + . . . (35)
Replacing the functions in E of Eq. (25) with these formulas in M , we obtain
R ≈ E +
(e+
e3
4
)sinE +
e2
4sin 2E +
e3
12sin 3E
= M +
(2e+
e3
4
)sinE +
e2
4sin 2E +
e3
12sin 3E
8 M. Muller
≈M + 2e sinM +5
4e2 sin 2M + e3
(−1
4sinM +
13
12sin 3M
)+ e4 . . . (36)
In this series, we have only taken into account terms up to third power of e. (Thisapproximation is already very accurate, for e is about 0.0167.) The angular velocity ofthe earth can now be obtained as the first derivative of R with respect to time
ω(t) =dR
dt=
dR
dM
dM
dt=
2π
T
dR
dM
≈ 2π
T
[1 + 2e cosM +
5
2e2 cos 2M + e3
(−1
4cosM +
13
4cos 3M
)]. (37)
This function is plotted in Fig. 3. It is mainly determined by the first variable term of theseries (2e cosM). The factor 2π/T is the mean angular velocity. The deviations amountto about ±3.5% (≈ 2e).
Fig. 3. The angular velocity of the earth as a function of time. On average it is about2π/365.25 per day or 360/365.25 ◦/day = 0.986 ◦/day.
3. The inclined plane of the ecliptic
3.1. The earth in space
Since the equation of time is to be examined during a year, the earth can be supposedto always have the same direction in space, i.e., we will not take into account precessionand nutation.
We can describe the direction of the axis of the earth with two angles. The first is theangle between the axis and the norm of the orbit (ε). The second is the angle P formedby the major axis of the orbit and the projection of the axis of the earth onto the plane ofthe orbit (see Fig. 4). At present, ε measures about 23.45◦, P is about 12.25◦. (P is alsothe angle which is covered by the earth between the beginning of winter (21st December)and the arrival of the earth at perihelion (2nd January).)
THE EQUATION OF TIME — A PROBLEM IN ASTRONOMY 9
Fig. 4. The angle P and the inclination of the axis of the earth at the winter solstice.
3.2. The projection of ecliptic angles onto the equatorial plane
The angles which the sun appears to cover relative to the earth are equal to those theearth actually covers relative to the sun. Since the corresponding angles parallel to theequatorial plane are needed for the measurement of time, the ecliptical angles have to beprojected onto the equatorial plane. This results in angle widening or shortening. For aninfinitesimally small angle the projection factor (the deformation factor) is determinedby the angle parameters ϕ and ε in Fig. 5, which represents the geocentric view. ϕ ismeasured from the winter solstice.
We get the following relations from Fig. 5:
tanϕ = cos ε tanϕp, tan(ϕ+ ω) = cos ε tan (ϕp + ϕp) . (38)
The projection factor f for a small angle can be calculated as a function of ϕ and ε:
f(ϕ) = limω→∞
(ωp
ω
)= lim
ω→0
arctan[
tan(ϕ+ω)cos ε
]− arctan
(tanϕcos ε
)ω
. (39)
Using the series (22) derived in Sec. 2.1 (ε ≡ α), we get an approximation which convergesvery quickly
f(ϕ) = f(R + P ) = 1 + 2 tan2 ε
2cos 2(R + P )
+2 tan4 ε
2cos 4(R + P ) + 2 tan6 ε
2cos 6(R + P ) + . . . (40)
The projection factor f is plotted as a function of time in Fig. 6. This graph resemblesa cosine function since it is mainly determined by the first variable term of the approxi-mation. The amplitude of the variation (2 tan2 ε
2) amounts to 81
2% and, thus, is greater
than that of the angular velocity (compare with Fig. 3).
10 M. Muller
Fig. 5. Geocentric view of the projection. In a short time interval, the sun has coveredthe angle ω. ωp is the orthogonal projection of ω.
Fig. 6. The projection factor as a function of time. The maxima are at the solstices,the minima at the equinoxes.
The extremes are situated at the beginning of the seasons. At the summer and wintersolstice, the sun reaches, respectively, its highest and lowest position. Here, an ecliptic an-gle is stretched maximally. At the vernal and autumnal equinox, the sun stands verticallyabove the equator. Here, ecliptic angles are shortened maximally.
THE EQUATION OF TIME — A PROBLEM IN ASTRONOMY 11
4. The calculation of the equation of time
4.1. Derivation of a more accurate approximation
In order to calculate the equation of time we need the projection of the true anomalyRp as a function of time
Rp =∫
dRp =∫f(R+P )dR =
∫ [1+2
∞∑i=1
tan2i ε
2cos 2i(R+P )
]dR (41)
= R +∞∑i=1
1
itan2i ε
2sin 2i(R + P ) + const. (42)
The constant term will be determined later. Let us replace R by M with Eq. (36)
R = M + 2e sinM +5
4e2 sin 2M + e3
(−1
4sinM +
13
12sin 3M
). . . (43)
Thus, Rp becomes a function of M and hence of time.
Rp = M + 2e sinM +5
4e2 sin 2M + e3
(−1
4sinM +
13
12sin 3M
)+ . . .
+ tan2 ε
2sin
{2(M + P ) + 2
[2e sinM +
5
4e2 sin 2M
+e3(−1
4sinM +
13
12sin 3M
)+ . . .
]}+
1
2tan4 ε
2sin
{4(M + P ) + 4
[2e sinM +
5
4e2 sin 2M
+e3(−1
4sinM +
13
12sin 3M
)+ . . .
]}+
1
3tan6 ε
2sin
{6(M + P ) + 6
[2e sinM +
5
4e2 sin 2M
+e3(−1
4sinM +
13
12sin 3M
)+ . . .
]}+ . . .+ const. (44)
The expansion of the sine functions yields
Rp ≈M + tan2 ε
2(1− 4e2) sin 2(M + P ) + 2e sinM
−2e tan2 ε
2sin(M + 2P ) + 2e tan2 ε
2sin(3M + 2P )
+1
2tan4 ε
2sin 4(M + P ) +
5
4e2 sin 2M − 2e tan4 ε
2sin(3M + 4P )
+2e tan4 ε
2sin(5M + 4P ) +
13
4e2 tan2 ε
2sin(4M + 2P )
+1
3tan6 ε
2sin 6(M + P ) + const. (45)
The equation of time is defined by
equation of time = “true equatorial sun angle”– “mean equatorial sun angle” (geocentric view)
= −(true projected anomaly – mean anomaly)(heliocentric view)
= M −Rp.
(46)
12 M. Muller
Fig. 7. The determination of the constant of integration Rp −M = P p − P .
Let us now determine the constant of integration. A commonly used definition impliesthat the “mean sun” arrives at the vernal equinox at the same time as a “dynamic sun”that runs in the ecliptic at a constant speed and leaves the perihelion at the same time asthe real sun (see [2]). Because of this definition the angles of the two regularly runningsuns to the vernal equinox are always equal. Consequently, the angle P between theperihelion and the direction earth–winter solstice in Fig. 7 is equal to the angle of the“dynamic sun” on the equator to the projected winter solstice direction. The constant ofintegration can now be determined if we examine the passage through the perihelion (seeFig. 7):
M −Rp = P − P p = P −∫ P
0f(ϕ)dϕ
= P −∫ P
0
(1 + 2
∞∑i=1
tan2i ε
2cos 2iϕ
)dϕ = −
∞∑i=1
1
itan2i ε
2sin 2iP. (47)
If we set t = 0 in Eq. (44) we will find that the constant of integration vanishes. Now wecan calculate the coefficients in Eq. (45) with ε = 23.45◦ and e = 0.0167. Finally we get
equation of time = M −Rp ≈ −591.7 sin 2(M + P )− 459.6 sinM
+19.8 sin(M + 2P )− 19.8 sin(3M + 2P )− 12.8 sin 4(M + P )
At present, the angle P measures about 12.25◦.Figure 8 shows the equation of time for a whole period of one year. In winter, the
value of the equation of time decreases mostly because the angular velocity of the earthand the projection factor reach their maximum. The opposite holds true between thepassage through the aphelion and the beginning of autumn. Afterwards, the equation oftime itself reaches an extreme value. In the time span between those extrema, especiallyin summer, the lower angular velocity and the greater projection factor compensate eachother.
Figure 9 (taken from [2]) shows the observed phenomenon of the equation of time. Inintervals of ten days, a picture of the sky was taken at 8.14 (mean solar time). If the sunmoved at a constant speed through the sky, all pictures of the sun would lay on a straightline that would be perpendicular to the daily path of the sun. In winter and in summer,
THE EQUATION OF TIME — A PROBLEM IN ASTRONOMY 13
Fig. 8. The equation of time with its two main terms. The difference between themaximum in October (≈ 16 min) and the minimum in February (≈ −14 min) is abouthalf an hour.
Fig. 9. The analemma (after [2]).
the sun has not reached as far as one would expect (it is below the “average line” (2)).Therefore, the value of the equation of time is negative (see Fig. 8). On the other hand,the sun advances faster from April to June and at the end of the year (it is above the“average line”) and the value of the equation of time is positive. These variations producethe form of a stretched and inclined “eight” in the picture.
4.2. The equation of time as a function of its parameters
The equation of time is determined by the following parameters:
• the eccentricity of the orbit of the earth
• the angle between the ecliptic and the equatorial planes
14 M. Muller
• the angle P between the winter solstice and the perihelion relative to the sunor: the time span ∆t from the beginning of winter to the passage through theperihelion
The last two parameters change gradually by nutation and precession. It is thereforeinteresting to examine the influence of each parameter. Figures 10–12 show how theequation of time changes when one parameter is varied.
1. parameter: the eccentricity. If e = 0 a regular variation results that is caused bythe inclination of the ecliptic plane. The deviations of the apparent solar time from themean solar time increase with growing e in winter and autumn. Thus, the yearly variationbecomes dominant. Since at the perihelion and aphelion the equation of time is only afunction of the ecliptic inclination and the angle P , all plots have the same value at thesetwo points.
2. parameter: the inclination of the ecliptic. ε = 0 yields a plot which is symmetric tothe passage through the aphelion. The greater ε the more dominant the variation with aperiod of half a year. All plots have four common points at the beginning of each season,for the equation of time depends only on the two other parameters there (eccentricityand P ). As the projection from the ecliptic plane onto the equatorial plane does notchange the polar angle relative to the winter solstice, ε does not influence the value of theequation of time at the beginning of a season.
3. parameter: the time interval between the beginning of winter and the passagethrough the perihelion. If ∆t = 0 the two main variations vanish both at the beginningof winter and summer (because winter begins when the earth passes the perihelion; theaphelion is the summer solstice). Therefore, the resulting function is symmetric and theextreme values are in autumn and winter. If ∆t increases, the two components tend tocompensate each other in winter whereas the negative value in summer begins to dominate.
Fig. 10. 1. parameter: the eccentricity. + + + e = 0.000, ◦ ◦ ◦ e = 0.005,2 2 2 e = 0.010, • • • e = 0.015, ××× e = 0.020.
Fig. 12. 3. parameter: the time interval between the beginning of winter and thepassage through the perihelion. + + + ∆t = 0 days, ◦ ◦ ◦ ∆t = 20 days, 2 2 2 ∆t =40 days, • • • ∆t = 60 days, ××× ∆t = 80 days.
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5. Construction of a model
As one might have some difficulty in imagining the matters treated herein concern-ing the solar system we decided to construct a model which illustrates the precedingexplanations.
5.1. The orbit of the earth
As to the revolution of the earth, the following points have to be taken into account:
1. The orbit of the earth has to be elliptical and the sun should be in a focus of theellipse.
2. The speed of the earth in its revolution is higher at the perihelion than at theaphelion.
Because of the first feature, the earth cannot be fixed to an arm as it is mostly seen inother models.
A simple solution for this problem is to run a vertical axis (which represents the earth)in an elliptic orbit by two elliptical hollow tracks in parallel planes.
This axis is pushed by a horizontal stick but it is not fixed to it. Thus, the distancebetween the axis and the rotary centre of the pushing stick can vary. If this stick turnsregularly, the speed of the axis in the hollow tracks is not constant; when the distancebetween the rotary centre and the axis increases, the axis will be pushed at a higher speed.
These considerations serve to find a solution for the second point. We need a higherspeed at the perihelion than at the aphelion. Therefore, the rotary centre of the pushingstick has to be placed further away from the perihelion than from the aphelion (seeFig. 13).
Fig. 13. The pushing stick in four positions.
5.2. The rotation of the earth
The drive of the rotation of the earth has to be fixed since it will be connected tothe revolution drive. On the other hand, the earth (or rather the vertical axis) revolvesaround the sun. So there has to be a flexible connection between the rotation drive and
THE EQUATION OF TIME — A PROBLEM IN ASTRONOMY 17
the axis in the hollow tracks. In our model, we use a chain for this purpose. In order touse the chain always to its entire length we can take advantage of a special feature of theellipse.
The sum of the distances between the foci and any point on the periphery of the ellipseis constant throughout. If we lead the chain around the foci to the axis in the hollowtracks the chain will always be tight (see Fig. 14).
Fig. 14. The leading of the chain.
For the chain of the rotation drive, we have to put one axis into each focus. It willmake drive only one of these two axes. Now we have to decide where to place the rotarycentre of the revolution drive. In Fig. 14 you can see that it is only possible to place it intothe hatched area because otherwise it would interfere with the chain of the rotation drive.Let us put the rotary axis at the most ideal place, that is the focus near the aphelion.There, it also serves as an axis for a sprocket wheel of the chain. The revolution drivebeing placed in this focus, the rotation drive has to be located in the other focus. (Thesprocket wheel at the focus of the revolution drive must not be fixed to the axis becausetheir rotary speeds are not equal.)
Both drives are connected with a transmission that realizes a ratio of approximately1:300.
5.3. The tilt of the axis of the earth
With a universal joint, we fix a second, tilted axis to the vertical axis that revolvesaround the sun. The inclined axis is held in its correct position (with a tilt of about 23.5◦)by a prop, which still allows the earth to rotate (see Fig. 15).
Fig. 15. Fixing of the axis of the earth in a position of a tilt of about 23.5◦.
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Fig. 16. Device for the right position of the axis of the earth.
The axis of the earth is fixed now with a tilt of about 23.5◦, but it is able to swivelround. Figure 16 illustrates how the prop can be controlled during a revolution so thatthe special direction of the axis of the earth is always the same.
