This paper was awarded in the II International Competition (1993/94) "First Step to Nobel Prize in Physics" and published in the competition proceedings (

M. MÜLLER

Gymnasium Münchenstein, Grellingerstrasse 5,

4142 Münchenstein, Switzerland

4142 Münchenstein, Switzerland

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 mathematical 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 fixed in such a way that it does
not deviate too much from the apparent solar time. The deviations between
apparent solar time and mean solar time are described by the *equation
of time*

The derivation, suitable approximations and relevant aspects of the equation of time are discussed in this paper. The derivation does not account for minor effects due to the gravitational fields of the moon and the planets. In principle, therefore, a comparison of the results of such an idealized equation of time with the actual observations can be used to estimate the magnitudes of these effects. Furthermore, parameters of the orbit of the earth, such as its eccentricity, can be verified or calculated. It should be mentioned that the 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 the apparent motion of the sun:

- According to Kepler's second law, the angular velocity of the earth relative to the sun varies throughout a year.
- Equal angles which the sun in its apparent movement goes through in the ecliptic do 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 the daily 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 sun runs faster than at the aphelion. Equal angles on the plane of the ecliptic do not correspond to equal angles on the equatorial plane. In this figure there is but .

**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 the sun, 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 out per time interval is constant .

Hence, during the time period *t* the radius vector from the sun to the
earth sweeps out an area of

(

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 here with *R* (see Fig. 2).

Fig.2.AnglesR,MandEat a specific time point. The affinity factor between the elliptical orbit and the circle going through the perihelion and the aphelion is given byb/a. As the angular velocity of the ``real earth'' and the``mean earth'' are both constant, the ratio between the two hatched areas is the same as between the areas of the circle and the ellipse, viz., 1:b/a.

Let us imagine a ``mean earth'' which has also a revolution time
*T* and is running at a constant speed on a circular orbit with the sun at
its centre. This ``mean earth'' would cover an angle, called *``mean
anomaly''* (*M*), in the same period of time as the 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 the angular velocity of the ``mean earth'' is
constant and its revolution lasts one year (*T*), *M* satisfies the simple
equation

where

We get the following relations from Fig. 2

Let us write Eq. (5) with

The differentiation of Eq. (6) yields

Since this expression is a periodic function of

where

and

Replacing sine and cosine by the complex exponential function yields

The contour of integration is twice the unit circle. The denominator has the roots

Now, Eq. (11) can be written in the form

As commonly known, it is

Therefore, we can develop the fractions of the integrand (13) into convergent series and go on calculating only with terms of the form , because all other terms contribute zero to the integral.

The integrand (13) becomes

Expressions in only result if the exponent is an integer. Consequently, the integral is zero if

The integration of this term yields

The coefficients

Consequently, the Fourier series only consists of cosine terms with an even coefficient in the argument

In Eq. (5), we have to set

With

The angle

On the other hand, the law of areas implies

The comparison with Eq. (26) yields

Equation (28) is called Kepler's equation. It is not possible to solve this equation for

According to Lagrange, any function

In the case of Kepler's equation (28), we set

To be able to express

Replacing the functions in

In this series, we have only taken into account terms up to third power of

This function is plotted in Fig. 3. It is mainly determined by the first variable term of the series . The factor is the mean angular velocity. The deviations amount to about % ().

Fig.3.The angular velocity of the earth as a function of time. On average it is about per day or /day = /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 supposed to always have the same direction in space, i.e., we will not take into account precession and nutation.

We can describe the direction of the axis of the earth with two angles. The
first is the angle between the axis and the norm of the orbit
(). The second is the angle *P* formed by the major axis of
the orbit and the projection of the axis of the earth onto the plane
of the orbit (see Fig. 4). At present, measures about
, *P* is about . (*P* is also the angle which is
covered by the earth between the beginning of winter (21st December) and
the arrival of the earth at perihelion (2nd January).)

**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 the earth actually covers relative to the sun. Since the corresponding angles parallel to the equatorial plane are needed for the measurement of time, the ecliptical angles have to be projected onto the equatorial plane. This results in angle widening or shortening. For an infinitesimally small angle the projection factor (the deformation factor) is determined by the angle parameters and in Fig. 5, which represents the geocentric view. is measured from the winter solstice.

Fig.5.Geocentric view of the projection. In a short time interval, the sun has covered the angle . is the orthogonal projection of .

Fig.6.The projection factor as a function of time. The maxima are at the solstices, the minima at the equinoxes.

We get the following relations from Fig. 5:

The projection factor

Using the series (22) derived in Sec. 2.1 ( ), we get an approximation which converges very quickly

The projection factor

The extremes are situated at the beginning of the seasons. At the summer and winter solstice, the sun reaches, respectively, its highest and lowest position. Here, an ecliptic angle is stretched maximally. At the vernal and autumnal equinox, the sun stands vertically above the equator. Here, ecliptic angles are shortened maximally.

**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 anomaly as a function of time

The constant term will be determined later. Let us replace

Thus, becomes a function of

The expansion of the sine functions yields

The equation of time is defined by

Let us now determine the constant of integration. A commonly used definition implies that 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 as the real sun (see [2]). Because of this definition the angles of the two regularly running suns to the vernal equinox are always equal. Consequently, the angle

If we set

At present, the angle

Fig.8.The equation of time with its two main terms. The difference between the maximum in October ( min) and the minimum in February ( min) is about half an hour.

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 earth and the projection factor reach their maximum. The opposite holds true between the passage through the aphelion and the beginning of autumn. Afterwards, the equation of time itself reaches an extreme value. In the time span between those extrema, especially in summer, the lower angular velocity and the greater projection factor compensate each other.

Figure 9 (taken from [2]) shows the observed phenomenon of the equation of time. In intervals of ten days, a picture of the sky was taken at 8.14 (mean solar time). If the sun moved at a constant speed through the sky, all pictures of the sun would lay on a straight line that would be perpendicular to the daily path of the sun. In winter and in summer, 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 produce the 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
- the angle
*P*between the winter solstice and the perihelion relative to the sun

or: the time span from the beginning of winter to the passage through the perihelion

1. parameter: the eccentricity. If *e*=0 a regular variation results that
is caused by the inclination of
the ecliptic plane. The deviations of the apparent solar time from the mean
solar time increase with growing *e* in winter and autumn. Thus, the yearly
variation becomes dominant. Since at the perihelion and aphelion the
equation of time is only a function of the ecliptic inclination and the
angle *P*, all plots have the same value at these two points.

2. parameter: the inclination of the ecliptic. yields a
plot
which is symmetric to the passage through the aphelion. The greater
the more dominant the variation with a period 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
(eccentricity and *P*). As the projection from the ecliptic plane onto the
equatorial plane does not change the polar angle relative to the winter
solstice, does not influence the value of the equation of
time at the beginning of a season.

Fig.10.1. parameter: the eccentricity. +++ e=0.000, ,

, , x x x e=0.020.

Fig.11.2. parameter: the inclination of the ecliptic. , , , , .

Fig.12.3. parameter: the time interval between the beginning of winter and the passage through the perihelion. days, days, days, days, days.

3. parameter: the time interval between the beginning of winter and the passage through the perihelion. If the two main variations vanish both at the beginning of winter and summer (because winter begins when the earth passes the perihelion; the aphelion is the summer solstice). Therefore, the resulting function is symmetric and the extreme values are in autumn and winter. If increases, the two components tend to compensate each other in winter whereas the negative value in summer begins to dominate.

**5. Construction of a model**

As one might have some difficulty in imagining the matters treated herein concerning the solar system we decided to construct a model which illustrates the preceding explanations.

**5.1. The orbit of the earth**

As to the revolution of the earth, the following points have to be taken into account:

- The orbit of the earth has to be elliptical and the sun should be in a focus of the ellipse.
- The speed of the earth in its revolution is higher at the perihelion than at the aphelion.

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 distance between the axis and the rotary centre of the pushing stick can vary. If this stick turns regularly, the speed of the axis in the hollow tracks is not constant; when the distance between 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 higher speed at the perihelion than at the aphelion. Therefore, the rotary centre of the pushing stick has to be placed further away from the perihelion than from the aphelion (see Fig. 13).

**5.2. The rotation of the earth**

The drive of the rotation of the earth has to be fixed since it will be connected to the revolution drive. On the other hand, the earth (or rather the vertical axis) revolves around the sun. So there has to be a flexible connection between the rotation drive and the axis in the hollow tracks. In our model, we use a chain for this purpose. In order to use the chain always to its entire length we can take advantage of a special feature of the ellipse.

The sum of the distances between the foci and any point on the periphery of the ellipse is constant throughout. If we lead the chain around the foci to the axis in the hollow tracks the chain will always be tight (see Fig. 14).

For the chain of the rotation drive, we have to put one axis into each focus. It will make drive only one of these two axes. Now we have to decide where to place the rotary centre of the revolution drive. In Fig. 14 you can see that it is only possible to place it into the 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 drive being placed in this focus, the rotation drive has to be located in the other focus. (The sprocket wheel at the focus of the revolution drive must not be fixed to the axis because their rotary speeds are not equal.)

Both drives are connected with a transmission that realizes a ratio of approximately 1:300.

**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 revolves around the sun. The inclined axis is held in its correct position (with a tilt of about ) by a prop, which still allows the earth to rotate (see Fig. 15).

The axis of the earth is fixed now with a tilt of about , but it is able to swivel round. Figure 16 illustrates how the prop can be controlled during a revolution so that the special direction of the axis of the earth is always the same.

Figure 17 shows the entire model.

- 1
- W. Schaub,
*Vorlesungen über sphärische Astronomie*, Akademischer Verlag Geest und Portig, 1950 - 2
- R. Sauermost,
*Lexikon der Astronomie: Die grosse Enzyklopädie der Weltraumforschung in zwei Bänden*, Herder Verlag, Freiburg im Breisgau 1989/90 - 3
*Brockhaus Enzyklopädie in zwanzig Bänden*, F.A. Brockhaus, Wiesbaden 1968

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