Local solar time is measured by a sundial. When the center of the sun is on an observer’s meridian, the observer’s local solar time is zero hours (noon). Because the earth moves with varying speed in its orbit at different times of the year and because the plane of the earth’s equator is inclined to its orbital plane, the length of the solar day is different depending on the time of year. It is more convenient to define time in terms of the average of local solar time. Such time, called mean solar time, may be thought of as being measured relative to an imaginary sun (the mean sun) that lies in the earth’s equatorial plane and about which the earth orbits with constant speed. Every mean solar day is of the same length.¹

In mathematical terms this means the following: Let e = (e₁, e₂, e₃) be the canonical base of ³ and think the origin to be positioned at the center of the earth. Then, choosing the initial time and the length of a (mean) solar day appropriately, the law of motion of the mean sun is given by

(1)

,

(2)

,

,

Local Mean time (or local civil time) has the convenient property, that,

if it is noon at Greenwich, it is 1 p.m. at a position 15^o east of Greenwich, or more general:

Thus x_m satisfies the functional equation

(3)

Theorem 1. Let y : ³ be given such that y(s) /= 0 for all real s and such that (3) holds for all , , s and some ] - /2, /2[. Let furthermore

(4)

Then there is some constant vector c ³, c /= 0, such that

(5)

Conversely we have that every y given by (5) satisfies (3).

Proof.  (3) is equivalent to But one easily can see that Thus (3) is equivalent to which for s = 0 and c = y(0) gives the desired result. The remaining part is obvious.     

Remark 1. The result of this theorem says, that every object behaving like the mean sun with respect to (3) is given by a clockwise rotation with center at some point of the earth’s axis and rotational velocity of one full rotation per day.

Now we come to a problem for which it is important to take the earth’s radius into account. Satellite dishes point to positions given by geostationary satellites, and thus have the desired property of fixed positions invariant with respect to time. Thus, if y(s) is the position of the satellite at time s, the property required is that the direction vector belonging to , i.e., the unit vector corresponding to this vector, expressed in local coordinates be independent of s:

(6)

Theorem 2. If for y : ³, /= 1 for all s , the expression is independent of s for at least three different points P (_i, _i), i = 1, 2, 3, then y itself is constant. Moreover every constant vector of length different from 1 satisfies (6).

Proof.  (6) implies for some unit vector u(, ). Thus for i = 1, 2, 3, where the g_i are straight lines in ³.

If, say, u(_i, _i) and u(_j, _j) are linearly independent, the intersection of g_i and g_j contains at most one point. Since y(s) g_i g_j this implies that y(s) equals the point of intersection of those two lines for all s. Thus y has to be constant.

The (only) other possibility left means that the two vectors u(₁, ₁), u(₂, ₂) as well as the vectors u(₁, ₁) and u(₃, ₃) are linearly dependent. But then all three lines g₁, g₂, g₃ coincide, since they have a point in common and since they have proportional direction vectors. Thus the three different points P (₁, ₁), P (₂, ₂), P (₃, ₃) of the same length are collinear, which is impossible.     

Now, as we have been successful with taking into account the radius of the earth in solving this problem, we want to reconsider, in the same spirit, the original problem. We again suppose this radius r to be of length 1 and we suppose for y that /= 0 for all s. The functional equation then reads as

(7)

, , s

/

< < /

Theorem 3. Assume that (7) holds for all , , s and some -/2 < < /2, where y : ³ is such, that /= 1 for all s. Then there is some constant vector c ³, /= 1, such that where U is given by (4). Moreover any y of the above form solves (7) for all , , s and all -/2 < < /2.

Proof.  Fixing ] - /2, /2[ we put c(s, ) := and u(s, ) := /c(s, ). This means

(8)

where c(s, ) > 0 and u(s, ) is a unit vector. Then (7) and the relation at the beginning of the proof of the first theorem imply

(9)

Using (9) with = 0 gives

(10)

Moreover P (, ) = U()P (0, ) and U( + ) = U()U(). Thus (8) reads as

(11)

We also have, by (8) for = 0

(12)

For we consider the straight line determined by y(0) and P (, ). Note that any may contain at most two unit vectors, one of them being P (, ).

Thus for any fixed real t there is some such that P (, )_t. Then obviously y(0) _t; and it is the only point in that intersection of the two lines and _t, since P (, )_t:

(13)

On the other hand we have by (8) with replaced by - t and s replaced by t

where we also used (9) with replaced by t, s by 0 and by - t. We also have

where we used (10) with s = 0 and = t.

Thus

as desired. (The second part of the theorem is obvious.)     

Remark 2. A good description on the relation between the real and the mean sun may be found in [1], one may also consult [4]. Generalizations of the questions treated here, especially conneted with the results of the first theorem, may be found in [2] and [3].

References