The spin rate of the earth varies constantly.
Daily changes are associated with atmospheric winds; long-term changes
are related to lunar and solar tidal friction and other slowly changing
geophysical parameters. The changed in the earth's spin rate reported
here have occurred over periods measured in centuries and are based on
observations in historical astronomical texts. The (negative)
secular spin acceleration was -19.8 parts per billion per century
around the year 600 AD and is now -8.6 parts per billion. These changes
in spin rate are due to contributions from tidal friction and from an
effect proportional to the square of the time-varying magnetic dipole
of the earth. When these contributions aresubtracted from the observed acceleration, a residual contribution of +41 parts per billion.
One face of the earth is closer to the moon than its
center is, so the moon's gravitation tends to pull that face away from
the center. Similarly, the center is closer than the opposite face, so
the moon's gravitation tends to pull the center away from the opposite
face. As a result, the earth tends to take on an ellipsoidal shape
However, the earth does not always present the same
face to the moon. Instead, it rotates with respect to the earth-moon
line one time in a lunar day, which is about 25 hours. As the earth
tries to maintain the ellipsoidal figure that is demanded by the moon's
gravitation, each point in it goes up and down twice in a lunar day,
giving rise to the lunar tide. There are two tides in a lunar day.
If the tidal motion took place without friction, the
tidal bulges would be directly under and directly opposed to the moon
(Fig. 1). In that configuration, the gravitational force would point
directly along the earth-moon line, and there would be no effect on the
rate of the earth's rotation.
In the actual case, there is friction in the tidal
motion, with the result shown in Fig. 2. The motion lags the
tide-raising force, so that the bulges are displaced in the direction
of rotation. The gravitational force is no longer along the earth-moon
line, passing through the center of the earth, and the gravitation
exerts a torque on the earth. 1 think the reader can see from the
figure that the torque is in the direction opposed to the earth's
In other words, friction in the lunar tide tends to retard the earth's spin.
Figure 1 — The tide-raising force. Because the moon's gravitation varies with distance, it tends
to pull the near side of the earth away from the center and to pull the center away from the far side.
Figure 2 — Tidal friction. As the earth moves up and down in response to the tide raising force.
As a result, a torque tends to slow up the earth's rotation.
Of course, there is an equal but opposite torque
acting on the moon. The direction of this torque is such that it
increases the angular momentum of the moon. However, in order to
increase its angular momentum, the moon must move into a larger orbit,
in which it has a smaller angular velocity. Therefore, tidal friction
decreases both the spin angular velocity of the earth and the orbital
angular velocity of the moon.
There is also a solar tide and friction that affects
the spin of the earth and its orbital motion around the sun Although
the sun's gravitational effect on the earth is much larger than the
moon's, the solar distance is so much greater that the solar tide is
actually less than the lunar tide. Friction in the solar tide is large
enough that we have to take account of it, but the lunar tide dominates
Tidal friction is increasing the length of the day by about a millisecond in a century.
THE NEED FOR OLD DATA
By modern astronomical data, I mean data obtained
since the introduction into astronomy of the telescope and pendulum
clock, which happened about three centuries ago. By old data, I mean
data obtained without the use of the telescope and pendulum clock.
The modern data are so accurate that we readily
obtain fairly accurate values for the acceleration of the moon and of
the earth's spin. Spencer-Jones [1
] analyzed a large
volume of data and obtained -22 seconds of arc per century per century
for the acceleration of the moon. (I will take the second of arc per
century per century for the standard unit of the moon's acceleration
and will omit the units in the rest of the article.) In a later work
, p. 457), I used some additional data and
obtained -28. We may take it that the acceleration of the moon is
reasonably well known and that the difference between -22 and -28
indicates the accuracy with which it is known.
No source except tidal friction has
been suggested for the acceleration of the moon, so I tentatively take
-28 for the lunar acceleration as a measure of lunar tidal friction. By
using the known astronomical constants, we can then estimate the effect
of lunar tidal friction on the rotation of the earth. From this, in
turn, we can use the relations between the astronomical parameters of
the sun and moon and estimate the effect of solar tidal friction on the
earth's spin acceleration. From these considerations, we find (Ref. 3
, p. 219) that the total contribution of tidal friction to the acceleration of the earth's spin is -32.6 parts in 109
per century if the lunar acceleration is -28.
In the rest of this article, I will use n'M
for the lunar acceleration when it is given in the units stated. I will use ó
for the secular acceleration of the earth's spin when stated in parts in 109
per century. Thus, for the contributions from tidal friction, we have n'M
= -28 and ó
We can also measure ó
independently. The results [4
] are that ó
is an order of magnitude larger than -32.6, that it changes at
irregular intervals that average about 4 years, and that it is about as
likely to be positive as negative. The latter fact alone tells us that
most of the contribution to ó does not come from tidal friction; ó
would have to be negative if that were so.
That fact also tells us that we must average ó
over a long time period if we are to learn anything about tidal
friction. To get an average that is significant to a size of 1, we must
average over about three centuries; this means that the modern data, in
spite of their accuracy, can contribute only one data point to our
study of tidal friction. To learn more about y
, we must have recourse to old data.
Old data are not very accurate, but they do not need
to be in order for us to learn something useful about the
accelerations. The value of an observation depends on the geocentric
angular velocity of the body being observed. The moon has by far the
largest angular velocity of any object in the heavens, so that lunar
observations are by far the most useful. In fact, for simplicity, I
will use only lunar observations in this study.
Old lunar observations give poorly conditioned equations for finding both n'M
so we cannot determine both of the accelerations with a satisfactory
accuracy from the old observations. However, we have a satisfactory
estimate of n'M
from new data, so we will use it in analyzing the old data.
ACCELERATIONS OF THE SUN AND MOON
If the earth's spin is accelerating, the length of
the day is not constant, and the day cannot be taken as the unit of
time. However, the acceleration of the spin has been known only
recently, and all old astronomical observations were made using the day
as the unit of time. In the time base in which the day is the unit, the
spin is exactly one rotation (with respect to the sun) per day, and it
is not accelerating.
In its place we have the acceleration of the sun. If
ó and hence the spin acceleration are negative, the length of the day
is increasing and the number of days in a year is getting smaller. That
is, the sun completes one full revolution around the earth in a smaller
number of days, so that it appears to be speeding up. Thus, when the
day is the unit of length, the sun has a positive acceleration if the
value of ó is negative.
The system of time in which the day is the unit of
time is called solar time. If we adopt a time system in which the sun
has no acceleration, that system is called ephemeris time. As far as we
know now, the planets do not have any acceleration in ephemeris time,
but the moon does. However, its acceleration with respect to ephemeris
time, which we have been calling n'M
is not the same as its acceleration with respect to solar time. In
solar time, it has an additional acceleration, positive just like the
sun's, that comes from the variation in the length of the day.
Thus the old observations were made using solar time
as the time base. This fact tells us how to estimate ó from an old
observation. We calculate the ephemeris time when the moon had the
observed position, using n'M
= -28 in
the calculations, which supplies the value of solar time when it had
the same position. The difference between ephemeris time and solar time
depends on the acceleration ó of the earth's spin, and we calculate
what value of ó leads to the required difference between ephemeris time
and solar time.
When the moon passes between an observer and the
sun, part of the sun is eclipsed. But because the moon is relatively
close to the earth, its direction at any given moment depends on where
the observer is. For some observers, the moon appears to miss the sun
completely, and, for others, the moon may appear directly in front of
the sun, and the sun may be totally eclipsed.
The distance to the moon is not constant but varies
by more than 5 percent from its average value. If the moon is close to
the earth when an eclipse occurs, it appears large enough to cover the
sun completely, and such an eclipse is called total. If the moon is far
from the earth, it is unable to cover the sun totally, and such an
eclipse is called annular because a small ring or annulus of the sun is
left visible at the height of the eclipse. About half of all eclipses
are annular somewhere.
An eclipse does not appear to be the same for all
observers. If an observer sees the sun and moon in a straight line at
the height of the eclipse, he will see either a total or an annular
eclipse. To cover both cases, let us say that he sees a central
eclipse. Because the sun and moon are so nearly the same apparent size,
an eclipse is central only in a rather narrow zone. For an observer
outside this zone, the eclipse is never central and is said to be
partial. For an observer far enough away from the central zone, the
moon misses the sun entirely and there is no eclipse. A particular
eclipse is visible over only a relatively small part of the earth, and
observers in most places will not see any eclipse, even though it may
be central to some observers.
This fact allows us to estimate v by using
statements that an eclipse of known date was seen at a particular
place. To simplify the explanation, let us suppose that a record says
that an eclipse of known date was total at the particular place. When
we calculate the circumstances of the eclipse using n'M
= -28 and ó
= 0, we will find that the eclipse was not total at the specified
place. To make it total, we have to rotate the earth without changing
the ephemeris time of the eclipse until the observer is brought into
the narrow zone within which the eclipse is total.
If the record does not say that the eclipse was
total, we do the same thing. If the eclipse was not actually total, we
will make an error in the resulting estimate of y\ however, the error
will average out if we use enough observations. Thus we can use records
of partial eclipses as if they were central and not make an error if we
have enough records.
A large eclipse, even though not central, is an
impressive sight, and the occurrence of eclipses was frequently
recorded in old annals, chronicles, and histories. Thus much of the
information we use in finding v does not come from astronomy at all but
from simple nontechnical sources. In fact, we have more old historical
records of eclipses than we have old astronomical observations of the
THE IDENTIFICATION GAME
Unfortunately, the method of using historical records of solar eclipses
was misused badly for over a century, during which time it became the
most popular method of finding the astronomical accelerations. Since
much of the resulting literature on the subject is still widely cited,
I want to caution the reader and tell him why the method so often used
In using a record that an eclipse was seen at a
known place, it is obviously crucial to have the date of the eclipse.
Of course, since there are usually only two solar eclipses in a year,
and since they will rarely be visible in the same place, we can
tolerate an uncertainty of a few years in the historical date of a
record and still determine which eclipse could have been seen at the
stated place. That is, we can usually determine the date exactly if the
record can be dated from historical evidence within a few years.
However, it became popular to use references to
eclipses that could not be dated closer than half a century. The date
was then "determined" by means of what I call "the identification
game", and the resulting record was used to estimate y
. The trouble with the procedure is that it was reasoning in a circle.
To play the identification game, the player started
by calculating the magnitudes of all eclipses in the possible time
frame that could have been seen at the place given in the record. The
eclipse with the largest calculated magnitude was taken to be the
An example of an eclipse that was widely used in the
astronomical literature is the so-called "eclipse of Archilochus".
Archilochus was a Greek soldier and a poet whose poems can be dated
only as being between about -710 and -640. Part of one of his poems is
translated in Ref. 5
: "Zeus...made night from midday, hiding the light of the shining sun, and sore fear came upon men."
There are several things wrong with using this
passage of poetry as an eclipse record in the astronomical literature.
For one thing, the passage does not say that the darkness was caused by
an eclipse. There is the phenomenon called a dark day, which probably
happens at a particular place as often as a central eclipse of the sun
does. A dark day is probably caused by weather conditions, and it is
just as impressive as an eclipse. However, let us grant that the
passage does refer to an eclipse and see where it leads us.
Those who want to use the passage as an eclipse
record are forced to assume that Archilochus could have written it only
if he had personally witnessed the eclipse. Since one of the
characteristics of a poet is his imagination, Archilochus could have
imagined the effect of seeing an eclipse if he had ever read or heard
about one. The eclipse Archilochus described does not have to be one
that was seen at all. Nonetheless, let us assume that the passage does
refer to an eclipse that Archilochus actually saw.
The biggest trouble in using an undated passage is that it is necessary to assume a value of ó
to use in calculating the magnitudes used in dating it, and the
identification depends on the value of v used. For example, if we take y
= -19, we conclude that the eclipse is that of -656 April 15; but if we take ó
= -22, we conclude that the eclipse is the one of -647 April 6.
Now, in using the method, we turn around and use the identification in finding y
. Since the identification we make is that of the largest eclipse, the value of ó
we will get from the identified eclipse is always close to the value of
ó we assumed in making the identification. Continuing with the eclipse
of Archilochus, we get ó
= -19.5 if we take the eclipse to be that of -656 April 15, and we get -22.2 if we take the eclipse to be that of -647 April 6.
Actually, those who used the identification game
used it to get the acceleration of the moon with respect to solar time,
but the principle is the same. In order to play the identification
game, it was necessary to assume a value for the acceleration with
respect to solar time; the player then used the identification to find
the acceleration he had assumed in the first place. It is this process
of reasoning in a circle that 1 wish to emphasize and not the specific
variable used. For simplicity, I will write as if the older work were
done in terms of ephemeris time, even though it was done in solar time.
I believe that Sir George Airy, who was the British
astronomer royal from 1835-81, was the first person to use the method
of reasoning in a circle [6
]. From then until 1970, when I pointed out the fallacy involved [8
], this method was the most popular, although not the only, one used to find the accelerations.
It is interesting that the value found for ó was
rather good, even though the method is reasoning in a circle that
cannot yield information. The reason for this is that writers before
Airy, using valid methods and data, had found a fairly accurate value
for the acceleration. The value was assumed in starting the reasoning
in a circle, which, in turn, necessarily yielded a value close to the
one that was used to start the process.
RESULTS FROM SOLAR ECLIPSES
We can now turn to valid results obtained by using solar eclipses. In two recent works [2
I have analyzed 631 records that tell us that an eclipse of the sun was
visible on a known date in a known place. The records fall into three
broad classes. One class says that the sun was totally obscured during
the eclipse. A second class says that stars (including planets in this
context) could be seen during the eclipse but does not say that the
eclipse was total. The third class merely says that the eclipse was
seen but makes no implication about its magnitude.
For eclipses since about 1000, we can calculate the
magnitude of the eclipse with no significant uncertainty. For such
records, I have studied the departure of the magnitude from unity,
which corresponds to a total eclipse. (The magnitude is the ratio of
the part of the solar diameter that is eclipsed to the entire
diameter.) For records that explicitly say that the eclipse was total,
the magnitude is actually less than unity in a large number of cases,
and the deviation from totality is 0.030 on a standard deviation basis.
For the records that say the stars could be seen, the corresponding
number is 0.051, and for the records that say nothing about the
magnitude, the standard deviation is 0.177.
This tells us, among other things, that we may not
use the identification game even to find the date of an eclipse, even
if we do not go on to find the acceleration from that date. The basic
assumption back of the identification game is that the recorded eclipse
was the one with the largest magnitude during the possible time period.
We see now that this is not necessarily so. For a record that says that
the eclipse was so large that stars could be seen, the magnitude can
easily be as small as 0.90. With this much range in the magnitude,
either date will fit the eclipse of Archilochus with either value of y
, and we do not get a unique choice for the date.
In fact, I do not know of any case in which we have
successfully identified an eclipse when the record leaves an
uncertainty of more than a few years on historical grounds alone.
Returning to the records, the dates range from - 719
February 22 to 1567 April 9.1 have divided the records into 16 time
bins, and I have analyzed the records from each time bin separately. In
doing so, I weight each record according to the information it gives
about the magnitude; I take the weight to be inversely proportional to
the square of the standard deviation of the magnitude that was stated
above for each class of record.
The results are shown in Fig. 3, where I give a
value and an error bar for ó as derived from the records in each time
bin. (The year for a plotted point is the average year for the data
used in getting that point.) Note that all the points through the year
1005 agree well with each other except for the point at the year 772.
Since the year 1000, though, ó shows a definite tendency to increase
algebraically with time; that is, to decrease in size.
The results just discussed were obtained from
qualitative records that merely say that a certain solar eclipse was
visible at a known point. Now we turn to quantitative observations in
which some position or phenomenon was measured quantitatively. The old
quantitative observations that I have discovered1 are summarized in
Table 1, where the observations have been grouped by type and, within
each type, by time bins like those used with the solar eclipses.
The first column in the table gives the average date
of the observations within a group, and the second column gives the
type of observation. The third column gives the value of ó inferred
from the observations in a group, and the fourth gives the standard
deviation of the inferred value. Some of the types of observation need
Figure 3 — ó as derived from historical records of solar eclipses. Note the definite tendency of ó to increase algebraically with time.
Table 1 — Quantitative observations.
||Type of Observation
||Moonrise and moonset
||Moonrise and moonset
||Moonrise and moonset
||Times of lunar eclipses
||Times of solar eclipse
||Moonrise and moonset
||Moonrise and moonset
||Times of solar eclipse
||Times of solar eclipse
||Times of solar eclipse
||Mean lunar elongation
||Magnitudes of solar eclipses
||Times of solar eclipses
||Times of lunar eclipses
||Lunar eclipse at moonrise
||Mean lunar elongation
||Time of lunar eclipse
||Magnitude of solar eclipse
||Mean lunar elongation
||Mean lunar elongation
||Measured lunar longitude
||Times of lunar eclipses
||Times of solar eclipses
||Modern solar data
The Babylonian month began at sunset on the first
day after a new moon that the moon could be seen in the western sky
after sunset. The Babylonian astronomers regularly measured the time
interval between moonset and sunset on that day. Similarly, near the
end of the month, they measured the interval between moonrise and
sunrise. Near the full moon, they measured the time intervals of the
four possible permutations of moonrise and moonset with sunrise and
sunset. Altogether, then, they measured an interval between moonrise or
moonset and sunrise or sunset six times each month, weather permitting.
The lengths of these intervals form the groups called "moonrise and
moonset" in Table 1.
Many astronomers recorded the time when the moon
passed a particular star or when it was a given distance from the star.
These measurements are the "lunar conjunctions" in Table I. There is
also one measured value of the lunar longitude in the table.
I believe that the times and magnitudes of eclipses
are obvious. This leaves the "mean lunar elongation" to explain. We
have a number of old tables of the sun and moon from which the values
arc taken. The tables include tables of the mean positions of the sun
and moon, along with tables or formulas for calculating the difference
between the mean position and the actual position at any time. The
tables of the mean positions had to be based on observations. I have
already remarked that only lunar observations are sensitive enough to
the accelerations to be useful in this study, so we omit the tables of
It is clear when we study the methods astronomers
used to construct their tables of the moon that they based them on
measurements of the lunar elongation, that is, the angular distance of
the moon from the sun. Hence, if we subtract the mean position of the
sun from that of the moon, we obtain the mean lunar elongation, which
represents observation. The date I assign to a value of the mean lunar
elongation is the approximate date of the observations used to
construct the tables, not the epoch to which the tables are referred.
Table 1 also contains a line called "modern solar data" that I will come back to.
The values and errors in Table 1 are plotted in Fig.
4, except for the modern solar data. When we compare Fig. 4 with Fig.
3, we see that the points in Fig. 4 have more scatter, in spite of
being based on quantitative observations. There are two reasons for
this. First, old quantitative astronomical observations were not very
accurate. Second, there are not as many of the quantitative
observations. Figure 4 is based on only 221 observations, while 631
observations were used in drawing Fig. 3.
Figure 4 — ó as derived from quantitative observations that involve the moon. The points show the same tendency
to change with time as those in Fig. 3, but the tendency is not as obvious to the eye.
In spite of the generally larger scatter, the error
bars are smaller in Fig. 4 than in Fig. 3 at two stages in history. One
is in the -4th century at the height of Babylonian astronomy. The
other is in the 10th century at the height of Islamic astronomy. At
those two stages in history, we have standard deviations in ó that are less than 1.
Also in spite of the larger scatter, we can see the same tendency in Fig. 4 as in Fig. 3. That is, ó
shows a definite tendency to increase algebraically with time.
Furthermore, the values of v from the two figures show excellent
In drawing Fig. 4, I have represented the types of
observation that have the same date in Table 1 by a single point. Thus,
for example, I have represented both the moonrise and moonset value and
the value from lunar conjunctions by the single value -16.0 ±
4.3 for the year -567. I have also represented the four consecutive
values with dates from 1221 through 1336 by a single value because of
the large standard deviation that each of the individual values has.
We are now ready to combine the results from Figs. 3
and 4 to obtain a single set of results from both the qualitative and
quantitative observations. In doing so, I have put all the observations
of both classes into 19 time bins, with dates ranging from - 660 to
1479. For all the observations in a single time bin, I have derived a
single estimate of ó and an associated standard deviation. The results
are plotted in Fig. 5.
In addition, Fig. 5 contains a point at the year
1790. This is from the line for modern solar data in Table 1 and is the
value of ó derived from modern observations of the sun made with the telescope and pendulum clock. It is
ó = -9.1
Note that we do not know the value of ó as accurately for modern times as we know it for the -4th century from Table I.
Figure 5 — ó as derived from all old observations involving the moon, except for the point at the year 1800.
That point is derived from modern observations of the sun. The curve is the best fitting quadratic.
The curve drawn in Fig. 5 is the quadratic function
of time that best fits the data. In deriving the best-fitting function,
it is desirable to take the origin of time to be at about the center
point of the data. I take this to be the year 600. If we let Ñ be the
time measured in centuries from the year 600, the best-fitting
ó = -19.86
± 0.83 + (0.487 ± 0.102) Ñ + (0.0229 ±
In the year 600, when Ñ = 0, we know ó with
an uncertainty of less than 1. It is hard to estimate the uncertainty
in other years because the uncertainties in the individual coefficients
are not independent. However, we see from Table 1 that the uncertainty
is less than 1 in the -4th century and in the 10th century, as I have
already commented. I think it is fair to say that Eq. 2 represents ó
with an uncertainty of less than 1 from -600 to 1200.
The estimate of the linear coefficient in Eq. 2 is
almost five times its standard deviation, so it is highly significant.
There can be little question that ó has changed by a large amount within historic times. The estimate of the quadratic coefficient is only about 11/2 times its standard deviation, which is not highly significant. That is, a linear variation of ó
with time fits the data almost as well as a quadratic variation.
Nonetheless, there is an independent reason for suspecting a quadratic
term, which I will take up in discussing the sources of y. Thus it is probable that the quadratic term is genuine.
The value of y from Eq. 2 has an extremum
near the year -460, when its value was about -22.4. Its value for the
year 2000 (C = 14) is about -8.6. Thus y has changed by a factor of almost three during historic times.
SOURCES OF THE ACCELERATION
Our first problem in looking for sources of the acceleration y
is to find a source that can vary significantly with time during a
period as short as historic time. All indications are that the oceans
have stayed nearly constant during that time. To be sure, there have
been slight changes in sea level and in the amount of ice that is
interacting with the oceans, but the changes could hardly have changed
tidal friction by a factor of three. Thus there is almost surely an
important source of y
other than tidal friction, even after we smooth out the violent
fluctuations that were described near the beginning of this article.
The only important geophysical property of the earth
that has changed by an important amount during historic time seems to
be its magnetic moment.
Smith  gives estimates of the magnetic moment from
about - 1050 to about 1600, and his estimates vary by a factor of two
or three during that time.
When we have spoken of the earth's spin acceleration
up to this point, we have tacitly meant the angular acceleration of the
crust, where the observers lived This angular acceleration is not
necessarily proportional to the time derivative of the earth's total
angular momentum, and the angular velocity of the mantle plus crust may
not be changing in the same way as the angular velocity of the core,
where most of the magnetic field originates.
In other words, the core and mantle may be
exchanging angular momentum through the agency of the magnetic field.
If they are, the exchange should take place mainly through induced
effects, so it should be proportional to the square of the magnetic
dipole moment. Accordingly, in Section X.5 of Ref. 3, I squared Smith's
values of the dipole moment M and passed a smooth curve through the
values. Finally, I fitted the values of y from Fig. 5 to a function of the form a + b M2 . The result is
y = 8.9 - 0.2268 M 2
The term -0.2268 M2 is the magnetic contribution to y while the constant 8.9 is the total contribution from all other effects.
Equation 3 gives a very good fit to the data. Further, both Eqs. 2 and 3 require y to have an extremum at about the same lime. This is a stronger reason for taking y to be a quadratic function of time than the statistical significance of the quadratic term in Eq. 2.
I calculated near the beginning of this article that the contribution of tidal friction to y,
in both the lunar and the solar tides, is -32.6. When we compare this
to the constant term in Eq. 3, we see that the total contribution of
all sources other than tidal and magnetic must be +41.5, which is
larger in magnitude than the contribution of tidal friction.
Many writers have suggested contributions to y
other than the tidal and the magnetic, but only three show promise of
contributing significantly to a value as large as 41.5. The first is a
change in the size of the earth's core. As the core grows, it means
that dense material migrates from the mantle, where it has a large
radius of gyration, to the core, where it has a small radius of
gyration. This decreases the earth's moment of inertia and thus
increases its angular velocity.
The second is a change in the amount of glaciation.
Most glaciers are found at high latitudes, where they have a small
radius of gyration. If a glacier melts, in whole or in part, its water
runs into the sea and increases sea level all over the earth. This
increases the radius of gyration of the earth and decreases its angular
The third change is cosmological in origin. It seems
to be well established that the universe is expanding, which lowers the
average density of matter in the universe. According to some theories
of gravitation, this causes a change in the constant of gravitation,
which, in turn, causes a change in the length of the year without
changing the length of the day. Thus the earth's spin velocity would
appear to change if the year is the unit of time.
We do not have the basic information that is needed
to calculate the contributions that these changes may make to y. All we can say is that the contributions may each be of the order of 10 and that each may be positive.
We have analyzed 852 observations that involve the
moon, with dates ranging from -719 to 1567. We have also analyzed
observations of the sun made with the telescope and pendulum clock in
modern times. As a result, we have been able to find y, the acceleration of the earth's spin, as a function of time over the past 2700 years.
The result is given in Eq. 2. ó has varied quadratically with time, having an extremum about the year -460. Its value at that time was about - 22.4 parts in 109 per century, and its value at the present time is about -8.6 parts in 109 per century.
ó contains one important contribution that is
produced by, or is at least correlated with, the earth's magnetic
field. This contribution accounts for the time dependence of y. The remaining contributions are essentially constant and amount altogether to 8.9 parts in 109 per century. One contribution is tidal friction, which amounts to -32.6 parts in 109 per century.
This leaves +41.5 as the contribution to ó from all
other sources. At the present time, we do not have the information
needed to tell us where that contribution comes from. It probably
arises from some unknown mixture of changes in the size of the core, in
the amount of glaciation, and in the size of the gravitational constant.
Exciting though it would be, I do not believe that
we can contribute to the question of a change in the gravitational
constant by studying the earth's spin. There are too many uncertainties
in the other sources of the earth's spin acceleration. The question of
a changing gravitational constant will probably be settled by the laser
ranging of the moon, which has been going on for about a decade. The
precision of the data is such that we can probably settle the question
in another few decades.
REFERENCES and NOTE
1. Spencer-Jones, "The Rotation of the Earth, and the Secular
Accelerations of the Sun, Moon, and Planets," Mon. Not. R. Astron. Soc.
99, 541-558 (1939).
2. R. R. Newton, The Moon's Acceleration and Its Physical Origins. Vol.
1, The Johns Hopkins University Press, Baltimore and London
3. R. R. Newton, The Moon's Acceleration and Its Physical Origins, Vol.
2, The Johns Hopkins University Press. Baltimore and London
4. W. Markowitz, "Sudden Changes in Rotational Acceleration of the
Earth and Secular Motion of the Pole, "in Earthquake Displacement
Fields and Rotation of the Earth, L. Mansinha, D. E. Smylie, and A. E.
Beck, eds., D. Reidel, Dordrecht, Holland, pp. 69-81 (1970).
5. J. K. Fotheringham, "A Solution of Ancient Eclipses of the Sun," Mon. Not. R. Astron. Soc. 81, 104-126 (1920).
6. G. Airy, "On the Eclipses of Agathocles, Thales, and Xerxes," Philos. Trans. R. Soc., London. Ser. A 143, 179-200 (1853).
7 G. Airy, "On the Eclipse of Agathocles, the Eclipse at Larissa, and
the Eclipse of Thales, with an Appendix on the Eclipse at Stiklastad,"
Mem. R. Astron. Soc. 26, 131-152 (1858).
8. R. R. Newton, Ancient Astronomical Observations and the
Accelerations of the Earth and Moon, The Johns Hopkins University
Press, Baltimore and London (1970).
9. P. J. Smith, "The Intensity of the Ancient Geomagnetic Field: A
Review and Analysis," Geophys. J. R. Astron. Soc. 12, 321-362 (1967).
l0. There is one complication. The values of M2 are instantaneous values, while the values of ó are average values of the type I call time-squared averages. By M2, I actually mean the time-squared average of the square of the dipole moment (see Ref. 3 for details).
ROBERT R. NEWTON is a research physicist who has spent most of his
career in fundamental studies of the mechanics of flight of
missiles, earth satellites, and spacecraft, and of the motions of the
earth, moon, and planets. He was born in Tennessee and earned his B.S.
in electrical engineering and M.S. in physics at the University of
Tennessee. During World War II, he carried out pioneering studies on f
the exterior ballistics of rockets and coauthored an authoritative book
on the subject. After receiving his Ph.D. in physics from Ohio
State University (1946), Dr. Newton joined the Bell Telephone
Laboratories, but soon returned to academia as professor of physics,
first at the University of Tennessee (1948-55) and then at Tulane
University (1955-57). During that period, he continued his research in
Dr. Newton joined APL in 1957, in time to contribute
to the Laboratory's space program from its inception. In 1959, when the
Space Department was formed, Newton became supervisor of the Space
Research and Analysis Group (later Branch) and served in that capacity
until 1983. He played a vital leadership role, both technical and
administrative, in the early days of the Department. He personally led
the most difficult theoretical tasks of precisely determining the
orbits of earth satellites from Doppler measurements and, from their
orbits, determining the geographic variation of the gravitational fine
structure of the earth's gravitational field. His skill in difficult
analysis and his insistence on the highest standards of rigor and
accuracy so improved our knowledge of the earth's gravitational field
and of other, time-dependent, forces acting on earth satellites that it
soon became possible to predict satellite orbits with high accuracy, an
essential requirement for the highly successful Transit satellite
navigation system. This pioneering work in both analysis and
computation is documented by over 50 publications by Newton and his
collaborators, most notably W. H. Guier and S. M. Yionoulis, in the
decade 1958-67. In addition to providing the basis for accurate
analysis and prediction of satellite orbits, the work was an
outstanding contribution to geodesy, improving the knowledge of the
shape of the earth — the geoid or equipotential surface — by orders of magnitude.
After the theory and methodology for solving the
difficult technical problems of satellite flight had been basically
established, Newton (although continuing to direct and supervise
improved analysis and computations) turned his personal research
attention to basic unsolved problems in geophysics. He first used
satellite data to determine the parameters of the earth's crustal
tides; this, in turn, led to a study of the secular accelerations of
the earth and moon. He found that, for the study of long-term
variations in the rotation of the earth and of the orbital motions of
the moon and the planets, it was advantageous to use ancient
astronomical records, the longer time base more than compensating for
the lower precision and accuracy in the ancient observations. So, in
the past 18 years, Newton has become a scholar of ancient astronomy
and has pioneered in the application of old data in determining the
variation in the motions of the earth, the moon, and the other planets
over millennia. This work is documented in eight books and numerous
shorter publications, culminating in the two-volume work, The Moon's
Acceleration and Its Physical Origins (1979; 1984).
In the course of these studies, Newton became a
superb scientific detective, analyzing both internal and external
evidence to determine the probable reliability and accuracy of ancient
observations. He discovered numerous errors and discrepancies in both
observations and analysis and, most notably, was forced to the
conclusion that Claudius Ptolemy had fabricated all the data he claimed
to have measured himself and much of the data he attributed to others.
That shattering conclusion—for Ptolemy was the most distinguished
name in astronomy prior to Copernicus and his work had been thought to
be both the summary and epitome of Greek science — was thoroughly
documented in The Crime of Claudius Ptolemy (1977), Newton's best
known and most controversial work. The smashing of an idol was not
readily accepted by many historians of science, but the rigor and logic
of Newton's analysis are prevailing. (Since this has been Newton's best
selling book, one cannot help wondering how many purchasers thought
they were acquiring an ancient Egyptian whodunit!)
Newton was a frequent contributor of articles to the
Johns Hopkins APL Technical Digest and served on the Editorial Board
for over two years (1982-84). We shall miss his contributions and his
Dr. Newton stepped down from his management position
in 1983 and retired from the Laboratory at the end of 1984 after seeing
through the press his most recent publication, The Origins of Ptolemy's
Astronomical Tables (1985). We salute his long and exceptionally
distinguished career, which has brought great credit to both himself
and the Laboratory.