Static Method for Measuring the Acceleration of Gravity

M. Polikarpov, Moscow

Measurements of gravity on the Earth’s surface are of great importance both for solving the fundamental problem of higher geodesy—the determination of the figure of the Earth—and for solving geophysical questions relating to the theory of isostasy, the structure of the upper layers of the Earth’s crust, etc. The majority of determinations of gravity made on the Earth’s surface up to the present time have been carried out by the dynamic method—with the aid of pendulum oscillations. This method has become widely used in determining gravity on land and also, at the suggestion of Vening Meinesz,* on the water surface. Meinesz swung pendulums on submarines in a submerged state. The pendulum method makes it possible to measure gravity with an accuracy of up to several units in the third decimal place, but it is very cumbersome and requires considerable time.

Attempts have repeatedly been made to apply static methods to the measurement of gravity. Among such attempts are the works of Hecker and Briggs.

Hecker*** (in 1901–1908), during a voyage along

* Prey. The theory of isostasy, its development and results, “Advances in Physical Sciences,” 6, issue 1.
* M. Polikarpov, On isostasy and gravity. “Advances in Physical Sciences,” 6, issues 4–5.
** Collection of abstracts and translated articles on geodetic questions. Supplement to parts XIX and XX of the “Notes of the Military-Topographic Department,”

For the Atlantic, Indian, and Pacific Oceans and the Black Sea he applied the method indicated by the Norwegian Mohn, based on determining the atmospheric pressure simultaneously by two methods: with an ordinary mercury barometer and from observations of the boiling point of a liquid—hypsothermometry. The first method gives a result that depends on the magnitude of the force of gravity at the given place; the second does not depend on it; from a comparison of these data the sought magnitude of the force of gravity can be obtained. All of Hecker’s attempts to increase the accuracy of the method, however, were not crowned with success; the accuracy of these measurements of gravity at sea remained considerably below that of the corresponding determinations on land. True, this circumstance did not prevent the well-known geodesist Helmert from drawing, on the basis of Hecker’s observations, the conclusion that the deviations of gravity over the oceans from the normal values measured on the continents are insignificant. Later attempts to improve Hecker’s method did not yield a significant increase in accuracy, but they did considerably complicate the observations themselves. This is fully confirmed by Duffield’s recently published experiments.*

Briggs, following Maskar’s idea,* proposed measuring changes in gravity by the height of a column of mercury balancing the pressure of one and the same mass of gas at constant temperature. To carry out his idea, Maskar constructed an instrument of the nature of a siphon barometer, the short closed limb of which contained carbon dioxide under a pressure sufficient to balance a column of mercury of 1 m. The chief difficulties, in Maskar’s opinion, lay beyond the possibility of accurately reading the level of the mercury and in determining the temperature. Taking into account all the difficulties noted by Maskar, Briggs accordingly modified the instrument, completely isolating it from the air, and used, to maintain a constant tempera-

* W. G. Duffield, Monthly Notic. R. Astr. Soc. Geophys. Suppl., Vol. I, No. 5, 1924.
* Proc. of the Nat. Ac. of Sc. U. S. A., Vol. 2, No. 7, 1916.
** C. R. 95, 1330.

STATIC METHOD OF MEASUREMENTS

…temperature of melting ice. The gas enclosed in the instrument was automatically brought to a constant volume, and since the temperature in this case is constant, the measurements were made at constant pressure. The measurement itself was reduced to noting the position of the upper end of the mercury column. Observations with this instrument were carried out on board ship on the voyage from Sydney to Australia to San Francisco via Wellington in New Zealand, and a second time on the voyage from New York to San Francisco via Panama. Comparing the results obtained by Briggs with Hecker’s measurements (corrected for the effect of the ship’s motion, the so-called “Eötvös effect”), one must give preference to the first method.

In the very recent period the German geophysicist Haalck modified and considerably improved Briggs’s method and developed a fully rational means of measuring the force of gravity. Even the preliminary experiments with Haalck’s instrument gave such promising results that one may hope that in the very near future the difficult problem of the static* determination of the force of gravity will receive its full and precise solution.

Haalck’s idea consists in the following.

If a mercury column of height $h$ is balanced by the pressure of a certain enclosed mass of gas occupying a volume $v$, then

\[ p = h\sigma g, \]

where $g$ is the acceleration of gravity, $\sigma$ is the density of mercury. The differential changes are written in the following form:

\[ \frac{dp}{p}=-\frac{dv}{v}+\alpha dt=\frac{dh}{h}+\frac{d\sigma}{\sigma}+\frac{dg}{g}, \tag{1} \]

where $\alpha$ is the coefficient of expansion of the gas, and $dt$ is the change of temperature.

It follows from this equation that, in order to determine the change in the force of gravity with an accuracy of up to 1 milligal,

\[ \text{* H. Haalck, Ein statischer Schwerkraftmesser, Z. Geophys. VII, Heft 1/2, 1931.} \]

\((0.001\ \text{cm}/\text{sec}^{2})\), it is necessary to measure the quantity \(\dfrac{dh}{h}\) with an accuracy up to \(10^{-6}\).

In order to achieve this, one may widen the vessel containing the mercury at one or both ends and thereby create as large a mercury surface \(F\) as possible (Fig. 1). Onto the mercury at both ends a light liquid is poured, enclosed in a capillary of cross-section \(q\), so that a change \(dh\) in the height of the mercury levels causes a displacement of the liquid meniscus by \(dh\dfrac{F}{q}\).

Fig. 1.

This method of obtaining the necessary magnifications was used in 1876 by Simpson in his experiments with the so-called botometer. The solution of the question of the sensitivity of the instrument does not present great difficulties, since its sensitivity depends only on the dimensions of the instrument and can attain any magnitude.

Considering the influence of temperature changes, on the basis of equation (1) we come to the conclusion that a temperature change of

\[ dt=\left(0.000293+293\,\frac{dv}{v}\right)^{\circ}\mathrm{C} \]

(approximately of the order of \(0.001^{\circ}\mathrm{C}\)) causes the same displacement of the meniscus as a change in the force of gravity by 1 milligal. This circumstance is the chief reason for the difficulty of the problem; at first glance it may seem that it is technically insurmountable, since achieving such constancy of temperature is impossible (because of the difficulty of measuring temperature with the required accuracy and the complexity of constructing the corresponding temperature compensators). The difficulties are further increased by the fact that not only temperature changes, but also temperature inequalities within the volume of mercury, from which it is very difficult to free oneself, have a great influence, and they cause

extremely nonuniform displacements of the meniscus of the liquid; especially important is the fact that it is impossible to measure, with the required accuracy, the actual temperature of the gas \(v\). Thus the question of applying the barometric principle to the static measurement of the force of gravity is chiefly a thermal problem.

Theoretical considerations lead to the conclusion that this problem can be solved by the following method of compensation: the volume \(v\) must be made as large as possible and must be divided into as great a number as possible of separate parts; the same applies also to the volume \(v_1\), with the parts of the volumes \(v\) and \(v_1\) alternating with one another. This is achieved by making the vessels \(v\) and \(v_1\) in the form of coils, the individual turns of which alternate with one another; at the same time the materials and dimensions of the vessels must be chosen so that the temperature effects compensate one another.

Fig. 2.

This temperature compensation will take place between the alternating individual parts of the spaces \(v\) and \(v_1\). Hence the following advantages result: first, the disturbances caused by temperature inequalities will be less noticeable, since the temperatures of the individual parts of the spaces will quickly equalize and thermal equilibrium will be better maintained; second, the errors that nevertheless occur must be referred to the total volume. Then only the mean value of all the effects of the individual parts enters into the measurements, and the reliability of the temperature compensation is the greater, the larger the number of separate elements taken. Thus the reliability of temperature compensation increases with an increase in the volume of the instrument.

In Fig. 2 is given the general appearance of the first of the constructed instruments with which the practical measurements were carried out; its height is about 160 cm, total weight 40–45 kg.

The basic equation for the calculation will be written as follows:

\[ dg=C_1\,(dx'-dx)+C_2dt, \tag{2} \]

where \(dx'\) is the displacement of the lower meniscus, and \(dx\) is the displacement of the upper meniscus; \(C_1\) and \(C_2\) are constants determined by the dimensions of the instrument. The constant \(C_1\) can be calculated from the dimensions of the instrument and was found to be equal to

\[ C_1=0{,}0175\ \text{to}\ 0{,}0187. \]

But it can also be obtained simply by an experimental method from observations of the displacement of the menisci caused by a definite inclination of the instrument, at constant temperature:

\[ dg=g(\cos\delta-1)=C_1\,(dx'-dx), \tag{3} \]

where \(\delta\) is the angle by which the instrument is deflected from the vertical.

The inclination of the instrument is produced by the setting screws (100 turns of the screw correspond to \(82'\) of angle of inclination).

From the equations written above, for sufficiently small inclinations it follows that:

\[ C_1=\frac{\delta^2}{2(dx'-dx)}. \tag{3a} \]

The calculation of \(C_1\) must be carried out in the following way. Let us denote by \(\delta_1\) the inclination in one direction, and by \(\delta_2\) that in the other; let the corresponding differences be \((dx'-dx)_1\) and \((dx'-dx)_2\). Then from equation (3a):

\[ \delta_1=\delta_2\sqrt{\frac{(dx'-dx)_1}{(dx'-dx)_2}}, \]

moreover \(\delta_1+\delta_2=164\).

In one of the experiments, for the given values of \(\delta_1\) and \(\delta_2\)

the following values of \(C_1\) were obtained from equation (3a):

Mean \(C_1=0,01835\pm0,00008\).

Thus, the experimental determination of the value of one division of the instrument scale agrees well with the value calculated from the dimensions of the apparatus. A displacement of the vernier by \(1\ \mathrm{mm}\) corresponds to a change in the force of gravity by \(1,83\) milligals.

The greatest unreliability in the measurement is introduced by the temperature term \(C_2 dx\). If no temperature compensation is introduced at all, then \(C_2\) is approximately equal to 3–4, i.e. a change in temperature by \(1^\circ\mathrm{C}\) corresponds to a change in the force of gravity by 3000–4000 milligals; complete compensation should make \(C_2=0\). It is very difficult to introduce complete temperature compensation or entirely avoid temperature inequalities in the instrument. However, the following series of experiments gives very encouraging results.

The instrument was installed in the large hall of the Geodetic Institute (in Potsdam) without any special temperature protection and remained there for a whole week, during which continuous readings were taken. The curves obtained (Fig. 3) give the course of the instrument readings (in milligals) as a function of the temperature of the surrounding air (in the case); of course, these temperatures do not coincide with the true temperature inside the instrument. The difference between the night and day temperatures was rather considerable; in the morning hours a sharp change in temperature occurred, as is seen from the curve. The first measurements on August 25 were omitted, since the instrument had only just been installed; the measurements on August 26 at about 18 hours were also omitted, since the instrument was opened from above. Subsequent measurements showed that the readings depend on temperature to a very slight degree; the deviations from the mean values are small, especially if one takes into account that without temperature compensation a change in temperature by \(1^\circ\) corresponds to a change in gravity of 3000–4000 milligals. In the instrument under investigation the magnitude of the oscillations is nevertheless rather large and is equal to 30–40 milligals, but it should be noted that, with increasing temperature, the reading decreases, whereas in the absence of temperature compensation the opposite phenomenon should have been observed; consequently, at the present time the instrument is already overcompensated.

Fig. 3.

The factor \(C_2\) is determined experimentally, and the term \(C_2 dt\) is a correction term. The introduction of this correction encounters difficulties, since with every change of temperature inequalities arise, remaining outside

control, and we therefore do not know what temperature to introduce into the calculations, and what the true temperature is of the separate parts of the volumes \(v_1\) and \(v_2\). It would be possible to reduce the temperature term \(C_2\,dt\) by placing the instrument in a space in which a constant temperature was maintained by means of a thermostat allowing fluctuations within the limits of \(0.25—0.1^\circ\).

Next, a series of experiments was carried out in order to determine the effect of transporting the instrument from place to place. The instrument was moved and carried up and down the institute stairways. At the same time, as a result of shaking, bubbles formed in the capillaries, which were eliminated by means of a special device. In these experiments with transport, the equality of the readings rarely changed by more than 10 milligals. An example of such a series of measurements with transport, after which the instrument was again set up at the same point, is given in the following table (experiment of September 29, 1930).

Mean \(+4\)

The maximum deviation reached 9 milligals; the mean error of an individual observation in this series was \(\pm 5\) milligals. Only if there was a bubble at the ends of the capillary during the reading was too small a value obtained. This inaccuracy of the reading may be explained partly by the influence of temperature, since there was no protective device of any kind against the influence of temperature, and partly

wetting of the glass walls, since when the liquid drained off, part of it remained on the walls. Shaking did not make the reading worse; on the contrary, it helped the accuracy of reading the meniscus. Each measurement took at most 2–5 min, and the readings had to be multiplied by 0.0183, so that the result was obtained immediately and even a non-specialist could work with the instrument.

The influence of tilts is very small, since the angle of inclination of the instrument enters into the calculations in the form of cos. An inclination of the apparatus by \(5'\) from the vertical gives an error of 1 milligal. Tilts greater than \(20—25^\circ\) should be avoided, since then the liquid may flow through the capillary into the space \(v_1\); still stronger tilts may cause destruction of the instrument itself.

The general results of these experiments indicate the possibility of determining the force of gravity with an accuracy up to 10 milligals. Although the instrument is still completely unsuitable for practical measurements in the field, considerable changes in gravity can already be measured with it. Measurements were made of the difference in gravity between the upper and lower stops of the elevator in the radiotelegraph tower.

The difference in height was 120 m, so that the gravity at the top was 37 milligals less than below. In order to get rid of disturbances caused by traffic, the first experiments were carried out at midnight. The observing conditions at this time of year were quite unfavorable, since the main condition for good operation of the instrument—constancy of temperature—was not fulfilled. The experiments were conducted on October 2 (Berger and Jung participated in them) at the following temperatures:

In the elevator the temperature during the measurements rose, owing to the presence of a large number of people, to \(16^\circ\) C and then again fell to \(14^\circ,5\) C. Of course, under such conditions

one could not expect great accuracy of the measurements. In Fig. 4 the graphical results of the measurements are presented: the observation time \(T\) is plotted along the abscissa axis, the readings \(x' - x\) along the ordinates; the readings below are denoted by points, those above by circles.

The measurements showed that the force of gravity at the bottom of the tower is considerably greater than at the top. The magnitude of this difference fluctuated rather strongly owing to considerable temperature changes, which followed an extremely irregular course. At 2 h. 10 min. a new adjustment of the instrument was made, since the meniscus had gone out of the field of view. During the subsequent measurements the elevator door was open, and the influence of the penetrating cold air affected the measurements. The oscillations experienced by the tower did not affect the accuracy of the observation. Only the measurements from 1 h. 15 min. to 3 h. 10 min. were taken into consideration, when the temperature effects were small; the following results were obtained:

The actual difference must be 37 milligals. Taking into account the unfavorable temperature conditions, the results should be considered quite successful.

The experiments carried out indicate the desirability of further development of static instruments. Here it should be noted that the dimensions of the first instrument were chosen so that changes in the force of gravity of 1 milligal could be read. The existing sources of error, due to uneven wetting of the capillary walls and to temperature changes, can be considerably reduced if the dimensions of the instrument itself are increased. Therefore the construction has now been begun of an instrument, technically more advanced, in which the following improvements have been introduced.

Fig. 4.

The sensitivity has been increased 13-fold (1 milligal corresponds to a displacement of the meniscus of 6 mm). At the same time the source of errors caused by uneven adhesion of the liquid to the capillary walls during the reverse return of the meniscus will be weakened. It is also important that, with an increase in the dimensions of the instrument, a more thorough temperature compensation will be achieved, since the number of separate alternating volumes will increase and their arrangement will become more symmetrical. In addition, there will be a device for precise adjustment of the temperature compensation by empirical means, and the limits within which measurements can be made will be extended to 1000 milligals. The dimensions of the instrument under construction are as follows: height 1.8 m, diameter 60 cm, weight 100–150 kg. A further increase in the reliability of the measurements will be achieved by surrounding the apparatus first with a heat absorber, then with a space,

equipped with an automatic thermoregulator maintaining a constant temperature. Small deviations of the temperature from the mean value will be equalized by the heat absorber—water. For practical measurements in the field, the instrument is to be mounted in an automobile on a gimbal suspension; when measurements are made on a ship or aircraft, a device must be introduced for damping oscillations caused by vertical accelerations.

On the basis of the results of measurements carried out thus far, with the introduction of a number of technical improvements into the apparatus, one may hope that in the very near future the problem of the static measurement of the force of gravity will be solved, i.e., that it will be possible to attain a measurement accuracy of no less than 1 milligal.