The mass of the moon relative to the mass of the earth. General characteristics of the Moon

Story moon mass estimates is hundreds of years old. A retrospective of this process is presented in an article by foreign author David W. Hughes. The translation of this article was made to the extent of my modest knowledge of English and is presented below. Newton estimated the mass of the moon at twice the value now accepted as plausible. Everyone has their own truth, but there is only one truth. point in this question we could put the Americans with a pendulum on the surface of the moon. They were there ;) . The same could be done by telemetry operators on the orbital characteristics of the LRO and other ISLs. It is a pity that this information is not yet available.

Observatory

Measuring the Mass of the Moon

Review for the 125th anniversary of the Observatory

David W. Hughes

Department of Physics and Astronomy, University of Sheffield

The first estimate of the lunar mass was made by Isaac Newton. The meaning of this quantity (mass), as well as the density of the Moon, have been the subject of discussion ever since.

Introduction

Weight is one of the most inconvenient quantities to measure in an astronomical context. We usually measure the force of an unknown mass on a known mass, or vice versa. In the history of astronomy, there was no concept of "masses", say, the Moon, the Earth, and the Sun (MM M , M E , M C) until time Isaac Newton(1642 - 1727). After Newton, fairly accurate mass ratios were established. So, for example, in the first edition of the Beginnings (1687), the ratio M C / M E \u003d 28700 is given, which then increases to M C / M E \u003d 227512 and M C / M E \u003d 169282 in the second (1713) and third (1726) publications, respectively, in connection with the refinement of the astronomical unit. These relationships highlighted the fact that the Sun was more important than the Earth and provided significant support for the heliocentric hypothesis. Copernicus.

Data on the density (mass/volume) of a body helps to estimate its chemical composition. The Greeks more than 2200 years ago obtained fairly accurate values ​​for the sizes and volumes of the Earth and the Moon, but the masses were unknown, and the densities could not be calculated. Thus, even though the Moon looked like a sphere of stone, it could not be scientifically confirmed. In addition, the first scientific steps towards elucidating the origin of the moon could not be taken.

By far the best method for determining the mass of a planet today, in the space age, relies on the third (harmonic) Kepler's law. If the satellite has a mass m, revolves around the Moon with mass M M , then

where a is the time-averaged average distance between M M and m, G is Newton's constant of gravity, and P is the period of the orbit. Since M M >> m, this equation gives the value of M M directly.

If an astronaut can measure the acceleration of gravity, G M, on the lunar surface, then

where R M is the lunar radius, a parameter that has been measured with reasonable accuracy since Aristarchus of Samos, about 2290 years ago.

Isaac Newton 1 did not measure the mass of the Moon directly, but attempted to estimate the relationship between the solar and lunar masses using sea tide measurements. Even though many people before Newton assumed that tides were related to the position and influence of the moon, Newton was the first to look at the subject in terms of gravity. He realized that the tidal force created by a body of mass M at a distance d proportional M/d 3 . If this body has diameter D and density ρ , this force is proportional to ρ D 3 / d 3 . And if the angular size of the body, α , small, tidal force is proportional to ρα 3. So the tide-forming power of the Sun is slightly less than half of the lunar one.

Complications arose because the highest tide was recorded when the Sun was actually 18.5° from the syzygy, and also because the lunar orbit does not lie in the plane of the ecliptic and has an eccentricity. Taking all this into account, Newton, on the basis of his observations, that “Up to the mouth of the River Avon, three miles below Bristol, the height of the rise of water in the spring and autumn syzygies of the luminaries (according to the observations of Samuel Sturmy) is about 45 feet, but in quadratures only 25 ”, concluded, “that the density of the substance of the Moon to the density of the substance of the Earth is related as 4891 to 4000, or as 11 to 9. Therefore, the substance of the Moon is denser and more earthy than the Earth itself”, and “the mass of the substance of the Moon will be in the mass of the substance of the Earth as 1 in 39.788” (Beginnings, Book 3, Proposition 37, Problem 18).

Since the current value for the ratio between the mass of the Earth and the mass of the Moon is given as M E / M M = 81.300588, it is clear that something went wrong with Newton. In addition, a value of 3.0 is somewhat more realistic than 9/5 for the syzygy height ratio? and quadrature tide. Also Newton's inaccurate value for the mass of the Sun was a major problem. Note that Newton had very little statistical precision, and his quoting of five significant figures in M ​​E /MM M is completely unsound.

Pierre-Simon Laplace(1749 - 1827) devoted considerable time to the analysis of tidal heights (especially in Brest), concentrating on the tides in the four main phases of the moon at both solstices and equinoxes. Laplace 2, using a short series of observations from the 18th century, obtained an M E /MM M value of 59. By 1797, he corrected this value to 58.7. Using an extended set of tidal data in 1825, Laplace 3 obtained M E /M M = 75.

Laplace realized that the tidal approach was one of many ways to figure out the lunar mass. The fact that the Earth's rotation complicates the tidal models, and that the end product of the calculation was the Moon/Sun mass ratio, obviously bothered him. Therefore, he compared his tidal force with the results of measurements obtained by other methods. Laplace 4 writes down the M E /MM M coefficients as 69.2 (using d'Alembert coefficients), 71.0 (using Bradley's Maskeline analysis of nutation and parallax observations), and 74.2 (using Burg's work on the lunar parallax inequality). Laplace apparently considered each result equally credible and simply averaged the four values ​​to arrive at an average. “La valeur le plus vraisembable de la masse de la lune, qui me parait resulted des divers phenomenes 1/68.5” (ref 4, p. 160). The average ratio M E /M M equal to 68.5 is repeatedly found in Laplace 5 .

It is quite understandable that by the beginning of the nineteenth century, doubts about the Newtonian value of 39.788 should have arisen, especially in the minds of some British astronomers who were aware of the work of their French colleagues.

Finlayson 6 returned to the tidal technique and when using the syzygy measurement? and quadrature tides at Dover for the years 1861, 1864, 1865, and 1866, he obtained the following M E /M M values: 89.870, 88.243, 87.943, and 86.000, respectively. Ferrell 7 extracted the principal harmonics from nineteen years of tidal data at Brest (1812 - 1830) and obtained a much smaller ratio M E / M M = 78. Harkness 8 gives a tidal value M E /M M = 78.65.

So-called pendulum method based on the measurement of acceleration due to gravity. Returning to Kepler's third law, taking into account Newton's second law, we obtain

where aM is the time-averaged distance between the Earth and the Moon, P M- lunar sidereal period of revolution (i.e. the length of the sidereal month), gE acceleration due to gravity on the Earth's surface, and R E is the radius of the earth. So

According to Barlow and Brian 9 , this formula was used by Airy 10 to measure M E /M M, but was inaccurate due to the smallness of this quantity and accumulated - the accumulated uncertainty in the values ​​of the quantities aM , gE, R E, and P M.

As telescopes became more advanced and the accuracy of astronomical observations improved, it became possible to solve the lunar equation more accurately. The common center of mass of the Earth/Moon system moves around the Sun in an elliptical orbit. Both the Earth and the Moon revolve around this center of mass every month.

Observers on Earth thus see, during each month, a slight eastward and then a slight westward shift of an object's celestial position, compared to the object's coordinates that it would have had the Earth not had a massive satellite. Even with modern instruments, this movement is not detectable in the case of stars. It can, however, be easily measured for the Sun, Mars, Venus, and asteroids that pass nearby (Eros, for example, is only 60 times further away than the Moon at its closest point). The amplitude of the monthly shift of the position of the Sun is about 6.3 arcseconds. Thus

where a C- the average distance between the Earth and the center of mass of the Earth-Moon system (this is about 4634 km), and a S is the average distance between the Earth and the Sun. If the average Earth-Moon distance a M it is also known that

Unfortunately, the constant of this “lunar equation”, i.e. 6.3", this is a very small angle, which is extremely difficult to measure accurately. In addition, M E / M M depends on an accurate knowledge of the Earth-Sun distance.

The value of the lunar equation can be several times greater for an asteroid that passes close to the Earth. Gill 11 used 1888 and 1889 positional observations of asteroid 12 Victoria and a solar parallax of 8.802" ± 0.005" and concluded that M E /M M = 81.702 ± 0.094. Hinks 12 used a long sequence of observations of asteroid 433 Eros and concluded that M E /M M = 81.53±0.047. He then used the updated solar parallax and the corrected values ​​for asteroid 12 Victoria by David Gill and obtained a corrected value of M E /M M = 81.76±0.12.

Using this approach, Newcomb 13 derived M E /M M =81.48±0.20 from observations of the Sun and planets.

Spencer John s 14 analyzed observations of the asteroid 433 Eros as it passed 26 x 10 6 km from Earth in 1931. The main task was to measure solar parallax, and a commission of the International Astronomical Union was set up in 1928 for this purpose. Spencer Jones found that the lunar equation constant is 6.4390 ± 0.0015 arcseconds. This, combined with a new value for the solar parallax, resulted in a ratio of M E /M M =81.271±0.021.

Precession and nutation can also be used. The pole of the Earth's axis of rotation precesses around the pole of the ecliptic every 26,000 years or so, which also manifests itself in the movement of the first point of Aries along the ecliptic at about 50.2619" per year. The precession was discovered by Hipparchus more than 2000 years ago. small periodic motion known as nutation, found James Bradley(1693~1762) in 1748. Nutation mainly occurs because the plane of the lunar orbit does not coincide with the plane of the ecliptic. The maximum nutation is about 9.23" and a complete cycle takes about 18.6 years. There is also additional nutation produced by the Sun. All of these effects are due to moments of forces acting on the Earth's equatorial bulges.

The magnitude of the steady-state lunisolar precession in longitude, and the amplitudes of the various periodic nutations in longitude, are functions of, among other things, the mass of the Moon. Stone 15 noted that the lunisolar precession, L, and the nutation constant, N, are given as:

where ε=(M M /M S) (a S /a M) 3 , a S and a M are the average Earth-Sun and Earth-Moon distances;

e E and e M are the eccentricities of the earth's and lunar orbits, respectively. The Delaunay constant is represented as γ. In the first approximation, γ is the sine of half the angle of inclination of the lunar orbit to the ecliptic. The value of ν is the displacement of the node of the lunar orbit,

during the Julian year, in relation to the line of equinoxes; χ is a constant that depends on the average perturbing force of the Sun, the moment of inertia of the Earth, and the angular velocity of the Earth in its orbit. Note that χ cancels out if L is divisible by H. Stone substituting L = 50.378" and N = 9.223" got M E / M M = 81.36. Newcomb used his own measurements of L and N and found M E / M M = 81.62 ± 0.20. Proctor 16 found that M E /M M = 80.75.

The motion of the Moon around the Earth would be exactly an ellipse if the Moon and Earth were the only bodies in the solar system. The fact that they are not leads to the lunar parallax inequality. Due to the attraction of other bodies in the solar system, and the Sun in particular, the moon's orbit is extremely complex. The three largest inequalities to be applied are due to evection, variation, and the annual equation. In the context of this paper, variation is the most important inequality. (Historically, Sedilloth says that the lunar variation was discovered by Abul-Wafa in the 9th century; others attribute this discovery to Tycho Brahe.)

The lunar variation is caused by the change that comes from the difference in solar attraction in the Earth-Moon system during the synodic month. This effect is zero when the distances from the Earth to the Sun and the Moon to the Sun are equal, in a situation occurring very close to the first and last quarter. Between the first quarter (through the full moon) and the last quarter, when the Earth is closer to the Sun than the Moon, and the Earth is predominantly pulled away from the Moon. Between the last quarter (through the new moon) and the first quarter, the Moon is closer to the Sun than the Earth, and therefore the Moon is predominantly pulled away from the Earth. The resulting residual force can be decomposed into two components, one tangent to the lunar orbit and the other perpendicular to the orbit (ie, in the Moon-Earth direction).

The position of the Moon changes by as much as ±124.97 arcseconds (according to Brouwer and Clements 17) from the position it would have if the Sun were infinitely far away. It is these 124.9" that are known as the parallax inequality.

Since these 124.97 arcseconds correspond to four minutes of time, it should be expected that this value can be measured with sufficient accuracy. The most obvious consequence of the parallax inequality is that the interval between the new moon and the first quarter is about eight minutes, i.e. longer than from the same phase to the full moon. Unfortunately, the accuracy with which this quantity can be measured is somewhat diminished by the fact that the lunar surface is uneven and that different lunar edges must be used to measure the lunar position in different parts of the orbit. (In addition to this, there is also a slight periodic variation in the apparent half-diameter of the Moon due to the changing contrast between the brightness of the Moon's edge and the sky. This introduces an error that varies between ±0.2" and 2", see Campbell and Neison 18).

Roy 19 notes that the lunar parallax disparity, P, is defined as

According to Campbell and Neyson,18 the parallax inequality was established as 123.5" in 1812, 122.37" in 1854, 126.46" in 1854, 124.70" in 1859, 125.36" in 1867, and 125.46" in 1868. Thus, the Earth/Moon mass ratio can be calculated from observations of parallax inequalities if other quantities, and especially solar parallax (i.e. a S) are known. This has led to a dichotomy among astronomers. Some suggest using the Earth/Moon mass ratio from the parallax inequality to estimate the average Earth-Sun distance. Others propose to evaluate the former through the latter (see Moulton 20).

Finally, consider the perturbation of planetary orbits. The orbits of our nearest neighbors, Mars and Venus, which are under the gravitational influence of the Earth-Moon system. Due to this action, orbital parameters such as eccentricity, node longitude, inclination, and perihelion argument change as a function of time. An accurate measurement of these changes can be used to estimate the total mass of the Earth/Moon system, and by subtraction, the mass of the Moon.

This suggestion was first made by Le Verrier (see Young 21). He emphasized the fact that the motions of the nodules and perihelions, although slow, were continuous, and thus would be known with increasing accuracy as time went on. Le Verrier was so fired up with this idea that he abandoned observations of the then transit of Venus, being convinced that the solar parallax and the Sun/Earth mass ratio would eventually be found much more accurately by the perturbation method.

The earliest point comes from Newton's Principia.

The accuracy of the known lunar mass.

Measurement methods can be divided into two categories. Tidal technology requires special equipment. A vertical pole with graduations is lost in the coastal mud. Unfortunately, the complexity of the tidal environment around the coasts and bays of Europa meant that the resulting lunar mass values ​​were far from accurate. The tidal force with which bodies interact is proportional to their mass divided by the cube of the distance. So be aware that the end product of the calculation is actually the ratio between the lunar and solar masses. And the relation between the distances to the Moon and the Sun must be precisely known. Typical tidal values ​​of M E / M M are 40 (in 1687), 59 (in 1790), 75 (in 1825), 88 (in 1865), and 78 (in 1874), highlighting the difficulty inherent in interpretation. data.

All other methods relied on accurate telescopic observations of astronomical positions. Detailed observations of stars over long periods of time have led to the derivation of constants for precession and nutation of the Earth's axis of rotation. They can be interpreted in terms of the ratio between lunar and solar masses. Accurate positional observations of the Sun, planets and some asteroids over several months have led to an estimate of the distance of the Earth from the center of mass of the Earth-Moon system. Careful observations of the position of the Moon as a function of time during the month have led to the amplitude of the parallactic inequality. The last two methods, together, relying on measurements of the Earth's radius, the length of the sidereal month, and the acceleration of gravity on the Earth's surface, led to an estimate of the magnitude of , rather than the mass of the Moon directly. Obviously, if known only to within ± 1%, the mass of the Moon is indeterminate. To obtain the M M / M E ratio with an accuracy of, say, 1, 0.1, 0.01%, it is required to measure the value with an accuracy of ± 0.012, 0.0012, and 0.00012%, respectively.

Looking back over the historical period from 1680 to 2000, it can be seen that the lunar mass was known ± 50% between 1687 and 1755, ± 10% between 1755 and 1830, ± 3% between 1830 and 1900, ± 0.15% between 1900 and 1968, and ± 0.0001% between 1968 and present. Between 1900 and 1968 the two meanings were common in serious literature. The lunar theory indicated that M E /MM M = 81.53, and the lunar equation and the lunar parallax inequality gave a somewhat smaller value of M E /MM M = 81.45 (see Garnett and Woolley 22). Other values ​​have been cited by researchers who have used different solar parallax values ​​in their respective equations. This minor confusion was removed when the light orbiter and command module flew well-known and well-measured orbits around the moon during the Apollo era. The current value of M E /M M = 81.300588 (see Seidelman 23), is one of the most accurately known astronomical quantities. Our exact knowledge of the actual lunar mass is clouded by uncertainties in Newton's constant of gravity, G.

Importance of the lunar mass in astronomical theory

Isaac Newton did very little with his newfound lunar knowledge. Even though he was the first scientist to measure the lunar mass, his M E / M M = 39.788 would seem to merit little contemporary commentary. The fact that the answer was too small, almost twice, was not realized for more than sixty years. Physically significant is only the conclusion that Newton drew from ρ M /ρ E =11/9, which is that "the body of the Moon is denser and more earthly than that of our earth" (Beginnings, Book 3, Proposition 17, Corollary 3).

Fortunately, this fascinating, albeit erroneous, conclusion will not lead conscientious cosmogonists into a dead end in an attempt to explain its meaning. Around 1830, it became clear that ρ M /ρ E was 0.6 and M E / M M was between 80 and 90. Grant 24 noted that "this is the point at which greater precision did not appeal to the existing foundations of science", alluding, that accuracy is unimportant here simply because neither astronomical theory nor the theory of the origin of the moon relied heavily on these data. Agnes Clerk 25 was more cautious, noting that "the lunar-terrestrial system... was a particular exception among bodies influenced by the Sun."

The Moon (mass 7.35-1025 g) is the fifth of ten satellites in the solar system (starting from number one, these are Ganymede, Titan, Callisto, Io, Luna, Europa, Saturn's Rings, Triton, Titania, and Rhea). Relevant in the 16th and 17th centuries, the Copernican Paradox (the fact that the Moon revolves around the Earth, while Mercury, Venus, Earth, Mars, Jupiter and Saturn revolves around the Sun) has long been forgotten. Of great cosmogonic and selenological interest was the ratio of masses “main / most massive-secondary”. Here is a list of Pluto/Charon, Earth/Moon, Saturn/Titan, Neptune/Triton, Jupiter/Callisto and Uranus/Titania, coefficients such as 8.3, 81.3, 4240, 4760, 12800 and 24600, respectively. This is the first indication of their possible joint origin by bifurcation through the condensation of body fluid (see, for example, Darwin 26, Jeans 27, and Binder 28). In fact, the unusual Earth/Moon mass ratio led Wood 29 to conclude that "indicates quite clearly that the event or process that created the Earth's Moon was unusual, and suggests that some weakening of the normal aversion to the involvement of special circumstances may be acceptable." in this issue."

Selenology, the study of the origin of the moon, became "scientific" with the discovery in 1610 by Galileo of the moons of Jupiter. The moon has lost its unique status. Then Edmond Halley 30 discovered that the lunar orbital period changes with time. This was not the case, however, until the work of G.Kh. Darwin in the late 1870s, when it became clear that the original Earth and Moon were much closer together. Darwin suggested that the early resonance-induced bifurcation, rapid rotation and condensation of the molten Earth led to the formation of the Moon (see Darwin 26). Osmond Fisher 31 and W.H. Pickering 32 even went so far as to suggest that the Pacific Basin is a scar that was left when the Moon broke away from the Earth.

The second major selenological fact was the Earth/Moon mass ratio. The fact that there was a violation of the meanings for the Darwin theses was noted by A.M. Lyapunov and F.R. Moulton (see, for example, Moulton 33). . Together with the low combined angular momentum of the Earth-Moon system, this led to the slow death of Darwin's theory of tides. It was then proposed that the Moon was simply formed elsewhere in the solar system and then captured in some complex three-body process (see eg C 34).

The third basic fact was the lunar density. The Newtonian value of ρ M /ρ E of 1.223 became 0.61 by 1800, 0.57 by 1850, and 0.56 by 1880 (see Brush 35). At the dawn of the nineteenth century, it became clear that the Moon had a density that was about 3.4 g cm -3. At the end of the 20th century, this value remained almost unchanged and amounted to 3.3437±0.0016 g cm -3 (see Hubbard 36). It is obvious that the lunar composition differed from the composition of the Earth. This density is similar to the density of rocks at shallow depths in the Earth's mantle and suggests that the Darwinian bifurcation occurred in a heterogeneous rather than homogeneous Earth at a time that occurred after differentiation and basic morphogenesis. Recently, this similarity has been one of the main facts contributing to the popularity of the ram hypothesis of lunar formation.

It was noted that the average density of the moon was the same like meteorites(and possibly asteroids). Gullemine 37 pointed density of the moon in 3.55 times more than water. He noted that "it was so curious to know the density values ​​of 3.57 and 3.54 for some meteorites collected after they hit the surface of the Earth". Nasmyth and Carpenter 38 noted that "the specific gravity of the lunar about the same as silicon glass or diamond: and oddly enough it almost coincides with the meteorites that we find lying on the ground from time to time; therefore, the theory is confirmed that these bodies were originally fragments of lunar matter, and probably were once ejected from lunar volcanoes with such force that they fell into the sphere of earth's gravity, and ultimately fell to the earth's surface.

Urey 39, 40 used this fact to support his theory of capture of lunar origin, although he was concerned about the difference between the lunar density and the density of certain chondrite meteorites, and other terrestrial planets. Epic 41 considered these differences to be insignificant.

findings

The mass of the moon is extremely uncharacteristic. It is too large to place our satellite comfortably among planetary captured asteroid clusters, like Phobos and Deimos around Mars, the Himalia and Ananke clusters around Jupiter, and the Iapetus and Phoebe clusters around Saturn. The fact that this mass is 1.23% of the Earth is unfortunately only a minor clue among many in support of the proposed impact-origin mechanism. Unfortunately, today's popular "Mars-sized body hits the newly differentiated Earth and knocks out a lot of material" theory has some petty problems. Even though this process has been recognized as possible, it does not guarantee that it is likely. like “why did only one moon form at that time?”, “why don't other moons form at other times?”, “why did this mechanism work on planet Earth, and not touch our neighbors Venus, Mars, and Mercury?” come to mind.

The mass of the Moon is too small to place it in the same category as Pluto's Charon. 8.3/1 The ratio between the masses of Pluto and Charon, a coefficient that indicates that the pair of these bodies is formed by a bifurcation of condensation, the rotation of an almost liquid body, and is very far from the value of 81.3/1 of the mass ratio of the Earth and the Moon.

We know the lunar mass to within one part of 10 9 . But we cannot help feeling that the general answer to this precision is “so what”. As a guide, or a hint about the origin of our heavenly partner, this knowledge is not enough. In fact, in one of the last 555-page volumes on the subject 42 , the index does not even include "lunar mass" as an entry!

References

(1) I. Newton, Principia, 1687. Here we are using Sir Isaac Newton's Mathematical Principles of Natural Philosophy, translated into English by Andrew Motte in 1729; the translation revised and supplied with an historical and explanatory appendix by Florian Cajori, Volume 2: The System of the World(University of California Press, Berkeley and Los Angeles), 1962.

(2) P.-S. laplace, Mem. Acad.des Sciences, 45, 1790.

(3) P.-S. laplace, Volume 5, Livre 13 (Bachelier, Paris), 1825.

(4) P.-S. laplace, Traite de Mechanique Celeste, Tome 3 (rimprimerie de Crapelet, Paris), 1802, p, 156.

(5) P.-S. laplace, Traite de Mechanique Celeste, Volume 4 (Courcicr, Paris), 1805, p. 346.

(6) H. P. Finlayson, MNRAS, 27, 271, 1867.

(7)W.E, Fcrrel, Tidal Researches. Appendix to Coast Survey Report for 1873 (Washington, D. C) 1874.

(8) W. Harkness, Washington Observatory Observations, 1885? Appendix 5, 1891

(9) C. W. C. Barlow ScG. H, Bryan, Elementary Mathematical Astronomy(University Tutorial Press, London) 1914, p. 357.

(10) G. B. Airy, Mem. ras., 17, 21, 1849.

(11) D. Gill, Annals of the Cape Observatory, 6, 12, 1897.

(12) A. R. Hinks, MNRAS, 70, 63, 1909.

(13) S. Ncwcomb, Supplement to the American Ephemeris for tSy?(Washington, D.C.), 1895, p. 189.

(14) H. Spencer Jones, MNRAS, 10], 356, 1941.

(15) E. J. Stone, MNRAS, 27, 241, 1867.

(16) R. A. Proctor, Old and Nets Astronomy(Longmans, Green, and Co., London), )