An estimation of the Moon radius by counting craters: a generalization of Monte-Carlo calculation of π to spherical geometry
aa r X i v : . [ phy s i c s . pop - ph ] J u l An estimation of the Moon radius by counting craters: ageneralization of Monte-Carlo calculation of π to sphericalgeometry J. S. Ardenghi †∗†
IFISUR, Departamento de F´ısica (UNS-CONICET)Avenida Alem 1253, Bah´ıa Blanca, Buenos Aires, ArgentinaAugust 1, 2019
Abstract
By applying Monte-Carlo method, the Moon radius is obtained by counting craters in a sphericalsquare over the surface of it. As it is well known, approximate values for π can be obtained bycounting random numbers in a square and in a quarter of circle inscribed in it in Euclidean geometry.This procedure can be extend it to spherical geometry, where new relations between the areas ofa spherical square and the quarter of circle inscribed in it are obtained. When the radius of thesphere is larger than the radius of the quarter of circle, Euclidean geometry is recovered and theratio of the areas tends to π . Using these results, theoretical deviations of π due to the Moonradius R are computed. In order to obtain this deviation, a spherical square is selected located ina great circle of the Moon. The random points over the spherical square are given by a specificzone of the Moon where craters are distributed almost randomly. Computing the ratio of the areas,the deviation of π allows us to obtain the Moon radius with an intrinsic error given by the finitenumber of random craters. The Moon is always a subject of intense debate, for instance, it is the cause of many natural phenomena,the most common of which are solar eclipses and ocean tides. In turn, it is a source for testing theoriesof gravitation and to investigate geophysical phenomenas ([1] and [2]). Moreover, there are severalrelational properties between the Moon and the Earth, such as the instantaneous distance betweenthem or the relative rotation, as well as intrinsic properties of the Moon, such as the Moon diameter,its excentricity or the more perplexing long-standing puzzle of its origin, that requires sophisticatedand non-sophisticated experimental procedures in order to obtain approximate values or conceptualexplanations (see for instance [3], [4], [5],[6], [7] and [8]). In turn, the Moon is suitable to desing simpleexperimental procedures to obtain accurate information of different parameters of it ([9]), or for exampleby measuring the Moon’s orbit by using a hand-held camera (see [10]). In the same line of thought,this work introduces a novel procedure to estimate the Moon’s radius without using any sophisticatedmeasuring device. Briefly, the method consists in using the Monte-Carlo method to obtain π usingrandom points (see [11] and [12]) generalized to spherical geometry. In this manuscript, the spherewill be the Moon and the random points will be the craters on it. The quarter of circle inscribed in asquare on the plane, where the random points are located and which is used in the typical Monte-Carlo ∗ email: [email protected], fax number: +54-291-4595142 In [12], page 302 there is a plot of π in a sphere, where the value is computed with the ratio of the cimcurference ofa circle to its diameter, measured on the curved surface of the sphere. π becausespherical geometry in this limit tends to flat geometry. This leads to the following conclusion: bycounting random points over quarter of circles and squares in spherical geometry we obtain deviationsfrom π by computing the areas ratio and these deviations are a function of the radius of the sphere.Interestingly, we can apply the Monte-Carlo method over a quarter of circle in the spherical surfacewithout knowing the sphere radius but it can be obtained by counting random points on it. Of course,to do so we must know the function that relates the deviation from π with the spherical surface radius.Using the craters of the Moon as a random points in a particular spherical square inscribed on it, wecan obtain the deviation from π and by using the theoretical function that relates the deviation from π with the Moon’s radius, we can deduce the Moon’s radius with an accuracy related to the numberof craters considered. Should be stressed that altough the conceptual procedure is simple, sphericalgeometry introduce subtetlies concerned with circle and squares inscribed over the surface of a spherethat must be taken into account in order to obtain the correct limit when the flat geometry is recovered,which can be implemented easily by taking the infinite limit of the sphere radius. Then, in order to beconsise and self-contained, this mansuscript will be organized as follows: In Section II, the function thatrelates the deviation from π in the Monte-Carlo method applied to spherical geometry and the sphericalsurface radius is obtained. In Section III we used the results obtained in Section II to determine theMoon’s radius with the respective error by considering random craters in a particular zone of the Moonsurface. In last section, the conclusions are presented and in Appendix some mathematical details areshown. The Monte-Carlo method is a general method that allows us to obtain specific results by manipulatingrandom variables ([13], [14], [15], [20], [21], [16] and [17]). As it is well known, approximate values of π can be obtained by counting random numbers on a square and on a quarter of circle inscribed in it ina plane([18]). In this method, by considering a square of side L = r and a a quarter of circle of radius Figure 1: Left: Random points in the unit square. Inside the unit square, the inscribed quarter of circlecan be seen. Right: Value of Pi obtained as a function of N random points. The confidence boundsthat behaves as the square roots of N is shown. 2igure 2: Spherical square (blue line) located symmetrically with respect a great circle of the sphereand the quarter of circle inscribed in it (yellow line). The sides of the spherical square are shown as λ and 2 φ . As it was explained in the introduction, these two sides must be identical in order to have acorrect limit in flat geometry. r with center in one vertex of the square as it can be seen in figure 1, the ratio between the areas read A C A S = πr r = π N random points in the unit square [0 , × [0 , N and the area of the quarter-circle can be approximated by the number of random points N C thatlie inside it. By computing 4 N C /N , an approximate value of π can be obtained and the approximationcan be accurate by increasing N as it can be seen in figure 1, where in turn the confidence bounds areshown that scales as N − / for large N . The same argument can be applied in spherical geometry, wherewe can consider a square of side r inscribed in a sphere of radius R and a circle of radius r inscribedin the square as it can be seen in the figure 2, where the spherical square is located symmetricallywith respect a great circle of the sphere. This choice of the spherical square location is suitable whenthe limit of infinite radius is taken, because the ratio of areas tends to π and the radius of the circlemeasured over the surface of the sphere is identical to the length of the sides of the square.In order to obtain the square and circle areas, we can consider the surface element area dA = R sin θdθdϕ in spherical coordinates, where R is the sphere radius, ϕ is the azimuth and θ is the anglewith respect the z axis. By using spherical coordinates it is possible to show that the area of the squareof radius r in a sphere of radius R reads (see Appendix) A S = 2 rR tan( r R ) (2)and the area of a circle of radius r inscribed in a sphere of radius R reads A C = π R h − cos( rR ) i (3)when r/R → A S = r and A C = πr as it is expected and eq.(1) holds. This limitcorresponds to the case in which R > r (the radius of the sphere is much larger than the radius ofthe circle inscribed in the sphere) and the Euclidean geometry is an accurate approximation. Then, by3 .5 1.0 1.5 2.0 2.5 3.0 rR A c A S Figure 3: Monte-Carlo method to obtain deviations of π as a function of rR in spherical geometry, where r is the radius of the circle and the side of the spherical square and R is the radius of the sphere.using eq.(2) and eq.(3), the factor 4 A C /A S reads4 A C A S = πRr sin( rR ) (4)and the result depends on the ratio r/R as it can be seen in the figure 3, where 4 A C /A S is plottedagainst rR , where the limit r/R → A C A S ∼ π − π ( r R ) + 215 π ( r R ) + O (cid:16) ( r R ) (cid:17) (5)The last equation shows the deviations from π in the Monte-Carlo method applied in spherical geometryand this deviation can be expanded in powers of r R , which implies that by knowing how much we havedeviate from π then we can deduce R by an appropiate choice of r . We will apply the results obtainedin this section in a particular case where nature provide us with natural random points in a sphericalsurface: the Moon and its craters. For instance, in order to see the physical implication of last equation,in figure 4, two different zones where the Monte-Carlo method can be implemented in a spherical surface.In this figure, we are showing the Moon’s surface, but any spherical surface is valid for the discussion.The two different zones are given with the respective r radius. It is instructive to note that for thesmallest r and considering that R = 1737 km for the Moon’s radius, r/R ∼ .
08 and for the secondzone with largest r , r/R ∼ .
47. The first zone with smallest r implies a small deviation from π thanthe second zone when the ratios of areas is computed. In the small zone, the Monte-Carlo method isindistinguishable from the Monte-Carlo in flat geometry, A C A S ∼ π − π (0 . ∼ . π and in thesecond zone A C A S ∼ . π . As it was said in last section, there is an interesting example in nature where random points are inscribedin a spherical geometry: the Moon or any planet with a large numbers of craters in its surface. Thereare zones in the Moon where the craters are distributed almost randomly as it can be seen in figure5, where it is shown the Moon with selected spherical square obtained from Google Moon [23] and4igure 4: Different zones over a spherical surface where the Monte-Carlo method can be applied. As itcan be seen, for r = 140 km the Monte-Carlo approximates accurately to flat geometry when the ratioof areas are computed, where the result expected is 0 . π . The larger zone with r = 818 km deviatesfrom flat geometry and the Monte-Carlo method gives 0 . π .the craters are marked and can be seen in the Supplementary Material ([22]). The software used [23]allows to draw great circles, circles and spherical squares with the respective lengths and areas. Thezone in the Moon was chosen due to the fact it is the largest spherical square located symmetricallywith respect a great circle of the Moon with the largest number of almost random craters. Then, itis possible to implement the same procedure explained in the last section by counting craters inside aspherical square and in a quarter of circle inscribed in the surface of the Moon. The ratio obtained α = 4 N C /N depends on the radius of the quarter of circle r chosen and the Moon’s radius R M . Thismeans that we can obtain an estimated value of the Moon’s radius by simply counting craters in asquare over the surface. This could sound peculiar but is a natural consequence of the Monte-Carlomethod in other geometries besides Euclidean geometry.In order to do this, Google Moon [23] was used, where a detailed image is available and wherethe craters can be pointed by a mark. The suplemmentary material ([22]) contains the circle and thespherical square chosen and the marks of each crater. The yellow marks are the craters inside the circleand the green marks are the craters outside the circle and inside the square.In order to apply the method explained above, a square of side r = 1815 km was considered as it canbe seen in figure 6. The points where craters are found are marked (see supplementary material [22]).By counting the number of craters N C inside the quarter of circle inscribed in the spherical square andthe total number of craters N inside the square, the factor 4 N C /N can be computed and the valueobtained is α = 4 N C N = 4 × .
616 (6)where the total number of random craters is N = 1752.By using eq.(5) or by computing numerically the inverse function α , the Moon’s radius obtained is R M = 1820 km. By using the Taylor expansion up to second order of eq.(5), that is α ∼ π − π ( r R ) ,the value of the Moon’s radius reads R M = 1811 km.Due to the fact that we have used N = 1752 random points (1752 random craters) the Moon’sradius value obtained has an error. The same behavior is obtained for the Monte-Carlo method in flatgeometry to obtain π , as it was shown in the last section, where the confidence bounds scales as N − / .In order to obtain the associated error of R M that we call ∆ R M in terms of the number of craters5igure 5: Circle (red line) and square (blue line) inscribed in it considered for the measurement of thenumber craters. The image was obtained from Google Moon (see [23]).Figure 6: Craters found in the circle and square over the Moon (image obtained from Google Moon).6 P r ob a b ili t y d i s t r i bu t i on Figure 7: Distribution function of the possible results of α in the numerical simulation where N =1752 random points have been considered, which is the number of random craters considered in theexperimental procedure.considered, we have implemented numerically the Monte-Carlo method in spherical geometry with thesame parameters (inscribed radius circle r = 1815 km) and we have performed several calculationsof N C N S considering N = 1752 random points (for the numerical implementation see SupplementaryMaterial). The dispersion of the results can be seen in figure 7. As it happens with the Monte-Carlomethod to obtain π , the dispersion of the results depends only on the number of random points. Thesame is expected in spherical geometry, the dispersion of the results are independent of r and R M and only depends on the number of craters considered for the calculation. Nevertheless, constantdispersion in α = N C N S does not imply constant dispersion in r R M . We might note this by computingthe differential dα as dα = dαd ( r RM ) d ( r R M ) which implies that for constant dα we have that d ( r R M ) hasa dependence on r R M . Considering the differential dα as the error associated to α = N C N S written as∆ α and the differential dR M as the error in R M written as ∆ R M , a straighforward calculation givesthe error associated to the Moon’s radius∆ R M = 1 π (cid:12)(cid:12)(cid:12)(cid:12) r sin( rR M ) − R M cos( rR M ) (cid:12)(cid:12)(cid:12)(cid:12) − ∆ α (7)The error ∆ α is obtained from the tails of the dispersion of figure 7 when the Monte-Carlo method isimplemented numerically. Using that ∆ α = σ = 0 .
045 and using r = 1815 km in last equation the finalresult for the Moon radius is R M = 1820 ±
146 km (8)The obtained value for the Moon radius is accurate and the error contains the real value of the Moonradius.Should be stressed that a better approximation can be obtained for the mean Moon radius R M by considering other spherical squares located symmetrically with respect the great circle of the Moonwhere the craters are randomly distributed. A simple inspection using Google moon software showsthat the largest zone is the one considered in this work.7 Conclusions
In this work the Moon radius can be obtained experimentally by counting craters in an spherical squareand a quarter of circle inscribed in it. This procedure is the generalization of the Monte-Carlo methodto obtain π in Euclidean geometry to spherical geometry. By deviating from flat geometry, the ratio ofrandom points in a square and a circle inscribed in it gives a deviation from π . This deviation can berelated to the radius of inscribed circle and the radius of the sphere. In particular, the Moon containzones with random craters that can be used as random points over a surface sphere. By applying theMonte-Carlo method, that is, by counting craters inside the square and circle defined in the surface ofthe Moon, a deviation from π is obtained and this result is used to compute the Moon radius. Theobtained value is R M = 1820 ±
146 km and the real value of the Moon radius is inside the error.Although the method introduced in this work is not very precise, shows how the randomness can beuseful to obtain information about the underlying space in which this random phenomena occurs. Inturn, this method can give better approximations by considering several squares and circles inscribedin the Moon.
This paper was partially supported by grants of CONICET (Argentina National Research Council)and Universidad Nacional del Sur (UNS) and by ANPCyT through PICT 1770, and PIP-CONICETNos. 114-200901-00272 and 114-200901-00068 research grants, as well as by SGCyT-UNS., J. S. A. isa member of CONICET.
The line element in spherical coordinates with a fixed radius is d l = Rdφ b e φ + R sin φdλ b e λ . Consideringthat two of the four sides of the spherical square can be obtained with constant λ and the two remainingsides with constant φ , we obtain by integration of d l between − φ and φ in φ and between 0 and λ in λ , and making both results identical (see figure 2) r = 2 Rφ r = Rλ cos φ (9)where r is the length of the side of the spherical square and simultaneously is the radius of the circleinscribed in the square. Making both results identical we obtain that λ = 2 φ / cos φ (10)This last result is used in section I.In order to compute the area of the spherical square centered in a great circle of radius r inscribedin a sphere of radius R , the surface element must be used dA = R sin θdθdϕ where θ is the polar angleand ϕ the azimuthal angle. By using the longitude λ = ϕ and latitude φ = π − θ , the area of the squarereads A S = R Z π + φ π − φ sin θdθ Z λ dϕ = 2 R sin φ λ (11)In turn because the sides of the square must be identical to the radius r of the inscribed circle, then r = R φ and r = R cos( φ ) λ , which implies that λ can be determined as λ = 2 φ / cos( φ ).Introducing this result in last equation, the area of the spherical square reads A S = 4 R φ tan( φ ) (12)where φ = r R . The area of the inscribed circle can be computed by simply realizing that this circle isa spherical cap with an angle 2 φ between the rays from the center of the sphere to the pole and the8dge of the disk forming the base of the cap. This area reads A C = 2 πR [1 − cos(2 φ )] (13)using that r = R φ then φ = r R and the area of the square and area of the circle can be written interms of r and R as A S = 2 rR tan( r R ) (14)and A C = 2 πR h − cos( rR ) i (15)These results will be used in Section II. References [1] A. L. Hammond,
Science , 1289–1290 (1970).[2] A. H. Cook and J. Kovalevsky,
Philos. Trans. R. Soc. London, Ser. A , 573–585 (1977).[3] L. J. Pellizza, M. G. Mayochi and L. C. Brazzano,
Am. J. Phys. , 311 (2014)[4] M. C. Lo Presto, The Physics Teacher , 179 (2000);[5] J. Lincoln., The Physics Teacher , 492 (2018)[6] D. Stevenson, Physics Today , 11, 32 (2014),.[7] P.Gorenstein, World Book-Moon , Sydney, Australia: World Book Inc, 1992: 782-795.[8] E. J. Sartoon,
The Physics Teacher Am. J. Phys. , 317 (2014).[11] D. E. Simanek, The Physics Teacher , 68 (2014);[12] S. C. Bloch and R. Dressler, Am. J. Phys. , 298 (1999).[13] H. Brody, The Physics Teacher , 197 (1976);[14] D. Shirer, P. M. Ogden, Am. J. Phys. , 208 (1972);[15] A. C. Duffy, The Physics Teacher , 4 (1976).[16] I. ¨Oks¨uz, J. Chem. Phys. , (11), 5005–5012 (1984).[17] P. Mohazzabi, Am. J. Phys. (2), 138–140 (1998).[18] T. Williamson, The Physics Teacher , 468 (2013).[19] X. L. Lv, Y. Xie, H. Xie, New J. Phys. , (2018).[20] J. H. Mathews, Pi Mu Epsilon Journal , , 281-282 (1972)[21] P.G. Lowry, Creative Computing , Supplementary material: Numerical implementation of Monte-Carlo method in spher-ical geometry
As it was shown in Section II of the manuscript, the area of the square and the circle in sphericalgeometry reads A S = 2 rR tan( r R ) (16)and the area of a circle of radius r inscribed in a sphere of radius R reads A C = π R h − cos( rR ) i (17)when r/R → A S = r and A C = πr as it is expected and A C A S = π holds. Then, byusing eq.(17) and eq.(18), the factor 4 A C /A S reads4 A C A S = πRr sin( rR ) (18)and the result depends on the ratio r/R . In the limit r/R →
0, the function behaves as4 A C A S ∼ π − π ( r R ) + 215 π ( r R ) + O (cid:16) ( r R ) (cid:17) (19)The last equation shows the deviations from π in the Monte-Carlo method applied in spherical geometryand this deviation can be expanded in powers of r R , which implies that by knowing how much we havedeviate from π then we can deduce R by an appropiate choice of r .In order to implement the Monte-Carlo method in spherical geometry numerically, we can considerthe square defined by the angles φ ∈ [ − φ , φ ], λ ∈ [0 , λ ] in spherical coordinates, where φ is themaxium latitude and λ is the maximum longitude as it can be seen in the figure 8. The angles φ and λ are related by λ = 2 φ / cos φ (20)which is obtained due to the fact that the sides of the spherical square are identical, that is, r = 2 Rφ and r = Rλ cos φ . The relation r = 2 Rφ implies that φ can be written as φ = r R (21)which implies that the latitude φ depends on the ratio r/R . In turn, that the maximum value of theradius r of the inner circle is r max = πR , which implies a maximum value of φ max0 = π/
2. This resultis expected due to the fact that the radius r of the inscribed circle must be smaller than the perimeterof a great circle of the sphere, which is identical to πR . When r = πR , the area of the circle is halfthe area of the sphere. Moreover, the angle φ is the angle formed by the rays from the center of thesphere to the edge of the disk that the circle form in the sphere of radius R . Considering N randompoints in the interval φ ∈ [ − φ , φ ] and λ ∈ [0 , φ / cos φ ], deviations from π can be obtained whenthe ratio of the number of points that fall inside the circle to the total number of points is computedand this deviation depends exclusively on the ratio r/R . By knowing r and 4 N C /N , the value of R canbe obtained. Should be stressed that in order to pick a random point on any surface of a unit sphere,is not correct to consider the spherical coordinates θ and φ from uniform distributions θ ∈ [0 , π ] y φ ∈ [0 , π ], since the area element dA = R sin φdθdφ is a function of φ and then the points picked areclustered in the poles. The correct choice is by introducing the distribution sin ϕ for the azimuth.10igure 8: Spherical square (blue line) located symmetrically with respect a great circle of the sphereand the quarter of circle (yellow line) inscribed in it. The sides of the spherical square are shown as λ and 2 φ . These two sides must be identical in order to have a correct limit in flat geometry.By writing eq.(4) using eq.(21) we obtain that the function α ( φ ) reads α ( φ ) = π φ sin(2 φ ) (22)where φ = r R and α = A C A S that in the numerical implementation becomes N C N S where N C is thenumber of random points in the quarter of circle and N S is the number of random points inside thesquare, which is identical to the total number of random points N .In figure 9, the function α ( φ ) is shown in red and different numerical calculations show the dispersionof the result around the theoretical curve, where we have considered N = 1752 random points over thespherical square inscribed in the sphere. The dispersion of the results are independent of φ and onlydepends on the number of random points N , as it happens with the Euclidean Monte-Carlo methodto measure π . Nevertheless, constant dispersion in α does not implies constant dispersion in φ . Wemight note this by considering that dα is related to dφ as dα = dαdφ dφ which implies that for constant dα we have that dφ has a dependence on φ . This implies that once we obtain dφ in terms of φ , wecan obtain the error in R M by | dφ | = r R M dR M . In the same figure 9, the vertical black line indicatesthe theoretical value is φ T = r R = 0 . r = 1815 km and R = 1737 km is the actualMoon’s radius. The horizontal black line indicates the value α ( φ T ) and the dashed green lines indicates α ( φ T ) ± σ , where σ is obtained from a Gauss distribution of the possible α values when φ T = 0 . N = 1752 random points.The tails α ( φ T ) ± σ are shown in green dashed lines in figure 9. In turn, vertical dashed green linesare shown around φ V which indicates the dispersion expected in the φ axis. Finally, the violet horizontalline indicates the obtained experimental value α V = N C N S = 2 .
61 in the manuscript by counting cratersin the surface of the Moon, which gives α V = N C N S = 4 × = 2 .
616 and where we have N = 1752random craters and the radius of the circle inscribed in the Moon’s surface is r = 1815 km. The verticalviolet line is the experimental value by inverting the function α ( φ ), that is φ V = α − ( α V ). From thefigure 9 it can be seen that the experimental value obtained for φ V is inside the vertical dashed lines.By inverting eq.(22), the value obtained for φ V is φ V = 0 . R M = 182011 α Experimental value α v =4N c /N s Theoretical curve α ( φ )Theoretical value α (r/2R) Experimental value φ v Theoretical value r/2R
Figure 9: Theoretical and numerical simulation of the function α ( φ ) that accounts for the deviation of π when the geometry is curved. The horizontal axis is the angle φ = r R , where r is the radius of theinscribed circle and R is the sphere radius.km. Finally, in order to obtain the error in the R , we can consider that the possible values of α impliespossible values of φ implicitly through eq. (22) for a fixed number of random points N . We can callthe error of the variable α as ∆ α and the error of φ as ∆ φ and both are related through eq.(22) as∆ α = dαdφ | φ = φ T ∆ φ (23)then ∆ φ = ( dαdφ | φ = φ T ) − ∆ α (24)and by using that φ T = r R M ∆ R M = (cid:12)(cid:12)(cid:12)(cid:12) r φ T ( dαdφ | φ = φ T ) − (cid:12)(cid:12)(cid:12)(cid:12) ∆ α (25)The error in ∆ α obtained with N = 1752 random points used is ∆ α = σ = 0 . α given be σ to the dispersionin φ . Using that ∆ α = σ = 0 .
045 and that dα = dαdφ dφ and | dφ | = r R M dR M we obtain the error in R M as ∆ R M = (cid:12)(cid:12)(cid:12)(cid:12) r φ T ( dαdφ | φ = φ T ) − (cid:12)(cid:12)(cid:12)(cid:12) ∆ α (26)By using that dαdφ = πφ h cos(2 φ ) − φ sin(2 φ ) i and r = 1815 km, then the final result for the Moon radiusis R M = 1820 ±±