Updating a simple model of lunar recession
R. Caimmi
astro-ph.EP















lefteqnarray=0pt
    lefteqnarray*=0pt
        leftsubeqnarray=0pt
        
        
        
        
        
        
        
        
        
        
        Updating a simple model of lunar recession
            
        
        
        
        
        
        
        
        
        
         R. Caimmi[ Physics and Astronomy Department, Padua University,
        Vicolo Osservatorio 3/2, I-35122 Padova, Italy.   Affialiated up to September
        30th 2014.  Current status: Studioso Senior.   Current position: in retirement
        due to age limits.
        email: roberto.caimmi@unipd.it   
        fax: 39-049-8278212]
        
        
        
        
        agga
            
        =============================================================================================================================================================================================================================================================================================================================
        
        
        
        § 
        
        
        
        
        Abstract
        
        
        A classical model of lunar recession is reviewed and updated, where input
        parameters are taken or inferred from earlier astronomical computation of
        insolation quantities on Earth back to 0.25 Gyr.   Free parameters are (i)
        the transition age, t_a-t^†, where mean lunar recession velocity,
        ȧ, is suddenly lowered, and (ii) the merging age, t_a-t^,
        where Earth-Moon distance (EMD), a, drops to zero, which relates to
        mean lunar recession velocity just after transition, ȧ^†.
        Predicted mean EMD and length of day (LOD) slightly overstimate their
        counterparts from the above mentioned astronomical computation.   Predicted
        LOD, where the effect of atmospheric tides and nontidal processes is
        considered and supposed
        to be time independent, slightly understimates a linear interpolation from
        paleontological data related to Phanerozoic, inferred in earlier
        investigations.   If short-period (Δ t≈10^-3 Gyr) EMD
        fluctuations (Δ a≈∓0.5R_⊕), resulting from the above
        mentioned astronomical computation back to 0.25 Gyr, occur along the whole
        evolution of Earth-Moon system (EMS), then the predicted mean EMD
        is consistent with values inferred from three well studied
        paleontological data sets: Elatina-Reynella (ER), Big Cottonwood (BC), Weeli
        Wolli (WW), for t_a-t^†=0.25
        Gyr and t_a-t^=6 Gyr.   The same holds for LOD where, in
        addition, computed values are consistent with a linear interpolation from
        paleontological data related to Proterozoic, inferred in earlier
        investigations.   The situation is reversed if the effect of atmospheric tides
        and nontidal processes is taken into consideration and suppposed to be time
        independent, in the sense that fitting curves relate to merging age in
        advance with respect to formation age.   Accordingly, the above
        mentioned effect cannot be supposed as time independent during Proterozoic.
        In conclusion, the current model can be considered as a useful
        zeroth-order approximation to the evaluation of lunar recession and LOD.
        More accurate paleontological data, expecially in connection with error
        evaluation, would be desirable in view of improved results and further
        constraints involving both Astronomy and Paleontology.
        
        
         keywords -
        Earth-Moon system - tidal friction - lunar recession - length of day.
        
        
        § INTRODUCTION
        
        The evolution of Earth-Moon system (EMS) has been debated for centuries since
        the beginning of the scientific revolution.   Related references are mentioned
        in earlier investigations e.g.,
        <cit.><cit.><cit.><cit.> and shall not be repeated
        here.
        
        The driving
        mechanism lies in the lunar tidal friction, which acts along the following
        steps e.g., <cit.>.
        
          (1)  
        The gravitational force of the Moon raises a tidal bulge in the solid Earth
        and oceans.
        
          (2)
        Because of friction there is a delay in Earth's response, causing the tidal
        bulge to lead the Earth-Moon axis by a small angle.
        
          (3) 
        The Moon exerts a torque on the tidal bulge that retards Earth's rotation,
        thereby increasing the length of day (LOD).
        
          (4) 
        The torque that Earth's tidal bulge exerts on the Moon leads to an
        acceleration of the Moon's orbital motion, causing the Moon to recede from
        Earth.
        
        While lunar recession is almost entirely due to tidal friction, LOD can be
        affected, in addition, by other processes such as core-mantle friction,
        atmospheric tides, mantle convection, macrodiffusion at the core-mantle
        boundary e.g., <cit.><cit.><cit.>.   Tidal despinning is
        less or more effective according if continents are gathered to form a
        supercontinent, Pangaea say, or are spread over Earth surface, respectively
        e.g., <cit.><cit.>.
        
        Direct determination of lunar recession by laser ranging <cit.> yields
        a value that mildly changes back to 0.25 Gyr <cit.>.   Extrapolation to
        earlier ages would imply catastrophic Earth-Moon close approach within 2 Gyr
        or less <cit.><cit.><cit.>, which is in contradiction with
        inferred age of lunar rocks (brought back via Apollo missions) equal to about
        4.5 Gyr e.g., <cit.><cit.><cit.>.   Then the mean lunar
        recession before Phanerozoic (back to about 0.57 Gyr)
        had to be significantly lower than the current value and even the average
        value through Phanerozoic e.g., <cit.>.
        
        Paleotidal and paleorotational values can be inferred from fossil corals and
        molluscs e.g., <cit.><cit.><cit.><cit.><cit.><cit.><cit.>.
        More specifically, fine laminae are
        interpreted either as daily growth increments or as the record of the
        semidiurnal or diurnal tide, in accordance with the general growth habits of
        related current descendants.   Modulation of the fine banding by the
        fortnightly or monthly tidal cycles and by the yearly seasonal cycle allows
        estimates of the number of days per year, the number of days per month, and
        the number of months per year <cit.>.
        
        In general, paleontological data are presented with no reliable error bands
        for both related age and inferred quantities which are quite large anyway
        e.g., <cit.>.   An exception is concerned with at least three cases
        concerning precambrian cyclic rhytmites found on the following formations: (i)
        Weeli Wolli (WW), Western Australia, about 2.45 Gyr ago
        <cit.><cit.><cit.>; (ii) Big Cottonwood (BC), Utah, about
        0.90 Gyr ago <cit.><cit.>; (iii) Elatina and Reynella Siltstone
        (ER), South Australia, about 0.62 Gyr ago
        <cit.><cit.><cit.><cit.><cit.><cit.><cit.>.
        For further details, an
        interested reader is addressed to the parent paper <cit.>.
        
        A nontrivial question is how to consider the above mentioned paleontological
        data with respect to the remaining ones.   To this respect, an extreme case
        could be excluding the latter e.g., <cit.> and the opposite could be
        assigning equal statistical weight to all available data e.g., <cit.>.
        Comparison of model predictions to values inferred from observations would
        imply the former alternative, in that error bars are needed, while
        determination of empirical laws would prefer the latter alternative, where
        larger datasets are available.
        
        Under restrictive but plausible assumptions, the mean Earth-Moon distance
        (EMD), a, in the past can be expressed in terms of the age, t_a-t
        (t_a is the current EMS age), the current EMD, a_a,
        and the current mean lunar recession, <ȧ_a>, which implies catastrophic
        Earth-Moon collision within 2 Gyr or less
        <cit.><cit.><cit.>.  To avoid
        this discrepancy, a sudden reduction in mean lunar recession velocity at an
        assigned transition age is necessary <cit.><cit.>.
        Accordingly, model key parameters
        are the transition age, t_a-t^†, and the merging
        age, t_a-t^, where the latter has necessarily to take place in
        advance with respect to EMS formation age, t_a-t_i.
        
        In this view, the merging age is an artefact of the model, in the sense the
        Moon orbit becomes unstable when Roche limit or corotation limit is attained,
        and computation should be halted there.   In addition, EMS birth has to be
        conceived as the earliest configuration where (quasi) equilibrium holds.   To
        this respect, all proposed models of Moon formation e.g., <cit.> could
        be viable.
        
        Though the above mentioned classical model of lunar recession has
        been used in the past e.g., <cit.><cit.>, still detailed
        formulation and exploration of parameter space is lacking (to the knowledge of
        the author) and it could be a useful zeroth order approximation for more
        refined attempts.   In addition, Moon formation is never explicitly mentioned
        (to the knowledge
        of the author) in dealing with paleotidal and paleorotational values which, on
        the other hand, could strenghten mutual connection between astronomical and
        paleontological data.
        
        The current paper is aimed to review and update the classical model of lunar
        recession e.g., <cit.><cit.>, according to the above
        considerations.    The model and related input parameters are presented in
        Section <ref> after a number of basic considerations.   The results are
        shown in Section <ref> and compared
        with their counterparts inferred from both astronomical computations (0-0.25
        Gyr ago) <cit.> and paleontological data (0-2.5 Gyr ago)
        <cit.><cit.><cit.><cit.>.
        The discussion is performed in
        Section <ref> and the conclusion makes the subject of Section <ref>.
        
        
        
        § A CLASSICAL MODEL OF LUNAR RECESSION
        
        
         §.§ Boundary conditions
        
        Let a two-body system be considered where the secondary, S (in particular, a
        satellite) undergoes circular orbits around the primary, P (in particular, a
        planet).   Let the two bodies be rigid and spherical-symmetric.  If a_1 and
        a_2 are radii of different orbits, the application of Kepler third law
        yields:
              (P_1/P_2)^2=(a_1/a_2)^3  ;
        
        where P is the orbital period.   In presence of satellite (secular)
        recession, the indexes, 1 and 2, can be related to different ages e.g.,
        <cit.>.
        
        In the special case of negligible (with respect to unity) secondary-to-primary
        mass ratio, the lower limit of stable orbits, or Roche limit, is e.g.,
        <cit.> Chap.VIII 212:
               a_ R=2.4554(ρ_ P/ρ_ S)^1/3R_ P  ;    M_ S/M_ P≪1  ;
        
        where a_ R is the orbital radius when the Roche limit is attained,
        ρ is the mean density, R the radius and M the mass.   After
        performing little algebra, Eq. (<ref>) translates into:
               a_ R=2.4554(M_ P/M_ S)^
        1/3R_ S  ;    M_ S/M_ P≪1  ;
        
        in terms of masses instead of mean densities.   The numerical coefficient on
        the right-hand side of Eqs. (<ref>) and (<ref>) can be
        approximeted as 2^4/3≈2.5198.
        
        The orbital period of a stable (a_ S≥ a_ R) orbit is e.g.,
        <cit.>:
               P_ S=2π(a_ S^3/GM_ P)^1/2  ;    
        M_ S≪ M_ P  ;
        
        where G=6.67408·10^-8 cm^3/(g s^2) is the constant of
        gravitation.
        In the Roche limit (a_ S=a_ R) Eq. (<ref>) reads:
               P_ R=2π(a_ R^3/a_ S^3a_ S^3/GM_ P)^1/2=
        (a_ R/a_ S)^3/2P_ S  ;    
        M_ S≪ M_ P  ;
        
        in agreement with Eq. (<ref>).
        
        With regard to EMS, R_ S=R_◯=1378 km,
        M_ S/M_ P=M_◯/M_⊕=0.0123≪1,
        and Eq. (<ref>) reduces to:
               a_ R=18487.1467  km=2.8985 R_⊕  ;
        
        where R_◯, M_◯, R_ P=R_⊕, are Moon radius,
        Moon mass, Earth radius, respectively.
        
        The particularization of Eq. (<ref>) to the current EMS configuration,
        via Eq. (<ref>) yields:
               P_ R=6.9487  h  ;
        
        where a_◯=60.1426 R_⊕ and P_◯=27.4404 d have been
        inferred from an earlier investigation <cit.>.   If EMS evolution is
        intended as a sequence of (quasi) equilibrium configurations, then lunar
        distance and orbital period are confined above a threshold of about
        3R_⊕ and 7 h, respectively.
        
        Turning to the general case, let T_ P be the rotation period of the
        primary body.   The corotation condition is P_ S=T_ P e.g.,
        <cit.> which, via Eq. (<ref>), after little algebra reads:
               a_ C=(GM_ P/4π^2)^1/3T_ P^2/3  ;
        
        where a_ S=a_ C is the orbital radius in presence of corotation.
        If the Roche limit coincides with the corotation limit, a_ R=a_ C,
        then T_ P=P_ R via Eqs. (<ref>) and (<ref>).   A
        secondary body orbitating below the corotation radius will spiral in towards
        the primary due to tides e.g., <cit.>.
        
        With regard to EMS, M_ P=M_⊕=5.9726·10^27 g and
        Eq. (<ref>) reduces to:
               a_ C=0.795989R_⊕(T/ h)^2/3  ;
        
        where T=T_⊕ is LOD at an assigned age.
        
        For fluid masses in rigid rotation, a necessary condition for equilibrium
        (due to Poincaré) is e.g., <cit.> Chap. IX 240:
              Ω^2<Ω_ P^2=2π Gρ_ P  ;
        
        where Ω is the angular velocity and the index, P, denotes Poincaré
        limit.   Related Poincaré rotation period,
        T_ P=2π/Ω_ P, is inferred from Eq. (<ref>) as:
               T_ P=2π/(2π Gρ)^1/2=2π(2/3R^3/GM)^1/2  ;
        
        where the last equality holds for spherical bodies.   Shorter periods,
        T<T_ P, would imply fission or equatorial rings in centrifugal support
        e.g., <cit.> Chap. IX 235-240.   With regard to the Earth,
        R=R_⊕=6378.1366·10^5 cm and Eq. (<ref>) reduces to:
               T_ P=1.149728  h  ;
        
        which is a LOD lower limit for Earth stability.
        
        In summary, sequences of EMS equilibrium configurations must necessarily
        satisfy the following boundary conditions:
               a≥ a_ R=2.898518 R_⊕  ; 
               P≥ P_ R=6.948719  h  ;
        
        radius and period of lunar orbit above Roche limit;
               a≥ a_ C=0.795989 R_⊕  ;
        
        radius of lunar orbit above corotation radius;
               T≥ T_ P=1.149728  h  ;
        
        LOD above Poincaré limit.
        
        
        
         §.§ The model
        
        
        According to the classical model of lunar recession e.g.,
        <cit.><cit.><cit.><cit.><cit.><cit.>, the
        following restrictions hold.
        
          (1)   
        EMS evolution occurs along a sequence of (quasi) equilibrium configurations
        where the total angular momentum remains unchanged.
        
          (2)  
        The driving mechanism is due to semidiurnal tidal friction on solid Earth +
        ocean, which causes angular momentum transfer from Earth spin to Moon
        revolution, increasing both LOD and lunar recession.
        
          (3) 
        The tidal friction from the Sun is negligible with respect to the tidal
        friction from the Moon.
        
          (4)  
        Earth and Moon mass distribution are spherical-symmetric in absence of tidal
        effects.
        
          (5)   
        Earth and Moon mass and radius remain unchanged.
        
          (6)   
        The Moon retains syncronous rotation.
        
          (7)   
        Moon orbit retains circular and lying on the ecliptic plane.
        
        For further details, an interested reader is addressed to the above quoted
        papers.
        
        Within the framework of the classical model, EMD at the age, t_a-t,
        e.g., <cit.><cit.> and related recession velocity read:
              a/a_a=[1-13/2(t_a-t)<ȧ_a>/a_a]^2/13  ; 
              ȧ/<ȧ_a>=[1-13/2(t_a-t)<ȧ_a>/a_a]^-11/13  ;
        
        and the inverse function is:
               t_a-t=2/13a_a/<ȧ_a>[1-(a/a_a)^
        13/2]  ;
        
        where extrapolation to merging configuration, a=0, yields:
               t_a-t^=2/13a_a/<ȧ_a>  ;
        
        which marks the end of the sequence.
        
        The substitution of Eq. (<ref>) into (<ref>) after little algebra
        yields:
              a/a_a=[1-t_a-t/t_a-t^]^
        2/13=[(t_a-t^)-(t_a-t)/t_a-t^]^2/13  ;
        
        in terms of the merging age, t_a-t^.
        
        Merging configurations must be conceived as purely fictitious (unless EMS took
        birth from fission), keeping in mind sequences of (quasi) equilibrium
        configurations necessarily end by attaining the Roche limit.   On the other
        hand, the merging age cannot occur in advance of EMS formation, as:
               t_a-t_i≤ t_a-t^  ;
        
        and sequences for which Eq. (<ref>) is violated must be disregarded.
        
        Using current values of EMD and mean lunar recession velocity, listed in Table
        <ref>, the merging age via Eq. (<ref>) reads:
               t_a-t^=1.516617  Gyr  ;
        
        which is at odds with Eq. (<ref>), keeping in mind
        t_a-t_i≈4.5 Gyr e.g., <cit.>.
        The above discrepancy could be avoided, provided mean lunar recession velocity
        was subsantially reduced in the past with respect to the present e.g.,
        <cit.><cit.>.
        
        More specifically, the simplest option is a single discontinuity at a transition age, t_a-t^†, where ȧ(t_a-t^†) instantaneously
        drops to ȧ^†.   Accordingly, Eq. (<ref>) is valid back to
        t_a-t^†, as:
              a^†/a_a=[1-13/2(t_a-t^†)<ȧ_a>/a_a]^2/13  ;
        
        in earlier ages, t_a-t>t_a-t^†, Eqs. (<ref>), (<ref>),
        (<ref>), translate into:
              a/a^†=[1-13/2(t^†-t)<ȧ^†>/a^†]^2/13  ; 
               t^†-t^=2/13a^†/<ȧ^†>  ; 
              a/a^†=[1-t^†-t/t^†-t^]^
        2/13=[(t^†-t^)-(t^†-t)/t^†-t^]^2/13  ;
        
        where t^†-t is the age since the transition age, t_a-t^†.
        Accordingly, free parameters of the model are the transition age and the
        merging age.
        
        Under the restrictive assumption LOD changes only via tidal friction, using
        angular momentum conservation and Kepler's third law yields <cit.>
        
        [The term, (a_0/a)^6 has to be replaced by (a/a_0)^6 in Eq. (2)
        therein, to be consistent with next Eq. (3).]:
        
              (a/a_a)^1/2+C_2/13(a/a_a)^
        13/2=-C_1Ω/Ω_a+C_3  ; 
               C_1=1/4.93 ; C_2=(0.46)^2=0.2116 ; C_3=1+C_1+C_2/13=
        1.219117 ;
        
        where Ω is Earth angular velocity, C_1 is the current ratio of Earth
        spin angular momentum to Moon orbital angular momentum, C_2 is the current
        ratio of solar to lunar retarding torques acting on Earth, and C_3 is an
        integration constant e.g., <cit.>.
        
        In terms of LOD, T=2π/Ω, Eq. (<ref>) after some algebra reads:
              T/T_a=C_1{C_3-(a/a_a)^1/2[1+C_2/13(a/a_a)^6]}^-1  ;
        
        and LOD variation rate after some algebra can be inferred from
        Eq. (<ref>) as:
              Ṫ/T_a=1/21/C_1T^2/T_a^2(a/a_a)^-1/2ȧ/a_a[1+C_2(a/a_a)^6]
          ;
        
        where the current value is:
              Ṫ_a/T_a=1/21/C_1ȧ_a/a_a(1+C_2)  ;
        
        finally, dividing each side of Eq. (<ref>) by its counterpart of
        Eq. (<ref>) yields:
              Ṫ/Ṫ_a(T/T_a)^-2=ȧ/ȧ_a(a/a_a)^-1/21+C_2(a/a_a)^6/1+C_2  ;
        
        where ȧ_a=<ȧ_a> for sake of brevity.
        
        With regard to the transition age, t_a-t^†, Eqs. (<ref>) and
        (<ref>) after some algebra translate into:
              T/T^†=C_1^'{C_3^'-(a/a^†)^1/2[1+C_2^'/13(a/a^†)^6
        ]}^-1  ; 
              Ṫ/Ṫ^†(T/T^†)^-2=ȧ/ȧ^†(a/a^†)^-1/21+C_2
        (a/a^†)^6/1+C_2  ; 
               C_1^'=C_1T_a/T^†(a^†/a_a)^1/2 ; C_2^'=C_2(a^†/a_a)^6 ; C_3^'=C_3(
        a^†/a_a)^-1/2 ;
        
        where Eqs. (<ref>) and (<ref>) retain the formal expression of
        Eqs. (<ref>) and (<ref>), respectively, regardless of the
        transition age, provided related constants are redefined via
        Eq. (<ref>).
        
        In the special case of merging age, a=0, Eq. (<ref>) reduces to:
              T^/T_a=C_1/C_3=0.1663826  ;
        
        accordingly, LOD (under the influence of tidal friction) attains a minimum
        value as:
               T≥ T^=0.1663826 T_a=3.982278  h  ;
        
        regardless of the sequence under consideration.
        
        
        
         §.§ Input parameters
        
        
        For dealing with a self-consistent set of input parameters, values are taken
        or inferred
        from an earlier investigation <cit.> as listed in Table <ref>.
        
        
        EMS evolution therein is considered within the framework of a new solution for
        astronomical computation of insolation quantities on Earth spanning from
        -0.25 Gyr to +0.25 Gyr.   Related polynomial approximations for mean EMD
        and LOD read <cit.>:
        eq:apoa      a/R_⊕=∑_k=0^4A_k(t_a-t/ Gyr)^k  ; 
        eq:apob       A_0=a_a/R_⊕=60.142611 ;   A_1=-6.100887 ;  
        A_2=-2.709407 ; 
        eq:apoc       A_3=-1.366779 ;   A_4=-1.484062 ;
        eq:Tpoa      T/ h=∑_k=0^4B_k(t_a-t/ Gyr)^k  ; 
        eq:Tpob       B_0=T_a/ h=23.934468 ;   B_1=-7.432167 ;  
        B_2=-0.727046 ; 
        eq:Tpoc       B_3=-0.409572 ;   B_4=-0.589692 ;
        
        where a appears in place of <a> for sake of brevity and the terms related
        to k>4 are negligible in both cases.
        
        In particular, current mean lunar velocity recession and LOD variation rate
        are inferred
        from Eqs. (<ref>) and (<ref>), respectively, as listed in
        Table <ref>.   While the former result is consistent with the current
        mean lunar velocity recession inferred from lunar laser ranging,
        <ȧ_a>=(3.82∓0.07) cm/y <cit.>, the
        latter substantially exceeds LOD variation rates mentioned in different
        investigations e.g., Ṫ_a=(2.3∓0.1) ms/cy <cit.><cit.>;
        Ṫ_a=(2.50∓0.01) ms/cy <cit.>.
        
        A comparison between numerical integration and analytical approximation
        discloses the occurrence of short-period (Δ t≈10^-3 Gyr)
        EMD fluctuations (Δ a/R_⊕≈∓0.5) around the local mean
        value
        <cit.>.   Accordingly, EMD evolution lies within a band, centered on
        the local mean value, where the vertical width equals about one Earth radius.
        A choice of input parameters as listed in Table <ref> leaves two
        free parameters, namely the transition age, t_a-t^†, and the merging
        age, t_a-t^.
        
        Computer runs were performed for the following cases: (a)
        (t_a-t^†)/ Gyr=0.57; (t_a-t^)/ Gyr=1.516617 (no
        discontinuity in <ȧ>), 2.0, 2.5, ..., 4.5, and (b)
        (t_a-t^†)/ Gyr=0.25;
        (t_a-t^)/ Gyr=1.516617 (no discontinuity in <ȧ>),
        2.0, 2.5, ..., 4.5, 5.0, 6.0, 7.0, 10.0.   More specifically, a discontinuity
        in lunar recession velocity takes place (more or less) at the transition
        between Proterozoic and Phanerozoic in the former alternative, and at
        formation and deformation of Pangaea supercontinent in the latter alternative
        e.g., <cit.><cit.>.
        
        
        
        § RESULTS
        
        
        With regard to the past 0.25 Gyr, 0≤(t_a-t)/ Gyr≤0.25, 
        predicted mean EMD (in Earth radii), a/R_⊕, vs age,
        t_a-t, according to Eq. (<ref>), is shown in Fig. <ref> as a
        dashed line e.g., <cit.><cit.>.
        
        
        By comparison, mean EMD inferred from astronomical computation within the
        framework of a new solution for the insolation quantities on Earth, according
        to Eq. (<ref>), is shown in Fig. <ref> as a full line inside
        a band <cit.>.   The band vertical width (Δ a/R_⊕≈1)
        is owing to short-period (Δ t≈10^-3 Gyr) EMD fluctuations
        <cit.>.   An inspection of Fig. <ref> discloses
        Eq. (<ref>) is in close agreement with its counterpart,
        Eq. (<ref>), related to a more sophisticated model <cit.>.
        
        Dotted straight lines define a special point, (0.192150014, 58.85857007), on
        the central full curve, for which
        the ratio of mean EMD to lunar diameter reads a/(2R_◯)=108.   The
        occurrence of short-period fluctuations implies the above value can be
        attained within a range, 0.12(t_a-t)/ Gyr0.25.
        
        Predicted LOD due to tidal friction, T/h, vs age, (t_a-t)/Gyr, according
        to Eq. (<ref>), is shown in Fig. <ref> as a dashed line.
        
        
        By comparison, LOD inferred from astronomical computation within the
        framework of a new solution for the insolation quantities on Earth, according
        to Eq. (<ref>), is shown in Fig. <ref> as a full line
        <cit.>.   An inspection of Fig. <ref> discloses
        Eq. (<ref>) is in close agreement with its counterpart,
        Eq. (<ref>), related to a more sophisticated model <cit.>.
        
        Dotted straight lines define a special point, (0.192150014, 22.47646196), on
        the full curve, for which
        the ratio of mean EMD to lunar diameter reads a/(2R_◯)=108.   
        
        The dot-dashed line is a linear interpolation from paleontological data,
        T/h=T_a/h-4.98(t_a-t)/Gyr, shown in earlier investigations
        <cit.><cit.>.   A nonlinear trend also inferred from
        paleontological data <cit.><cit.> is shown as squares.
        
        Concerning whole EMS evolution, assumed to start from (t_a-t_i)/Gyr=4.57,
        and a transition age, (t_a-t^†)/Gyr=0.57, predicted EMD (in Earth
        radii), a/R_⊕, vs age, (t_a-t)/Gyr, is shown in Fig. <ref>
        as full lines
        for selected merging ages, (t_a-t^), or selected mean lunar
        recession velocities just after the transition age, <ȧ^†>, via
        Eq. (<ref>), as listed in Table <ref>.
        
        
        
        
        All features appearing in Fig. <ref>
        are included within 0≤(t_a-t)/Gyr≤0.25, but dotted straight lines
        define a special point, (0.198499271, 58.85857007), on the dashed curve
        therein, for which the ratio of mean lunar distance to lunar diameter reads
        a/(2R_◯)=108.   Dashed vertical straight lines mark the selected
        transition age, (t_a-t^†)/Gyr=0.57 (left) and the assumed EMS age,
        (t_a-t_i)/Gyr=4.57 (right).   The Roche limit,
        a_ R/R_⊕=2.898518, is defined by a horizontal straight line.
        Values inferred from paleontological data from earlier investigations
        <cit.> are plotted as diamonds, related to different authors when
        appropriate <cit.>.   Source of data (from the left to the
        right) are: ER, BC (lower value preferred), WW (upper value preferred)
        <cit.>.
        
        Predicted LOD due to tidal friction, T/h, vs age, (t_a-t)/Gyr, according
        to Eq. (<ref>), is shown in Fig. <ref> as full lines
        for a transition age, (t_a-t^†)/Gyr=0.57, and
        selected merging ages, (t_a-t^), or selected mean lunar
        recession velocities just after the transition age, <ȧ^†>, via
        Eq. (<ref>), as listed in Table <ref>.
        
        
        All features appearing in Fig. <ref>
        are included within 0≤(t_a-t)/Gyr≤0.25, but dotted straight lines
        define a special point, (0.198499271, 22.50719109), on the dashed curve
        therein, for which the ratio of mean lunar distance to lunar diameter reads
        a/(2R_◯)=108.
        
        Dashed vertical straight lines mark the selected
        transition age, (t_a-t^†)/Gyr=0.57 (left) and the assumed EMS age,
        (t_a-t_i)/Gyr=4.57 (right).   Symbol captions are as in
        Fig. <ref> (squares) and Fig. <ref> (diamonds).   Dashed
        horizontal straight lines mark the Roche limit,
        T_ R/ h=6.948719, the merging limit,
        T^/ h=3.982278, and the Earth Poincaré limit,
        T_ P/ h=1.149728.
        
        Dot-dashed lines are linear interpolations from paleontological data,
        T/h=23.934468-4.98(t_a-t)/Gyr, 0≤(t_a-t)/Gyr ≤0.64, and
        T/h=21.368068-0.97(t_a-t)/Gyr, 0.64≤(t_a-t)/Gyr ≤2.50<cit.><cit.>, extended back to (t_a-t_i)/Gyr=4.57.
        
        An inspection of Figs. <ref> and <ref> discloses the most
        accurately inferred ER value <cit.> cannot be fitted to model curves
        even in the extreme case of no lunar recession after transition age (full
        horizontal straight line).   Clearly the situation is reversed if the
        transition age is closer to the present.   Then a later transition age,
        arbitrarily chosen as t_a-t^†=0.25 Gyr, has to be considered.   The
        results are plotted in Figs. <ref> and <ref>, where captions
        are the same as in Figs. <ref> and <ref>, respectively.
        
        Curves related to very early merging ages,
        5(t_a-t^)/Gyr 10, are found to be consistent with ER
        and preferred WW value.   The related range of mean lunar recession velocity
        just after the transition age, via Table <ref> reads
        1.21<ȧ^†>/(cm/y) 0.60, about 3-6 times lower than
        the current value.
        
        
        
        
        
        Curves within a narrower range, 5(t_a-t^)/Gyr 7, are
        found to be
        consistent with both ER, preferred WW and unpreferred BC value <cit.>,
        or 1.2<ȧ^†>/(cm/y) 0.89 via Table <ref>,
        where <ȧ^†>/(cm/y) ≈4.5 just before transition.
        
        Further restriction arises in dealing with LOD, Fig. <ref>, where
        5(t_a-t^)/Gyr 6 is found following a similar
        procedure, or 1.2<ȧ^†>/(cm/y) 0.99, which could
        be shifted downwards if processes different from solid Earth + ocean tidal
        friction are effective in lowering LOD.
        
        
        
        § DISCUSSION
        
        Results from models of lunar recession need some caution, in the sense related
        computer runs start at present and go back into the past.   A necessary
        restrictive assumption is that minor mergers and close encounters due to
        large-mass asteroids had
        negligible effects on EMS evolution.   Models implying merging ages in advance
        with respect to formation age, t_a-t^<t_a-t_i≈4.5 Gyr, must
        be disregarded, which is the case of continuous mean lunar recession
        velocity, as shown in Figs. <ref> and <ref>.   Accordingly, key
        input parameters are the merging age and the transition age, t_a-t^†,
        where mean lunar recession velocity is suddenly reduced.
        
        Remaining input parameters are taken or inferred from a solution for
        astronomical computation of insolation quantities on Earth within the range,
        -0.25≤(t_a-t)/ Gyr≤+0.25<cit.>, to which the prediction of
        the current model <cit.><cit.> satisfactorily fit, as shown in
        Figs. <ref> and <ref>.   In particular, input current mean
        lunar recession velocity and LOD variation read <ȧ_a>=3.89 cm/y and
        Ṫ_a=2.68 ms/cy, respectively <cit.>, against
        <ȧ_a>=(3.82∓0.07) cm/y inferred from lunar laser ranging experiment
        <cit.> and Ṫ_a=(2.50∓0.01) ms/cy determined from solid Earth +
        ocean tidal friction <cit.>.
        
        Predicting LOD implies further problems, as lunar recession occurs only via
        tidal friction while LOD depends, in addition, on atmospheric tides and
        nontidal processes e.g., <cit.><cit.><cit.>.   The current
        model works in presence of tidal friction only, which implies a LOD lower
        limit, keeping in mind the above mentioned additional effects act in the
        opposite direction i.e. increasing Earth spin.   On the other hand, the model
        can be extended to the special case where the contribution from atmospheric
        tides and nontidal processes is time independent i.e. related LOD variation
        maintains the current value,
        ΔṪ_a=(-0.10-0.55) ms/cy =-0.65 ms/cy <cit.>.
        
        A comparison between model predictions and paleontological data inferred from
        fossil corals and molluscs e.g.,
        <cit.><cit.><cit.><cit.><cit.><cit.><cit.>
        can be made in a twofold manner, according if lunar recession or LOD are
        involved.
        
        With regard to lunar recession, empirical values are inferred from the
        following formations: WW <cit.><cit.><cit.>, BC
        <cit.><cit.>, ER
        <cit.><cit.><cit.><cit.><cit.><cit.><cit.>,
        where a collection appears in a later investigation <cit.>.   For a
        transition age fixed at the late Precambrian, t_a-t^†=0.57 Gyr, even
        if no lunar recession occurs at earlier ages, predicted EMD cannot fit to ER
        and WW data, while cases where the merging age is not in advance with respect
        to the formation age, t_a-t^≥ t_a-t_i≈4.5 Gyr, are
        marginally consistent with BC data, as shown in Fig. <ref>.
        
        Conversely, a transition age selected on the late Permian,
        t_a-t^†=0.25 Gyr, yields acceptable cases,
        5≤(t_a-t^)/ Gyr ≤7, where predicted EMD fits to ER, BC
        (unpreferred value) and WW (preferred value) data, as shown in
        Fig. <ref>.   In addition, acceptable cases cannot fit to BC
        (preferred value) data, which implies at least one model assumption has to be
        released, provided data under discussion are correct, namely mean lunar
        recession velocity underwent several (instead of a single) discontinuities in
        the past.
        
        The above mentioned discrepancy could be lowered by the occurrence of
        short-period (Δ t≈10^-3 Gyr) EMD fluctuations
        (∓Δ a≈0.5R_⊕), which were established back to 0.25 Gyr via
        astronomical computations <cit.>.   If fluctuations of the kind
        considered took place back to 2.50 Gyr, it cannot be excluded EMD inferred
        from ER and WW data relate to a fluctuation maximum i.e. in excess of about
        0.5 R_⊕ with respect to related mean EMD, and EMD inferred from BC data
        relate to a fluctuation minimum i.e. in defect of about 0.5 R_⊕ with
        respect to related mean EMD.   Accordingly, the preferred EMD inferred from
        BC data <cit.> should be incremented by about 1R_⊕, passing from
        (54.7∓0.7)R_⊕ to (55.7∓0.7)R_⊕, which could be consistent
        with the unpreferred EMD inferred from BC data, equal to 57.1R_⊕ where
        the error is not reported <cit.>.
        
        Earlier applications of the current model yield similar results e.g.,
        <cit.><cit.>, regardless of (slightly different) values of input
        parameters.   The current choice, listed in Table <ref>, makes a set of
        homogeneous data from a single source <cit.>, which ensures consistency
        with astronomical computations, as shown in Figs. <ref> and
        <ref>.
        
        With regard to LOD, in addition to ER, BC, and WW data, different linear
        interpolations are available for Proterozoic and Phanerozoic e.g.,
        <cit.><cit.>, as:
        eq:lipa       T=T_a-4.98(t_a-t)  ;    0≤ t_a-t≤ t_a-t^∗  ; 
        eq:lipb       T=T_a^'-0.97(t_a-t)  ;     t_a-t^∗≤ t_a-t≤ t_a-t_i
        
        where T_a=23.934468 h is the current (2000.0 JC date) LOD <cit.>,
        T is the LOD at the age, t_a-t, and t_a-t^∗=0.64 Gyr.
        
        The intercept, T_a^', can be determined keeping in mind
        Eqs. (<ref>) and (<ref>) both hold at the intersection point
        of related straight lines.   The result is:
               T_a^'=T_a-(4.98-0.97)(t_a-t^∗)=21.368068  h  ;
        
        and, in addition, T^∗=T_a-4.98(t_a-t^∗)=20.747268  h via
        Eq. (<ref>).
        
        Predicted LOD agrees to an acceptable extent with its counterpart inferred
        from astronomical computations <cit.>, with a maximum discrepancy of
        12 m at t_a-t=0.25 Gyr.   The contrary holds with respect to inferred
        (from paleontological data) LOD, Eq. (<ref>), where the maximum
        discrepancy amounts to 30 m at the same age, which is worsened considering a
        nonlinear trend, as shown in Fig. <ref>.
        
        This is why predicted LOD
        is affected only by tidal friction in models under discussion
        <cit.><cit.><cit.>, while inferred (from paleontological
        data) LOD also includes atmospheric tides and nontidal processes such as
        mantle convection and iron macrodiffusion, which act in opposite sense with
        respect to tidal friction e.g.,
        <cit.><cit.><cit.><cit.>.
        
        More specifically, the contribution from total solid Earth + ocean to current
        LOD variation is evaluated as (2.50∓0.01) ms/cy, and the contribution
        from atmospheric tides and nontidal processes as (-0.10∓0.01) ms/cy and
        (-0.55∓0.01) ms/cy, respectively <cit.>.   Then the global
        contribution to current LOD variation amounts to (1.85∓0.01) ms/cy.
        Accordingly, the input current LOD variation, Ṫ_a=2.68 ms/cy, relates
        to the contribution from total solid Earth + ocean, while empirical LOD
        variation inferred from linear interpolation,
        Ṫ_a=4.98 h/Gyr ≈1.79 ms/cy, relates to the global
        contribution.
        
        Under the restrictive assumption of time independent contribution from
        atmospheric tides and nontidal processes, Δ T/(t_a-t)= const, using the
        above mentioned values yields Δ T/(t_a-t)=0.65 ms/cy, or:
              Δ T/ h=65/36t_a-t/ Gyr  ;
        
        where T+Δ T is the LOD at the age, t_a-t, due to total solid Earth +
        ocean tidal friction (T) with the addition of atmospheric tides and
        nontidal processes (Δ T).   Related explicit form via 
        Eqs. (<ref>) and (<ref>) reads:
              T+Δ T/T_a=C_1{C_3-(a/a_a)^1/2[
        1+C_2/13(a/a_a)^6]}^-1+65/36t_a-t/T_a h/ Gyr  ;  
        which, in the special case of merging age, a=0, T=T^, reduces to:
               T^+Δ T^=C_1/C_3T_a+65/36
        (t_a-t^)=3.982659  h+65/36(t_a-t^)
         h/ Gyr  ;
        
        that is a straight line in the variables, T^ and t_a-t^.
        
        Turning to the general case, the curve described by  
        Eq. (<ref>) is plotted in Fig. <ref> as a dotted line, which
        is satisfactorily close to the interpolation line (dot-dashed) inferred from
        paleontological data, Eq. (<ref>).   Fitting to an inferred nonlinear
        trend <cit.><cit.>, plotted in Fig. <ref> as squares,
        would need several (instead of a single) discontinuities in LOD variation,
        taking due account of, say, Pangaea formation and deformation e.g.,
        <cit.><cit.>.
        
        An extension of inferred (from paleontological data) interpolation line to
        Archean and, fictiously, to EMS formation age, could be illustrative for
        comparison with predicted LOD, as shown in Figs. <ref> and
        <ref>.   To this aim, a choice must be done of what is the better
        empirical evidence among ER, BC, WW data <cit.> on one hand, and
        inferred linear trend considering the above mentioned data as representative
        as the remaining ones <cit.> on the other hand.   In fact, inferred
        linear interpolation lies below ER value and more or less in the middle of
        preferred and unpreferred BC and WW values.   Both alternatives shall be
        considered in the following.
        
        If the better empirical evidence of LOD relates to ER, BC and WW data, current
        models yield acceptable fits with regard to (i) late transition ages, and
        (ii) unpreferred BC value, as shown in Fig. <ref> for curves related
        to merging ages, 5≤(t_a-t^)/ Gyr ≤6.
        
        If the better empirical evidence of LOD relates to inferred linear
        interpolations, current models provide acceptable fits for late transition
        ages, as shown in Fig. <ref> for curves related to merging ages,
        (t_a-t^)/Gyr ≈6.
        
        In addition, errors on age of ER, BC and WW formation are not
        available e.g., <cit.> and for this reason an uncertainty of
        ∓0.05 Gyr has arbitrarily been assumed in plotting related values, even
        if larger uncertainties are probably expected.
        
        Keeping into due account
        the (time independent) effect of atmospheric tides and nontidal processes, the
        situation is worsened as shown in Figs. <ref> and <ref>, in the
        sense the merging age would be in advance with respect to the formation age.
        To avoid this inconsistency, the above mentioned effect should increase going
        back in time.
        
        
        
        
        
        In conclusion, predicted LOD satisfactorily fits to both ER, BC, WW data and
        inferred linear interpolations with regard to late transition ages,
        t_a-t^†≈0.25 Gyr, and merging ages,
        t_a-t^≈6 Gyr, leaving aside the effect of atmospheric tides
        and nontidal processes.   Suitable choices of the above mentioned input
        parameters could better reproduce empirical values, but the main result is
        current models are consistent with the data in spite of restrictive
        assumptions.
        
        On the other hand, a linear LOD-age relation is a zeroth-order approximation,
        as shown in Figs. <ref>, <ref>, <ref>, <ref>,
        <ref>, by comparison with data from earlier investigations
        <cit.><cit.>,
        plotted as squares.   A nonlinear trend could be owing to e.g., occurrence and
        persistence of a supercontinent associated to a deep superocean
        <cit.><cit.>.   Even in this context, the current model could fit
        to the data if several (instead of a single) transition ages are considered,
        where mean lunar recession velocity suddenly drops to a lower value.   To
        this aim, the effect (impacts or close encounters) of large-mass asteroids 
        should be analysed in detail.
        
        
        
        § CONCLUSION
        
        The evolution of Earth-Moon system (EMS) relates to several different
        disciplines, such as Astronomy, Geodesy, Geophysics, Oceanology, Geology,
        Paleontology, where comparison between theoretical predictions and results
        from data collections could imply constraints within all the involved fields.
        In this view, a classical model <cit.><cit.><cit.> has been
        reviewed and updated to many extents.
        
        First, a homogeneous set of input parameters has been taken from earlier
        astronomical computations on insolation quantities on Earth spanning from
        -0.25 Gyr to +0.25 Gyr <cit.>.
        
        Second, predicted Earth-Moon distance (EMD) for assigned transition age,
        t_a-t^†, and merging age, t_a-t^, has been considered
        keeping in mind acceptable cases necessarily imply merging age in advance
        with respect to formation age, t_a-t_i.
        
        Third, the presence of short-period (Δ t≈10^-3 Gyr) EMD
        fluctuations (Δ a≈∓0.5R_⊕) from astronomical
        computations <cit.>, if typical of the whole evolution, implies
        uncertainty bands around a(t) curves of thickness about ∓0.5R_⊕
        and values inferred from paleontological data affected by an additional error,
        Δ a≈∓0.5R_⊕, if related to mean (over a typical
        fluctuation period) EMD.
        
        The main results of the present paper can be summarized as follows.
        
          (1)  
        Predicted EMD and length of day (LOD) slightly overstimate their counterparts
        from astronomical computation of insolation quantities on Earth back to
        0.25 Gyr <cit.>.
        
          (2)  
        Predicted LOD, when a time independent effect via atmospheric tides and
        nontidal processes is considered <cit.>, slightly understimates a linear
        interpolation from paleontological data related to Phanerozoic era
        <cit.><cit.>.
        
          (3) 
        For assigned transition age and merging age, predicted EMD and LOD cannot fit
        to their counterparts inferred from ER, BC and WW data, unless unpreferred
        instead of preferred values <cit.> are considered for BC data.   Related
        difference in EMD appears comparable to amplitude in short-period fluctuations
        from astronomical computations during the late Phanerozoic <cit.>.
        Accordingly, a transition age, t_a-t^†=0.25 Gyr and a merging age,
        t_a-t^=6 Gyr, yield an acceptable fit.   The situation is reversed
        if a time independent effect via atmospheric tides and nontidal processes is
        considered, in the sense that fitting curves relate to merging age in  advance
        with respect to formation age.
        
          (4) 
        For assigned transition age, t_a-t^†=0.25 Gyr and merging age,
        t_a-t^=6 Gyr, predicted LOD is consistent with inferred linear
        interpolation to paleontological data <cit.><cit.>. The  situation
        is reversed  if a time independent effect via atmospheric tides and nontidal
        processes is considered, in the sense that fitting curves relate to merging
        age in  advance with respect to formation age.
        
        Even if LOD variation was not linear in time e.g., <cit.><cit.>,
        the current model remains valid as a zero-th order approximation.   Further
        refinement would be desirable in dealing with paleontological data, in
        connection with error evaluation.
        
        
        
        § ACKNOWLEDGEMENTS
        
        
        Thanks are due to S. Lambert for fruitful e-mail correspondance.
        
        asp14 E. Asphaug,
                Impact Origin of the Moon?,
                Annual Review of Earth and Planet Sciences 42 (2014), 551-578.
        bro84 P. Brosche,  
                Tidal friction in the Earth-Moon system,
                Philosophical transactions of the Royal Society of London,
                Ser. A 313 (1984), 71-75.
        dea11 C. Denis, K.R. Rybicki, A.A. Schreider, S. Tomecka-Suchoń, 
                P. Varga,   
                Length of the day and evolution of the Earth's core in the geological
                past,
                Astronomische Nachrichten 332 (2011), 24-35.
        deu90 F.-L. Deubner,
                     Discussion on Late Precambrian tidal rhythmites in South Australia and 
                    the history of the Earth's rotation,
                    Journal of the Geological Society of  London 147 (1990), 1083-1084.
        dia94 J.O. Dickey, P.L. Bender, J.E. Faller, X.X. Newhall, R.L.
                Ricklefs, J.G. Ries, P.J. Shelus, C. Veillet, A.L. Whipple, J.R.
                Wiant, et al,
                Lunar Laser Ranging: A Continuing Legacy of the Apollo Program,
                Science 265 (1994), 482-490.
        jea29 J. Jeans, 
                     Astronomy and Cosmogony (1929), 
                     Dover Publications, New York, 1961.
        kop74 Z. Kopal,
                The Moon in the Post-Apollo Era, 
                D. Reidel Publishing Company, Dordrecht,  1974.
        lam80 K. Lambeck,  
                The Eart's Variable Rotation, 
                Cambridge University Press, New York, 1980.
        laa04 J. Laskar, P. Robutel, F. Joutel, M. Gastineau, A.C.M.
                Correia, B. Levrard,  
                A long-term numerical solution for the insolation quantities of the
                Earth,
                Astronomy and Astrophysics 428 (2004), 261-285.
        maj95 E. Majewski,
                Thermodynamic Model of the Core-mantle Boundary,
               Publications of the Institute of Geophysics, Polish Academy of Sciences A 25 (1995), 1-59.  
        mal09 P.M. Mathews, S.B. Lambert,
                 Effect of mantle and ocean tides on the Earth's rotation rate,
                 Astronomy and Astrophysics 493 (2009), 325-330.
        scr78 C.T. Scrutton,  
                Periodic growth features in fossil organisms and the length of the day
                and month,
                in Tidal Friction and the Earth's Rotation, edited by P. Brosche and
                J. Sündermann, pp. 154-196, Springer-Verlag, New York, 1978. 
        soa96a C.P. Sonett, E.P. Kvale,  A. Zakharian, M.A. Chan, T.M.
                Demko, 
                Late Proterozoic and Palaeozoic tides, retreat of the Moon, and
                rotation of the Earth, 
                Science 273 (1996), 100-104.
        soa96b C.P. Sonett, A. Zakharian, E.P.  Kvale, 
                Ancient tides and length of day: Correction, 
                Science 274 (1996), 1068-1069. 
        sac98 C.P. Sonett, M.A. Chan, 
                Neoproterozoic Earth-Moon dynamics: Rework of the 900 Ma Big
                Cottonwood Canyon tidal rhythmites, 
                Geophysical Research Letters 25 (1998), 539-542.
        vaa98 P. Varga, C. Denis, T. Varga,
                Tidal Friction and its Consequences in 
                Palageodesy, in the Gravity Field
                Variations and in Tectonics,
                Journal of Geodynamics 25 (1998), 61-84.
        val06 P. Varga, K.R. Rybicki, C. Denis,  
                Comment on the paper "Fast tidal cycling and the origin of life" by
                Richard Lathe, 
                Icarus, 180 (2006), 274-276.
          wil89a G.E. Williams,  
                Late Precambrian tidal rhythmites in South Australia and the history
                of the Earth's rotation, 
                Journal of the Geological Society of  London 146 (1989), 97-111. 
        wil89b G.E. Williams, 
                Precambrian tidal sedimentary cycles and Earth's paleorotation, 
                Eos Transactions AGU 70 (1989), 33 and 40-41.
        wil89c G.E. Williams,  
                Tidal rhythmites: Geochronometers for the ancient Earth-Moon system, 
                Episodes 12 (1989), 162-171.
        wil90 G.E. Williams,  
                Key to the history of the Earth's rotation and the lunar orbit, 
                Journal of Physics of Earth 38 (1990), 475-491.
        wil91 G.E. Williams,  
                Upper Proterozoic tidal rhythmites, South Australia: Sedimentary features,
                deposition, and implications for the earth's paleorotation, 
                in Clastic Tidal Sedimentology, edited by D.  G. Smith G.E. Reinson, B.A. 
                Zaitlan, R.A. Rahmani, 
               Canadian Society of Petroleum Geologists Memoir 16 (1991), 161-177.
        wil94 G.E. Williams,  
                History of Earth's rotation and the Moon's orbit: A  key datum from Precambrian 
                tidal strata in Australia, 
               Australian Journal of Astronomy, 5 (1994),  135-147. 
        wil97 G.E. Williams,  
                Precambrian length of day and the validity of tidal rhythmite paleotidal values, 
                Geophysical Research Letters, 24 (1997), 421-424. 
        wil00 G.E. Williams,  
                Geological Constraints on the Precambrian History of Earth's Rotation
                and the Moon's Orbit,
                Reviews of Geophysics 38 (2000), 37-59.
        waz86 J.C.G. Walker, K.J. Zahnle,
                Lunar nodal tide and distance to the moon during the Precambrian,
                Nature 320 (1986), 600-602. 
        
        
        
        
        
        
        § APPENDIX
        
        
        § 108 AND 360: AN ELEGANT FORMULATION
        
        An elegant formulation of 108 and 360, from a purely numerical standpoint i.e.
        with no connection with physics, astronomy, geometry, reads:
               108=1^1·2^2·3^3  ; 
               360=1^5·2^3·3^2·5^1=2^3·3^2·(2+3)  ;
        
        the ratio of which is:
              108/360=3/10  ;
        
        involving both 3 and 10, highly relevant in ancient books of wisdom.
        
        
        
        § LENGTH VALUES
        
        Length values used for calculations are listed in Table <ref> together
        with related symbol and meaning.   Lengths are expressed in km.   Solar radius
        value reconciles results from different methods <cit.>.
