Energy production in varying α theories
Lucila Kraiselburd, Marcelo Miller Bertolami, Pablo Sisterna, Héctor Vucetich
aa r X i v : . [ g r- q c ] A p r Astronomy & Astrophysics manuscript no. 15970 c (cid:13)
ESO 2018October 19, 2018
Energy production in varying α theories Lucila Kraiselburd ⋆ , Marcelo Miller Bertolami , ⋆⋆ , Pablo Sisterna , and H´ector Vucetich Grupo de Gravitaci´on, Astrof´ısica y Cosmolog´ıa, Facultad de Ciencias Astron´omicas y Geof´ısicas, UniversidadNacional de La Plata, Paseo del Bosque S/N, cp 1900 La Plata, Argentina Instituto de Astrof´ısica La Plata Grupo de Evoluci´on estelar y pulsaciones, Facultad de Ciencias Astron´omicas y Geof´ısicas, Universidad Nacional deLa Plata, Paseo del Bosque S/N, cp 1900 La Plata, Argentina Facultad de Ciencias Exactas y Naturales, Universidad Nacional de Mar del Plata, Funes 3350, cp 7600 Mar delPlata, Argentinae-mail: lkrai, mmiller, [email protected], [email protected]
Preprint online version: October 19, 2018
Abstract
Aims.
On the basis the theoretical model proposed by Bekenstein for α ’s variation, we analyze the equations thatdescribe the energy exchange between matter and both the electromagnetic and the scalar fields. Methods.
We determine how the energy flow of the material is modified by the presence of a scalar field. We estimatethe total magnetic energy of matter from the “sum rules techniques”. We compare the results with data obtained fromthe thermal evolution of the Earth and other planets.
Results.
We obtain stringent upper limits to the variations in α that are comparable with those obtained from atomicclock frequency variations. Conclusions.
Our constraints imply that the fundamental length scale of Bekenstein’s theory “ ℓ B ” cannot be larger thanPlanck’s length “ ℓ P ”. Key words.
Bekenstein’s model–planetary heat flux
1. Introduction
The time variation in the fine structure constant has beenstudied several times since first being proposed by Gamow(1967). Observational upper bounds on its time variation aswell as several theoretical frameworks that consider α as adynamical field have been published (an exhaustive list canbe found in (Landau 2002; Uzan 2003) and references therein). Although still disputed, the claim that α was smaller inthe past is an exciting perspective, (Murphy et al. 2003).Beckenstein’s theory (Bekenstein 1982), which is basedon a number of minimal hypothesis of highly accepted phys-ical principles, is in a sense representative of many low en-ergy theories inspired by grand unification schemes. In thiswork, we derive equations that govern the energy exchangebetween matter, the scalar field, and the electromagneticfield. Although we do not analyze the precise mechanism ofenergy release, we assume that the work done by the scalarfield is radiated away in an efficient way, as for the roto-chemical heating of neutron stars due to the spin downof the star (Reisenegger 1995; Fernandez & Reisenegger2005).In section 2, we briefly review Beckenstein’s theory, aswell as the cosmological time evolution of α that it predicts.In section 3, we derive a generalized version of the Poynting ⋆ fellow of CONICET ⋆⋆ member of the Carrera del Investigador Cient´ıfico yTecnol´ogico, CONICET theorem for the electromagnetic field, and from the conser-vation of the total energy-momentum tensor we find howthe energy flow of matter is modified by the scalar field. Insection 4, we discuss the magnetic energy of matter usinga simple nuclear model. In section 5, we study the thermalhistory of the Earth in the presence of Bekenstein’s scalarfield. We also describe in section 6 the results we obtainedfor the outer planets. Finally in section 7 we summarize ourconclusions.
2. Time variation of α in Bekenstein’s formalism We briefly review Bekenstein’s formalism and its predic-tion for the cosmological time variation of α . Although weconsider galactic as well as terrestrial phenomena, we cannevertheless confidently assume that they track the cosmo-logical evolution of α , (Shaw & Barrow 2006).Bekenstein (1982) proposes to modify Maxwell’s theoryby introducing a field ǫ that dynamically describes the vari-ation of α . The foundational hypothesis are the following(Bekenstein 1982; Landau 2002):1. The theory must reduce to Maxwell’s when α = Cte.2. The changes in α are dynamical (i.e. generated by adynamical field ǫ ).3. The dynamics of the electromagnetic field, as well asthe ǫ field can be obtained from a variational principle.4. The theory must be locally gauge invariant. Kraiselburd et al.: Variation in α and energy production
5. The theory must preserve causality.6. The action must be time reversal invariant.7. Planck’s scale ℓ P is the smallest length available in thetheory.8. Einstein’s equations describe gravitation.String theories and the like in which there are otherfundamental length scales, force us to set aside condition7. These hypothesis uniquely lead to the action S = S em + S ǫ + S m + S G , (1)where S em = − π Z F µν F µν √− gd x, (2) S ǫ = − ~ c ℓ B Z ǫ ,µ ǫ ,µ ǫ √− gd x, (3) S m and S G are the matter and gravitational field actions,respectively, ℓ B is the so-called Bekenstein’s fundamentallength, and the metric here is ( − , , , F µν = 1 ǫ [( ǫA ν ) ,µ − ( ǫA µ ) ,ν ] (4)and the second kind of local gauge invariance implies that ǫA ′ µ = ǫA µ + χ ,µ , (5) ∇ µ = ∂ µ − e ǫA µ , (6)as the gauge transformation and covariant derivative ofthe theory, respectively. The last equation defines the localvalue of the elementary electric charge (coupling constant) e ( r , t ) = e ǫ ( r , t ) , (7)that is ǫ = (cid:18) αα (cid:19) . (8)In what follows, we neglect the small spatial variations in α and focus on the cosmological variation, as we are interestedin any secular energy injection of the scalar field on a planetsuch as the Earth. In our approximate analysis it is alsoenough to work in a flat space-time.The field equations for the electromagnetic field and for ǫ are (cid:18) ǫ F µν (cid:19) ,ν = 4 πj µ , (9a) (cid:3) ln ǫ = ℓ B ~ c (cid:20) ǫ ∂σ∂ǫ − ǫj µ A µ + 14 π ( A µ F µν ) ,ν (cid:21) = ℓ B ~ c (cid:18) ǫ ∂σ∂ǫ − F µν F µν π (cid:19) , (9b)where j µ = P ( e /cγ ) u µ ( − g ) − / δ [ x i − x i ( τ )], u µi is an“standard estimate” of the 4th velocity of each particle ac-cording to the model, and σ is the energy density of matter,Bekenstein (1982). (cid:3) is the covariant flat d’Alambertian (cid:3) φ = φ ,µ,µ = η µν φ ,µ,ν . (10) A note regarding the matter Lagrangian is in order: in(Bekenstein 1982, 2002), Bekenstein represents matter asan ensemble of classical particles. However, wherever quan-tum phenomena become important, as in white dwarfs orcondensed matter physics, this is not a realistic description.It is also an inaccurate description at high energies (or onsmall length scales) because fermions have a “natural lengthscale”, the particle Compton wave-length λ C = ~ /mc ,which makes quite unrealistic any classical model at higherenergies. In particular, several conclusions of Bekenstein(2002) have to be reconsidered.In Bekenstein (1982), it is shown that the cosmologicalequation of motion for ǫ is ddt (cid:18) a ˙ ǫǫ (cid:19) = − a ℓ B ~ c (cid:20) ǫ ∂σ∂ǫ − π (cid:0) E − B (cid:1)(cid:21) . (11)In the non-relativistic regime, E ≫ B and σ ∝ ǫ , hence ddt (cid:18) a ˙ ǫǫ (cid:19) = − a ζ c ℓ B ~ c ρ m c , (12)where ρ m is the total rest-mass density of electromagnet-ically interacting matter and ζ c is a parameter describingits “electromagnetic content”, which is essentially the frac-tional contribution of the electromagnetic energy to the restmass. A first estimation according to Bekenstein (2002) is ζ c ∼ . × − . (13)Following the standard cosmological model, we assume thatdark matter is to be electromagnetically neutral.Given that ρ m ∝ a − , we can integrate Eq.(12) obtain-ing ˙ ǫǫ = − ζ c (cid:18) ℓ B c ~ (cid:19) ρ m ( t − t c ) , which can be written, using the usual cosmological nota-tion, as follows˙ ǫǫ = − ζ c π (cid:18) ℓ B ℓ P (cid:19) H Ω B (cid:20) a a ( t ) (cid:21) ( t − t c ) . (14)The primordial nucleosynthesis standard model tells us thatthe integration constant t c must be very small in ordernot to spoil the agreement between theory and observation.Using WMAP values, we obtain the prediction for ( ˙ α/α ) of (cid:18) ˙ αH α (cid:19) = 1 . × − (cid:18) ℓ B ℓ P (cid:19) . (15)Any measurement with a precision such as σ ( ˙ α/H α ) ∼ − is difficult to achieve, so the comparison between the-ory and experiment is a difficult task.The same arguments can be applied to many the-ories with varying α , such as Kaluza-Klein (Landau2002) or string spired theories such as Damour-Polyakov’s(Damour & Polyakov 1994a,b).
3. Energy transfer in Bekenstein’s formalism
We study how energy is injected and then released in vary-ing α theories, in order to look for observable consequences raiselburd et al.: Variation in α and energy production 3 in the emission of astrophysical as well as geophysical sys-tems. The energy momentum tensor in Bekenstein’s theoryis defined as usual to be T µν = 2 c δSδg µν . (16)Using c = 1, the electromagnetic contribution then has thesame form as in Maxwell’s theory T em µν = 14 π h F µλ F ν λ − g µν F λσ F λσ i , (17)the difference lying in the connection between the vectorpotential and the field given in Eq.(4).The energy-momentum tensor of the scalar field ǫ is T µνǫ = ~ ℓ B (cid:18) ǫ ,µ ǫ ,ν ǫ − g µν ǫ ,α ǫ ,α ǫ (cid:19) . (18)In what follows, we use the redefined field ψ = ln ǫ. (19)As we consider local phenomena, we can work in a lo-cally inertial coordinate system. We denote the “field partof the energy-momentum tensor” as the scalar plus electro-magnetic energy momentum tensor T µν f = T µν em + T µνǫ . (20)In terms of ψ and replacing g µν with η µν , we obtain T µν f = 14 π (cid:20) F µλ F νλ − η µν F λσ F λσ (cid:21) + ~ ℓ B (cid:18) ψ ,µ ψ ,ν − η µν ψ ,α ψ ,α (cid:19) . (21)The divergence of T f is T µν f ,ν = 14 π (cid:20) F µα,ν F ν α + F µα F να,ν − η µν F αβ F αβ,ν (cid:21) + ~ ℓ B (cid:0) ψ ,µ,ν ψ ,ν + ψ ,µ ψ ,ν,ν − η µν ψ ,α,ν ψ ,α (cid:1) . (22)The equations of motion Eqs.(9) are F αν ,ν = 4 πe ψ j α + ψ ,ν F αν , (23a) ψ ,ν,ν = (cid:3) ψ = ℓ B ~ (cid:18) ∂σ∂ψ − F µν F µν π (cid:19) , (23b)which can be used in Eq.(22) obtaining T µν f ,ν = 14 π [ F µα,ν F να − F µα (cid:0) πe ψ j α + ψ ,ν F αν (cid:1) − η µν F αβ F αβ,ν ]+ ~ ℓ B [ ψ ,µ,ν ψ ,ν + ψ ,µ ℓ B ~ (cid:18) ∂σ∂ψ − F µν F µν π (cid:19) − η µν ψ ,α,ν ψ ,α ] . (24)This expression can be simplified using the homoge-neous Maxwell equations F αβ,γ = − F βγ,α − F γα,β , (25) which cancels out the first bracket. The first and last termin the second bracket also cancel out, thus we obtain forEq.(24) the expression T µν f ,ν = − e ψ j α F µα + ψ ,ν (cid:18) η µν ∂σ∂ψ + T µν em − π η µν F αβ F αβ (cid:19) . (26)We add to both sides of the equation the divergenceof the energy momentum tensor of matter T µν m ,ν in orderto find the energy transfer (according to the hypothesis 8in Sect. 2, we assume that Einstein’s equations hold un-modified for the gravitational field and hence that the totalenergy momentum tensor is conserved) T µν f ,ν + T µν m ,ν = 0= T µν m ,ν − e ψ j α F µα + ψ ,ν (cid:18) η µν ∂σ∂ψ + T µν em − π η µν F αβ F αβ (cid:19) . (27)This equation explicitly shows the energy transfer from thefield ǫ to matter T µν m ,ν = e ψ j α F µα − ψ ,ν (cid:18) η µν ∂σ∂ψ + T µν em − π η µν F αβ F αβ (cid:19) , (28)which is the source of any observable effect. From ψ ,ν = ǫ ,ν ǫ = 12 α ,ν α , (29)we find that the “machian” contribution to energy transferis given by T µν m ,ν ( machian ) = 12 α ,ν α (cid:18) η µν ∂σ∂ψ + T µν em − η µν π F αβ F αβ (cid:19) . (30)Using Bekenstein’s notation, that is, if the time-space com-ponents of e ψ F µν are identified with E while space-spacecomponents are identified with B , the contribution thentakes the form T ν m ,ν ( machian ) = − ˙ ψ ∂σ∂ψ + e − ψ ∇ ψ. S + ˙ ψe − ψ ( B + E )8 π + e − ψ ˙ ψ π ( B − E )= − ˙ ψ ∂σ∂ψ + e − ψ ∇ ψ. S + ˙ ψe − ψ B π , (31)where S = E × B π . Then, the component 0 of Eq.(28) reads T ν m ,ν = j . E − e − ψ B ˙ ψ π − e − ψ ∇ ψ. S + ˙ ψ ∂σ∂ψ . (32)An implicit assumption of our previous analysis and al-gebra is the generalized Poynting theorem. In its standardversion, it involves only electromagnetic terms, while in ourcase it also involves the interaction between the electromag-netic and scalar fields given by T µν em ,ν = 14 π [ F µα,ν F να − F µα (cid:0) πe ψ j α + ψ ,ν F αν (cid:1) − η µν F αβ F αβ,ν ] . (33) Kraiselburd et al.: Variation in α and energy production Using again Eq.(25), T µνem,ν = − e ψ j α F µα + ψ ,ν (cid:20) F µα F να π − η µν F αβ F αβ π + η µν F αβ F αβ π (cid:21) , (34) T µνem,ν = − e ψ j α F µα + ψ ,ν ( T µνem + η µν F αβ F αβ π ) . (35)Then, T ρem,ρ = − E · j + e − ψ ( E + B )8 π ˙ ψ + e − ψ S . ∇ ψ − e − ψ ˙ ψ π ( B − E ) , (36) T em ρ,ρ = ∂u em ∂t + ∇ .e − ψ ( E × B π )= − E · j + e − ψ E π ˙ ψ + e − ψ S . ∇ ψ, (37)where T em , = ( ∂u em ) /∂t , the electromagnetic energy is u em = e − ψ ( E + B ) / (8 π ), T em i,i = ∇ .e − ψ ( E × B π ) = ∇ .e − ψ S , and S is the Poynting vector. We note that thisresult is independent of the details of the gravitational andmatter Lagrangians, as well as their interacting terms withthe electomagnetic field. In particular, it holds indepen-dently of the details of the interaction of matter with thescalar field. We recall that the usual interpretation of thefirst term in the right hand side of Eq.(37) is the work doneby the electromagnetic field on matter. In the same fash-ion, we may interpret the second and last term as the workdone by the electromagnetic field on the scalar field. Ananalog phenomenon would be that given by the work doneby an increasing Newton constant G on a planet augmentsits pressure and thus compresses it (Jofr´e et al. 2006).We estimate the electrostatic contribution to the matterenergy. In a non-relativistic system such as a light atom ornuclei, the electromagnetic energy is given by the electro-static field that satisfies the equation ∇ · E e − ψ = 4 πρ , (38)where ρ is the reference charge density. In the limit where α varies only cosmologically we have ∇ · E = 4 πe ψ ρ , (39)whose solution is E = e ψ E , (40)where E is the electrostatic reference field defined for e ψ =1. We can evaluate the electromagnetic energy density tobe u em = e − ψ ( B + E )8 π = e ψ u em , (41)and the temporal variation˙ u em = 2 ˙ ψu em + e ψ ˙ u em = ˙ αα u em + e ψ ˙ u em . (42)If there were no scalar injection of energy and ˙ u em ≈
0, the Poynting theorem Eq.(37) and the energy variationgiven by Eq.(42) would lead to 2 ˙ ψu em = 2 ˙ ψe − ψ ( B + E )8 π = − j · E + ˙ ψe − ψ E π (43)or j · E = − B π ˙ ψe − ψ . (44)As we consider phenomena where the motion of mat-ter is negligible, taking the first index as 0 is equivalentto projecting along the fluid four-velocity. In addition, thetotal time derivative d/dt = ∂/∂t + v . ∇ will be equal tothe partial time derivative ∂/∂t . In the general case whenthere is viscosity and heat transfer, the right-hand side canbe written, in the non-relativistic limit, as T ν m ,ν = ∂∂t ( 12 ρv + u ) + ∇ . [ ρ v ( 12 v + w ) − v .σ ′ + J ] , (45)where w is the specific enthalpy, u is the internal energydensity, J is the heat flux, which can generally be writ-ten as − κ ∇ T , T is the temperature and κ is the thermalconductivity. Finally, ( v .σ ′ ) k stands for v i σ ′ ik , where σ ′ isthe viscous stress tensor, (Landau & Lifschitz 1987). As westated above, we neglect the velocity of the fluid, so obtain T ν m ,ν = ∂u∂t + ∇ · J . (46)A note of caution regarding the internal energy is inorder. We understand, as usual, “internal energy” as theenergy that can be exchanged by the system in the pro-cesses considered (heat exchange, radiative transfer, etc.),which will differ from what we understand by “rest mass”,which is the “non-convertible energy”. If the scalar fieldcan change the effective electric charge, then it can alterthe electromagnetic contribution to the rest mass, and con-sequently, this contribution will be no longer “rest mass”,but “internal energy”.The time variation in the internal energy u will havetwo contributions: one corresponding to the cooling pro-cess ∂u∂t | cooling and another one related to the interactionwith the scalar field ∂σ µ ∂t . This last term accounts for thedependence of the bulk of matter on the scalar field, whichis mainly given by the electromagnetic contribution to thenuclear mass. Equation (32) then will finally read ∂u∂t | cooling + ∂σ µ ∂t + ∇ · J = − B π ˙ ψe − ψ − e − ψ B ˙ ψ π − e − ψ ∇ ψ. S − ˙ ψ ∂σ∂ψ . (47)Since the scalar field is space independent, and giventhat the electromagnetic energy of matter is mainly ac-counted for the nuclear content, we assume that the follow-ing condition ∂σ∂ψ − ∂σ µ ∂ψ ≈ ∂u∂t | cooling + ∇ · J ≈ − e − ψ B ˙ ψ π . (48)This equation becomes clearer if we make a trivialchange to produce ∇ · J = − e − ψ B ˙ ψ π − ∂u∂t | cooling , (49) raiselburd et al.: Variation in α and energy production 5 which clearly shows that besides the standard cooling mech-anism of the body, there is a contribution from the partialrelease of the magnetic energy injected by the scalar field.We define ε a = 2 e − ψ B ˙ ψM a π ≈ αα B πM a , (50)to be equal to twice the energy production per mass unit ofany material substance a (using the approximation, e − ψ → ψ << thecooling term is not modified by the scalar field . The rea-sons for this assumption are fold: 1) as we have just shown,the electrostatic energy “injected” by the scalar field re-mains within the bulk matter (the cancellation of termsoccurring as seen in Eq.(48)), and 2) the thermal evolutionshould not change given the high thermal conductivity ofthe Earth and white dwarfs considered in this work. Thus,we expect the magnetic energy excess to be radiated away,increasing the heat flux J as shown in Eq.(49).
4. The electromagnetic energy of matter
As we mentioned in the Sect. 3, the only “input” is that de-rived from the magnetic field. Stationary electric currentsgenerated by charged particles and their static magneticmoments, and quantum fluctuations of the number den-sity are responsible for the generation of magnetic fields inquantum mechanics. These contributions have been studiedand calculated by Haugan & Will (1977) and Will (1981)from a minimal nuclear shell model using the following anal-ysis (for more details see Kraiselburd & Vucetich (2011)).The total magnetic energy of the nucleus can be writtenas E m ≃ c X α Z d x d x ′ h | j ( x ) | α i · h α | j ( x ′ ) | i| x − x ′ | , (51)where α runs over a complete set of eigenstates of the nu-clear Hamiltonian H . Neglecting the momentum depen-dence of the nuclear potential and assuming a constantdensity within the nucleus, we obtain the result h | j ( x ) | α i · h α | j ( x ′ ) | i ≃ | d α | ~ E α V N cos θ, (52)where d α is the dipole density, V N = π R N is the nuclearvolume, and θ is the angle between ˆ x and ˆ x ′ . Hence, E m ≃ P a E α | d α | ~ c R d x d x ′ cos θ | x − x ′ | V N . (53)In the last equation, R d x d x ′ cos θ | x − x ′| V N is equal to R N , and thefirst term can be computed from the connection betweenthe strength function and the photoabsorption cross-section σ α = 4 π ~ c E α | d α | . (54)From this, we easily find that X a E α | d α | = ~ c π R Eσ ( E ) dE R σ ( E ) dE · Z σ ( E ) dE = ¯ E Z σ ( E ) dE, (55) where ¯ E ∼
25 MeV is the mean absorption energy, which isroughly independent of A (number of nucleons).The cross-section satisfies the Thomas-Reiche-Kuhnsum rule Z σ ( E ) dE = (1 + x ) 2 π e ~ mc N ZA ≃ (1 + x )15 MeV mbarn A, (56)where x ∼ . E m = Z d x B π ≃ c Z d xd x ′ j ( x ) · j ( x ′ ) | x − x ′ |≃ π ¯ ER ( A ) ~ c Z σdE, (57)where R ( A ) is the nuclear radius. These quantities have thefollowing approximate representation R ( A ) = 1 . A fm , Z σdE ≃ . A MeV fm . (58)The fractional contribution of the magnetic energy to restmass energy is then ζ ( A ) ≃ E m A m A c ≈ . × − A − / . (59)Table 1 shows typical values of ζ ( A ) computed usingthe semi-empirical mass formula and the contribution ofneutrons and protons. Nucleus 10 ζ He . . . . . Table 1. ζ values for typical indoor stellar and planetaryelements
5. The Earth heat flux
There are several models that attempt to explain the aver-age rate of secular cooling of the Earth in terms of varia-tions in the composition of the mantle melts through time(Labrosse & Jaupart 2007). Constraints on these theoriesare set by terrestrial heat flow measurements on the sur-face.The contribution of ˙ α/α to the heat flux can be calcu-lated using the global heat balance for the Earth, assumingthat the machian contribution H C is the only extra energyproduction, M E C p dT m dt = − Q tot + H C + H G , (60)where M E is the Earth’s mass, C p is the average heat capac-ity of the planet, T m is the mantle potential temperature,and H G represents the heat generated by radioactive iso-topes. The total heat loss Q tot can be written as the sum Kraiselburd et al.: Variation in α and energy production of two terms, one that comes from the loss of heat in theoceans Q oc , and the other from continental heat loss Q cont .Using the results obtained by Labrosse & Jaupart (2007),we rewrite the total heat loss as Q tot ≈ M C p λ G T m , where λ G is the timescale constant for the secular Earth’s cooling.Assuming that the most abundant elements of the Earth areoxygen, silica, and iron, the “extra” energy contribution canbe written as H C = ¯ ζc H ˙ ααH , (61)where ¯ ζ is the mass-weighted averaged of the parameter ζ ( A ). Parameter
V alue H . × − s − M E . × KgC P / Kg − K¯ ζ . × − λ G . − Table 2.
Values of parametersFrom Eq.(14), we can describe the extra contribution asa function of time, writing a ( t ) a as a power series, (Weinberg1972) a ( t ) a ≈ H dt − q H dt ) + j H dt ) + · · · (62)and then making a Taylor series expansion up to third or-der of H C . Replacing this machian contribution in Eq.(60),solving it, and using the parameter’s values from Table 2,we find that the cosmological perturbation of the mantle’stemperature ∆ T m in terms of the time interval ∆ t and ˙ ααH is given by∆ T m ( t ) = 2 . × / Gyr ˙ αH α (∆ t ) − . × / Gyr ˙ αH α (∆ t ) + 3 . × / Gyr ˙ αH α ∆ t. (63)According to Labrosse & Jaupart (2007), the totalamount of cooling experienced by the Earth after an ini-tial magma ocean phase cannot exceed
200 K. Hence, in thepast 2 . T m <
200 K. With these restrictions, weobtain a bound for the time variation in α of, (cid:12)(cid:12)(cid:12)(cid:12) ˙ αH α (cid:12)(cid:12)(cid:12)(cid:12) < . × − . (64)Using this result in Eq.(15), we find that, (cid:18) ℓ B ℓ P (cid:19) < . , ℓ B ℓ P < . . (65)A different bound can be obtained by observing thatthe total radiated power of the Earth Q tot can be ex-plained by radiactive decay to within twenty per cent(Labrosse & Jaupart (2007)). The most recent data was estimated from an adjustment made with 38347 measure-ments. The methodology was to use a half-space cooling ap-proximation for hydrothermal circulation in young oceaniccrust; and in the remainder of the Earth’s surface, the aver-age heat flow of various geological domains was estimatedas defined by global digital maps of geology, and then aglobal estimate was made by multiplying the total globalarea of the geological domain, (Davies & Davies 2010).The result shows that Q tot ≈
47 TW ( in Table 3 thisestimate is separated into continental and oceanic contri-butions). Therefore, | Q mach | = | M E C P λ G T m ( t ) | < . Q tot . (66)In an interval of 2 . (cid:12)(cid:12)(cid:12)(cid:12) ˙ αH α (cid:12)(cid:12)(cid:12)(cid:12) < . × − , (67)and (cid:18) ℓ B ℓ P (cid:19) < . , ℓ B ℓ P < . . (68)
6. The heat fluxes of the outer planets
Jupiter, Saturn, Uranus, and Neptune are often called gasgiants. They are massive planets with a thick atmosphereand a solid core. Jupiter and Saturn are composed primar-ily of hydrogen and helium, while Uranus and Neptune aresometimes called ice giants, as they are mostly composed ofwater, ammonia, and methane ices. By comparising the ob-served bolometric temperatures of giant planets with thoseexpected when the planets are in thermal equilibrium withincident solar radiation, it is clear that all of these planetsexcept for Uranus have a significant internal heat source,(Irwin 2006).In the case of Jupiter, the residual primordial heat emit-ted is caused by the continued cooling and shrinking of theplanet via the
Kelvin-Helmholtz mechanism .Saturn must have also started out hot inside like Jupiteras the result of its similar formation. But being somewhatsmaller and less massive, Saturn was not as hot in the be-ginning of its life and has had time to cool. As a result, thisplanet has lost most of its primordial heat and there mustbe another source of most of its internal heat. This excessheat is generated by the precipitation of helium into itsmetallic hydrogen core. The heavier helium separates fromthe lighter hydrogen and drops toward the center. Smallhelium droplets form where it is cool enough, precipitateor rain down, and then dissolve at hotter deeper levels. Asthe helium at a higher level drizzles down through the sur-rounding hydrogen, the helium converts some of its energyinto heat, (Lang 2003).The low value of Uranus’ internal heat is still poorlyundertood. One suggestion is that chemical compositiongradients may act as inhibitors of heat transport from theplanet’s hot interior to the surface. Another hypothesis isthat it was hit by a supermassive impactor that caused it toexpel most of its primordial heat, leaving it with a depletedcore temperature. Uranus has as much as 4 M ⊕ of rockymaterials hence, part of the internal flux ( ≈ . W m − )comes from radioactive decay; Kelvin-Helmholtz mechanism would also be expected.Although Neptune is much farther from the Sun thanUranus, its thermal emission is almost equivalent. Several raiselburd et al.: Variation in α and energy production 7Part of the Earth Area (10 m ) Heat flow (TW) Mean heat flow ( mWm )Continent 2.073 14.7 70.9Ocean 3.028 31.9 105.4Global Total 5.101 46.7 91.6 Table 3.
Summary of the heat flow from Davies & Davies (2010) preferred estimatespossible explanations have been suggested, including radio-genic heating from the planet’s core, conversion of methaneunder high pressure into hydrogen, diamond and longer hy-drocarbons (the hydrogen and diamond would then rise andsink, respectively, releasing gravitational potential energy),and convection in the lower atmosphere that causes gravitywaves to break above the tropopause (Fortney & Hubbard2004; Hubbard 1978).We calculate the heat fluxes J ζ i for each planet usingthe equation of heat conduction1 r ddr (cid:18) Kr dTdr (cid:19) = − ερ, (69)where K is the effective thermal conductivity of the planetmaterial. The heat flux is J = − K dTdr . (70)If ¯ ε is the planet mean heat production, which is esti-mated from the results in Table 1 according to the chemicalcomposition of each planet, then J ( r ) = − K dTdr = 1 r Z ∞ ε ( r ′ ) ρ ( r ′ ) dr ′ = ¯ ε m ( r )4 πr , (71)hence the surface flux is J ζ i = − K dTdr (cid:12)(cid:12)(cid:12)(cid:12) ζ i = ¯ ε m ( R i )4 πR i , (72)which is the fundamental equation. Thus, we comparethe results of J ζ i with the observed fluxes obtained withVoyager 1, 2, and Cassini, (Pearl et al. 1990).The “3 σ ” upper bounds and the corresponding“( ℓ B /ℓ P )” bounds are Planet (cid:12)(cid:12)(cid:12) ˙ αH α (cid:12)(cid:12)(cid:12) ( ℓ B /ℓ P )Jupiter 2 . × − . . × − . . × − . . × − . (1) . × − . (2) . × − . Table 5.
Bounds from the outer planets and the Earth ( (1) results from Eqs.(64) and (65) and (2) from Eqs.(67) and(68)).
7. Conclusions
The energy exchange with ordinary matter in alternativetheories with new fields such as Beckenstein’s theory is aquite undeveloped field of subject. Using the field equationsand general hypothesis of the theory we have derived theenergy transfer between matter and fields. The Hypothesis8 in Sect. 2 is a probable key, because it states that thematter energy momentum tensor is the quantity that hasto be added to the field sector in order to make the totaltensor divergence free. We have also assumed that darkmatter is electrically neutral, have neglected the motion ofmatter in the bodies considered, and have found that thedynamical feature of the electric charge makes the atomicelectromagnetic energy part of the internal energy of thesystem. Eq.(48) shows that there is another contributionto the heat current in addition to the cooling of matter,which is given by the time variation in the scalar field andthe magnetic content of matter. We have also justified ourassumption that the matter cooling rate is not modifiedby the scalar field. Finally, using a minimal nuclear shellmodel we estimated the magnetic energy content of matter,thus permitting us to quantify the anomalous heat flux interms of the fundamental parameters of the theory and thechemical composition of the body.Our tightest constraint was obtained by analyzing thegeothermal aspects of the Earth, which are naturally themost clearly understood and reliably measured of our so-lar system, and the surface heat flux is very low. Ourbounds are comparable with that obtained in the labo-ratory by combining measurements of the frequences ofSr (Blatt et al. (2008)), Hg+ (Fortier et al. (2006)), Yb+(Peik et al. (2004)) and H (Fischer et al. (2004)) relativeto Caesium (Li et al. 2010; Uzan 2010), and only one orderof magnitude weaker than Oklo’s, which is the most strin-gent constraint on α ’s time variation up to date (Uzan 2010;Fujii et al. 2000)), and another found from measurementsof the ratio of Al+ to Hg+ optical clock frequencies over aperiod of a year (Rosenband et al. 2008; Li et al. 2010)(seeTable 6). The constraints we found depend on the cool-ing model of the Earth, but there is a general agreementabout the mechanisms behind it (Jessop (1990)). The outerplanets have provided us with additional constraints, whichare between the same and one order of magnitude weakerthan Earth’s, but are nevertheless valuable, as the chemicalcomposition and cooling mechanisms differ widely from ourplanet. The data set considered here is able to place inde-pendent constraints on the theory parameters. This analysismay be applied to other theories with extra fields that in-troduce extra “internal energies” to matter. We will reporton further work in future publications. Kraiselburd et al.: Variation in α and energy productionPlanet J obs (W / m ) M (Kg) R (m) J ζ i (W / m ) ˙ αH α Jupiter 5 . ± .
43 1 . × . × . × Saturn 2 . ± .
14 5 . × . × . × Uranus 0 . ± .
047 8 . × . × . × Neptune 0 . ± .
09 1 . × . × . × Table 4.
The observed heat flux, mass, radius, and calculated heat flux of the outer planets
Constraint (cid:12)(cid:12) ˙ αα (cid:12)(cid:12) (yr − ) Reference
ClocksCs (3 . ± . × − (1)ClocksHg (5 . ± . × − (1)Oklo (2 . ± . × − (2)J ⊕ . × − (3)J ⊕ II . × − (4)J Jup . × − (5)J Sat . × − (5)J Ur . × − (5)J Nep . × − (5) Table 6.
The table comparises different kinds of con-straints, the value of ˙ αα , and the reference. References (1)Li et al. (2010); (2)Fujii et al. (2000); (3) Eq.64; (4) Eq.67;(5) Table5 References
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