4.3 Stellar constraints

Early works, see Section IV.C of FVC [500], studied the Solar evolution in presence of a time varying gravitational constant, concluding that under the Dirac hypothesis, the original nuclear resources of the Sun would have been burned by now. This results from the fact that an increase of the gravitational constant is equivalent to an increase of the star density (because of the Poisson equation).

The idea of using stellar evolution to constrain the possible value of G was originally proposed by Teller [487Jump To The Next Citation Point], who stressed that the evolution of a star was strongly dependent on G. The luminosity of a main sequence star can be expressed as a function of Newton’s gravitational constant and its mass by using homology relations [224Jump To The Next Citation Point, 487]. In the particular case that the opacity is dominated by free-free transitions, Gamow [224] found that the luminosity of the star is given approximately by L ∝ G7.8M 5.5. In the case of the Sun, this would mean that for higher values of G, the burning of hydrogen will be more efficient and the star evolves more rapidly, therefore we need to increase the initial content of hydrogen to obtain the present observed Sun. In a numerical test of the previous expression, Delg’Innocenti et al. [140Jump To The Next Citation Point] found that low-mass stars evolving from the Zero Age Main Sequence to the red giant branch satisfy L ∝ G5.6M 4.7, which agrees to within 10% of the numerical results, following the idea that Thomson scattering contributes significantly to the opacity inside such stars. Indeed, in the case of the opacity being dominated by pure Thomson scattering, the luminosity of the star is given by L ∝ G4M 3. It follows from the previous analysis that the evolution of the star on the main sequence is highly sensitive to the value of G.

The driving idea behind the stellar constraints is that a secular variation of G leads to a variation of the gravitational interaction. This would affect the hydrostatic equilibrium of the star and in particular its pressure profile. In the case of non-degenerate stars, the temperature, being the only control parameter, will adjust to compensate the modification of the intensity of the gravity. It will then affect the nuclear reaction rates, which are very sensitive to the temperature, and thus the nuclear time scales associated to the various processes. It follows that the main stage of the stellar evolution, and in particular the lifetimes of the various stars, will be modified. As we shall see, basically two types of methods have been used, the first in which on relate the variation of G to some physical characteristic of a star (luminosity, effective temperature, radius), and a second in which only a statistical measurement of the change of G can be inferred. Indeed, the first class of methods are more reliable and robust but is usually restricted to nearby stars. Note also that they usually require to have a precise distance determination of the star, which may depend on G.

4.3.1 Ages of globular clusters

The first application of these idea has been performed with globular clusters. Their ages, determined for instance from the luminosity of the main-sequence turn-off, have to be compatible with the estimation of the age of the galaxy. This gives the constraint [140]

G˙∕G = (− 1.4 ± 2.1) × 10−11 yr−1. (147 )

The effect of a possible time dependence of G on luminosity has been studied in the case of globular cluster H-R diagrams but has not yielded any stronger constraints than those relying on celestial mechanics

4.3.2 Solar and stellar seismology

A side effect of the change of luminosity is a change in the depth of the convection zone so that the inner edge of the convecting zone changes its location. This induces a modification of the vibration modes of the star and particularly to the acoustic waves, i.e., p-modes [141Jump To The Next Citation Point].

Helioseismology. These waves are observed for our star, the Sun, and helioseismology allows one to determine the sound speed in the core of the Sun and, together with an equation of state, the central densities and abundances of helium and hydrogen. Demarque et al. [141] considered an ansatz in which G ∝ t−β and showed that |β| < 0.1 over the last 4.5 × 109 years, which corresponds to ˙ − 11 −1 |G ∕G | < 2 × 10 yr. Guenther et al. [240] also showed that g-modes could provide even much tighter constraints but these modes are up to now very difficult to observe. Nevertheless, they concluded, using the claim of detection by Hill and Gu [251], that |G ˙∕G | < 4.5 × 10− 12 yr−1. Guenther et al. [239] then compared the p-mode spectra predicted by different theories with varying gravitational constant to the observed spectrum obtained by a network of six telescopes and deduced that

|| || − 12 − 1 |G˙∕G | < 1.6 × 10 yr . (148 )
The standard solar model depends on few parameters and G plays a important role since stellar evolution is dictated by the balance between gravitation and other interactions. Astronomical observations determines GM ⊙ with an accuracy better than 10 −7 and a variation of G with GM ⊙ fixed induces a change of the pressure (P = GM 2∕R2 ⊙ ⊙) and density (ρ = M ⊙∕R3 ⊙). The experimental uncertainties in G between different experiments have important implications for helioseismology. In particular the uncertainties for the standard solar model lead to a range in the value of the sound speed in the nuclear region that is as much as 0.15% higher than the inverted helioseismic sound speed [335Jump To The Next Citation Point]. While a lower value of G is preferred for the standard model, any definite prediction is masked by the uncertainties in the solar models available in the literature. Ricci and Villante [436] studied the effect of a variation of G on the density and pressure profile of the Sun and concluded that present data cannot constrain G better than −2 10 %. It was also shown [335] that the information provided by the neutrino experiments is quite significant because it constitutes an independent test of G complementary to the one provided by helioseismology.

White dwarfs. The observation of the period of non-radial pulsations of white dwarf allows to set similar constraints. White dwarfs represent the final stage of the stellar evolution for stars with a mass smaller to about 10M ⊙. Their structure is supported against gravitational collapse by the pressure of degenerate electrons. It was discovered that some white dwarfs are variable stars and in fact non-radial pulsator. This opens the way to use seismological techniques to investigate their internal properties. In particular, their non-radial oscillations is mostly determined by the Brunt–Väisälä frequency

2 dlnP 1∕γ1∕ρ N = g ------------ dr
where g is the gravitational acceleration, Γ 1 the first adiabatic exponent and P and ρ the pressure and density (see, e.g., [283] for a white dwarf model taking into account a varying G). A variation of G induces a modification of the degree of degeneracy of the white dwarf, hence on the frequency N as well as the cooling rate of the star, even though this is thought to be negligible at the luminosities where white dwarfs are pulsationally unstable[54Jump To The Next Citation Point]. Using the observation of G117-B15A that has been monitored during 20 years, it was concluded [43] that
−10 −1 −11 − 1 − 2.5 × 10 yr < G˙∕G < 4.0 × 10 yr , (149 )
at a 2σ-level. The same observations were reanalyzed in [54] to obtain
˙ −11 −1 |G ∕G | < 4.1 × 10 yr . (150 )

4.3.3 Late stages of stellar evolution and supernovae

A variation of G can influence the white dwarf cooling and the light curves ot Type Ia supernovae.

García-Berro et al. [225Jump To The Next Citation Point] considered the effect of a variation of the gravitational constant on the cooling of white dwarfs and on their luminosity function. As first pointed out by Vila [518], the energy of white dwarfs, when they are cool enough, is entirely of gravitational and thermal origin so that a variation of G will induce a modification of their energy balance and thus of their luminosity. Restricting to cold white dwarfs with luminosity smaller than ten solar luminosity, the luminosity can be related to the star binding energy B and gravitational energy, E grav, as

dB-- ˙G- L = − dt + G Egrav, (151 )
which simply results from the hydrostatic equilibrium. Again, the variation of the gravitational constant intervenes via the Poisson equation and the gravitational potential. The cooling process is accelerated if G˙∕G < 0, which then induces a shift in the position of the cut-off in the luminosity function. García-Berro et al. [225] concluded that
− 11 −1 0 ≤ − G˙∕G < (1 ± 1 ) × 10 yr . (152 )
The result depends on the details of the cooling theory, on whether the C/O white dwarf is stratified or not and on hypothesis on the age of the galactic disk. For instance, with no stratification of the C/O binary mixture, one would require −11 −1 G˙∕G = − (2.5 ± 0.5) × 10 yr if the solar neighborhood has a value of 8 Gyr (i.e., one would require a variation of G to explain the data). In the case of the standard hypothesis of an age of 11 Gyr, one obtains that 0 ≤ − G˙∕G < 3 × 10 −11 yr− 1.

The late stages of stellar evolution are governed by the Chandrasekhar mass (ℏc ∕G)3∕2m −2 n mainly determined by the balance between the Fermi pressure of a degenerate electron gas and gravity.

Simple analytical models of the light curves of Type Ia supernovae predict that the peak of luminosity is proportional to the mass of nickel synthesized. In a good approximation, it is a fixed fraction of the Chandrasekhar mass. In models allowing for a varying G, this would induce a modification of the luminosity distance-redshift relation [227, 232, 435Jump To The Next Citation Point]. However, it was shown that this effect is small. Note that it will be degenerate with the cosmological parameters. In particular, the Hubble diagram is sensitive to the whole history of G (t) between the highest redshift observed and today so that one needs to rely on a better defined model, such as, e.g., scalar-tensor theory [435Jump To The Next Citation Point] (the effect of the Fermi constant was also considered in [194]).

In the case of Type II supernovae, the Chandrasekhar mass also governs the late evolutionary stages of massive stars, including the formation of neutron stars. Assuming that the mean neutron star mass is given by the Chandrasekhar mass, one expects that G˙∕G = − 2 M˙NS ∕3MNS. Thorsett [492Jump To The Next Citation Point] used the observations of five neutron star binaries for which five Keplerian parameters can be determined (the binary period P b, the projection of the orbital semi-major axis a sin i 1, the eccentricity e, the time and longitude of the periastron T0 and ω) as well as the relativistic advance of the angle of the periastron ω˙. Assuming that the neutron star masses vary slowly as MNS = M (N0)S − M˙NStNS, that their age was determined by the rate at which Pb is increasing (so that tNS ≃ 2Pb∕P˙b) and that the mass follows a normal distribution, Thorsett [492] deduced that, at 2σ,

−12 −1 G˙∕G = (− 0.6 ± 4.2) × 10 yr . (153 )

4.3.4 New developments

It has recently been proposed that the variation of G inducing a modification of the binary’s binding energy, it should affect the gravitational wave luminosity, hence leading to corrections in the chirping frequency [554]. For instance, it was estimated that a LISA observation of an equal-mass inspiral event with total redshifted mass of 105M ⊙ for three years should be able to measure G˙∕G at the time of merger to better than 10–11/yr. This method paves the way to constructing constraints in a large band of redshifts as well as in different directions in the sky, which would be an invaluable constraint for many models.

More speculative is the idea [25] that a variation of G can lead a neutron to enter into the region where strange or hybrid stars are the true ground state. This would be associated with gamma-ray bursts that are claimed to be able to reach the level of 10–17/yr on the time variation of G.

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