Two restrictions in the theories that include G(t) and c(t) varying with time

Much work has been done taking into account the possibility that the gravitational constant G may vary with cosmological time t (or with the cosmological scale factor a(t)). The same may be said about the speed of light c. We present here two important remarks on these subject. These remarks include G(t) and c(t) varying with time with the restriction 8πG/c ⁴=constant.     

Bianchi VI h with variable gravitational and cosmological constants

In order to study how the gravitational and the cosmological constants, G, Λ may vary, we consider two theoretical frameworks which are, a modification of the General Relativity and several scalar models (the standard, non-interacting and interacting models and their respective modifications to allow a G varying). We find exact self-similar solutions for the geometry Bianchi VI _h , (that is, the models: III, VI₀, and VI _h ,). Some physical and geometrical properties of the models are also discussed and we compare the obtained theoretical results with the current observational data. In the first of the theoretical models, we reach the conclusion that, from the structure of the field equations, the behaviour of Λ and G are related, but taking into account the observational data, we conclude that the Λ behaves as a positive decreasing time function while G is growing but in the long time regimen it tends to a constant value. In the scalar models, our solutions predict a “positive” dynamical cosmological constant in all the obtained solutions while the behaviour of G yields indeterminate, since its depends on a free parameter, G≈t ^2α , so it may be growing or decreasing as in the scalar-tensor theories.     

The stars in the Main Sequence and their dynamical parameters

The stars in the Main Sequence are seen as a hierarchy of objects with different massesM and effective dynamical radiiR _eff=R/α given by the stellar radii and the coefficients for the inner structure of the stars. As seen in a previous work (Paper I), during the lifetime in the Main SequenceR _eff(t) remains a near invariant when compared to the variation in the time ofR(t) and α(t). With such an effectiveR _eff one obtains the amounts of actionA _c(M), the effective densities ρ_eff(M)=ρ(M)α³(M), the densities of action and of energy (or mean presures in the stellar interior)a _c(M),e _c(M), and the potential energiesE _p(M). The amounts of action areA _c∝M ^k withk≈1.87 for the M stars,k≈5/3 for the KGF stars, andk≈1.83 for the A and earlier stars, representing very simples conditions for the other dynamical parameters. For instancek≈5/3 means a near invariant effective density α_eff for the KGF stars, while for such stars the mean densities and coefficients α present the strongest variations with masses ρ(M)∝M ^?1.81, α(M)∝M^0.6. The cases for the M stars (e _c(M)∝M ^?1) and for the A and earlier stars (betweena _c(M)=constant and α_eff(M)∝M ^?1) and also discussed. These conditions for the earlier stars also represent reasonable mean values for the whole stellar hierarchy in the range of masses 0.2M _⊙≤M≤25M _⊙. With all this, one can build ‘dynamical’ HR diagrams withA _c(M), E_p(M), α_eff M ^?p , etc., whose characteristics are analogous to these in the photometrical HR diagram. A comparison is made betweenA _c(M) from the models here and the HR diagram with the best known stars of luminosity classes IV, V, and white dwarfs. The comparison of the potential energiesE _p(M)∝M ^?p according to the stellar models used here and the observed frequency function ψ(M∝M _?q (number of stars in a given interval of masses) from different authors suggests the possibility that the productE _p(M)ψ(M) is a constant, but this must be confirmed with further studies of the function ψ(M) and its fine structure. There are analogies between the formulation used here for the stellar hierarchy and other physical processes, for instance, in modified forms of the Kolmogorov law of turbulence and in the formulation used for the hierarchy of molecular clouds in gravitational equilibrium. Besides, the function of actionA _c(M) for the stars has analogous properties to the relations of angular momenta and massesJ(M) for different types of objects. The cosmological implications of all this are discussed.     

Evolution of the number of accreting white dwarfs with shell nuclear burning and the SNe Ia rate

We analyze the time evolution of the number of accreting white dwarfs with surface shell hydrogen burning in semidetached and detached binaries. We consider the case where continuous star formation with a constant rate takes place in a stellar system over 10¹⁰ Gyr and the case of a starburst in which the same mass of stars is formed over 10⁹ Gyr. The evolution of the number of white dwarfs is compared with the evolution of the rate of events that are usually considered as SNe Ia and/or accretion-induced collapses, i.e., the accumulation of a Chandrasekhar mass by white dwarfs or the merger of white dwarf pairs with a total mass greater than or equal to the Chandrasekhar one. In stellar systems with a starburst, the supersoft X-ray sources observed at t = 10¹⁰ yr are most likely not the progenitors of SNe Ia. The same is true for a significant fraction of the sources in systems with a constant star formation rate. In both cases, the merger of white dwarfs is the dominant mechanism of SNe Ia. In symbiotic binaries, accreting CO dwarfs do not accumulate enough mass for an SNe Ia explosion, while ONeMg dwarfs finish their evolution by an accretion-induce collapse with the formation of a neutron star.     

Chemical evolution of the galaxy during its contraction

Models for the chemical evolution of the galaxy are constructed in which the time evolution is imposed by the contraction rate of the galaxy and present observations of stellar metal abundances as a function of height above the galactic plane. Stars with massm?3.5m _⊙ do not contribute to the metal enrichment of the interstellar gas, and we argue that the interstellar metal abundance at this epoch should be fairly insensitive to the size of the mass fraction of the galaxy that condenses into such stars. The birth rate for stars more massive than 3.5m _⊙ is assumed proportional toV _gal ^?n , whereV _gal is the contracting volume of the main body of the galaxy. If a dynamic time-scale is adopted for the contraction of the galaxy, then the assumed power-law birth rate yields suitable chemical evolution models only if observed Population II metals are synthesized in stars more massive than about 8.5m _⊙. This mass range is consistent with the predictions of current stellar-evolution theory. Provided the birth function does include stars more massive than 8.5m _⊙, the relation between the value of the parametern in the birth rate and the observed chemical evolution rate is not particularly sensitive to the specific form adopted for the initial mass spectrum, or to the proportionality constant in the birth rate. We find (i)n?1.4, in general, and (ii)n is close to 1.4 if the contraction of the galaxy to a heighth=400 pc above the plane occurs at close to the free-fall rate. These results are independent of the form of the initial mass spectrum, providedS ³ is small. HereS is the total mass fraction of the galaxy that cycles through stars during its contraction. Numerical models, with an explicit initial mass spectrum, indicate that the same restrictions on the values ofn apply approximately whenS ³ is not small. To introduce low mass stars, we allow the birth rate for stars more massive than 3.5m _⊙ to level off at a time intervalt _L just before the contraction of the galaxy stops, while the total birth rate remains a simple power law. We find that reasonable models are obtained witht _L ?1.5×10⁷ yr if the galaxy contracts at a dynamic rate. However, aside from these restrictions on the values ofn andt _L , there is no uniquely favored model. For any suitable model, the supernova rate must be small enough so that shock waves from neighboring supernovae do not collide during the adiabatic expansion stage. Otherwise, the interstellar gas would not have time to cool, and its high temperature would tend to impede both star formation and the rapid contraction of the galaxy. The supernova rates in the numerical models given here are small enough to avoid this problem, but large enough to achieve a uniform metal abundance on a time scale short compared to the chemical-evolution time scale. At the epoch considered here, the interstellar metal abundance is approximately less than 0.4Z _⊙, and the models are assumed to apply before galactic-scale inhomogeneities, such as the galactic nucleus, become important. Therefore, the chemical mixing time scales imply that most Population II stars of the same age should have approximately the same initial metal abundance, unless the clustering of supernova explosions associated with massive Population II stars is significant. It is shown that collisions between shock waves from neighboring supernovae can produce local regions of significantly enhanced density. The peak bolometric luminosity of the galaxy during its contraction is similar to that predicted by Partridge and Peebles (1967a), but it occurs during the final stages of contraction to the disc. Numerical models give values between 13 and 34 yr^?1 for the average number of supernova explosions per year during this bright phase. The X-ray luminosity of the galaxy from these supernovae may be comparable to that of Seyfert galaxies.     

Cosmic ray anisotropies observed late in the decay phase of solar flare events

Concurrent interplanetary magnetic field and 0.7–7.6 MeV proton cosmic-ray anisotropy data obtained from instrumentation on Explorers 34 and 41 are examined for five cosmic-ray events in which we observe a persistent eastern-anisotropy phase late in the event (t ? 4 days). The direction of the anisotropy at such times shows remarkable invariance with respect to the direction of the magnetic field (which generally varies throughout the event) and it is also independent of particle species (electrons and protons) and particle speed over the range 0.06 ? β ? 0.56. The anisotropy is from the direction 38.3° ± 2.4° E of the solar radius vector, and is inferred to be orthogonal to the long term, mean interplanetary field direction. Both the amplitude of the anisotropy and the decay time constant show a strong dependence on the magnetic field azimuth. Detailed comparison of the anisotropy and the magnetic field data shows that the simple model of convection plus diffusion parallel to the magnetic field is applicable for this phase of the flare effect. It is demonstrated that contemporary theories do not predict the invariance of the direction as observed, even when the magnetic field is steady; these theories need extension to take into account the magnetic field direction ψ varying from its mean direction ψ _o. It is shown that the late phase anisotropy vector is not expected to be everywhere perpendicular to the mean magnetic field. The suggestion that we are observing kinks in the magnetic field moving radially outwards from the Sun leads to the conclusion that the parallel diffusion coefficient varies as 1/cos² (ψ ? ψ _o). Density gradients in the late decay phase are estimated to be ≈ 700%∣AU for 0.7–7.6 MeV protons. A simple theory reproduces the dependence of the decay time constant on anisotropy; it also leads to a radial density gradient of about 1000%∣AU and diffusion coefficient of 1.3 × 10²⁰ cm² s^?1.     

Orbital variations and outbursts of the unusual variable star V1129 Centauri

The variable star V1129 Cen is classified in the GCVS as being of β Lyr type. Unusual for such stars, it exhibits outbursts roughly once a year, lasting for ∼ 40 days. For this reason, a relationship to the dwarf novae has been suspected. Here, for the first time a detailed analysis of the light curve of the system is presented. Based on observations with high time resolution obtained at the Observatório do Pico dos Dias and on the long term ASAS light curve the orbital variations of the system are studied. They are dominated by ellipsoidal variations and partial eclipses of a probably slightly evolved F2 star in a binary with an orbital period of 21^h 26^m. Comparison with the characteristics of dwarf novae show that the observational properties of V1129 Cen can be explained if it is just another dwarf novae, albeit with an unusually bright and early type mass donor which outshines the accretion disk and the mass gainer to a degree that many normal photometric and spectroscopic hallmarks of cataclysmic variables remain undetected.     

Modeling the luminosity function of galactic low-mass X-ray binaries

The evolution of the family of binaries with a low-mass star and a compact neutron star companion (low-mass X-ray binaries (LMXBs) with neutron stars) ismodeled by the method of population synthesis. Continuous Roche-lobe filling by the optical star in LMXBs is assumed to be maintained by the removal of orbital angular momentum from the binary by a magnetic stellar wind from the optical star and the radiation of gravitational waves by the binary. The developed model of LMXB evolution has the following significant distinctions: (1) allowance for the effect of the rotational evolution of a magnetized compact remnant on themass transfer scenario in the binary, (2) amore accurate allowance for the response of the donor star to mass loss at the Roche-lobe filling stage. The results of theoretical calculations are shown to be in good agreement with the observed orbital period-X-ray luminosity diagrams for persistent Galactic LMXBs and their X-ray luminosity function. This suggests that the main elements of binary evolution, on the whole, are correctly reflected in the developed code. It is shown that most of the Galactic bulge LMXBs at luminosities L _x > 10³⁷ erg s^?1 should have a post-main-sequence Roche-lobe-filling secondary component (low-mass giants). Almost all of the models considered predict a deficit of LMXBs at X-ray luminosities near ～10^36.5 erg s^?1 due to the transition of the binary from the regime of angular momentum removal by a magnetic stellar wind to the regime of gravitational waves (analogous to the widely known period gap in cataclysmic variables, accreting white dwarfs). At low luminosities, the shape of the model luminosity function for LMXBs is affected significantly by their transient behavior-the accretion rate onto the compact companion is not always equal to the mass transfer rate due to instabilities in the accretion disk around the compact object. The best agreement with observed binaries is achieved in the models suggesting that heavy neutron stars with masses 1.4–1.9M _⊙ can be born.     

Holographic dark energy with time depend gravitational constant in the non-flat Ho?ava-Lifshitz cosmology

We study the holographic dark energy on the subject of Hořava-Lifshitz gravity with a time dependent gravitational constant G(t), in the non-flat space-time. We obtain the differential equation that specify the evolution of the dark energy density parameter based on varying gravitational constant. We find out a relation for the state parameter of the dark energy equation of state to low red-shifts which containing varying G corrections in the non-flat space-time.     

Variable cosmological “constant” and motion of a test particle in a cloud of dust

We find that Einstein’s like field equations with coordinate-dependent cosmological “constant” Λ(x ⁱ ) imply a non geodesic law of motion for test particles moving in a continuous distribution of incoherent matter (“dust”). The deviation from the geodesic law depends on the derivatives ?Λ/? x ⁱ and, in the weak field approximation, causes an anomalous acceleration A～(Vc ²/γ ρ)?Λ/? t+(c ⁴/γ ρ)?Λ/? r where V=dr/dt, c=the speed of light, γ=8π G with G=the gravitational coupling, ρ=the mass density of the cloud, r and t are the radial and time coordinate respectively. Reasonable assumptions on Λ=Λ(t) give A<10^?8 cm/s² when ρ>10^?29 g/cm³ i.e. in all known astrophysical systems. A possible connection with the anomalous Pioneer acceleration is shortly discussed in the case of a cosmological “constant” coupled to matter.     

Search for and study of photometric variability in magnetic white dwarfs

We report the results of photometric observations of a number of magnetic white dwarfs in order to search for photometric variability in these stars. These V-band observations revealed significant variability in the classical highly magnetized white dwarf GRW+70?8247 with a likely period from several days to several dozen days and a half-amplitude of about 0_. ^m 04. Our observations also revealed the variability of the well-known white dwarf GD229. The half amplitude of its photometric variability is equal to about 0_. ^m 005, and the likely period of this degenerate star lies in the 10–20 day interval. This variability is most likely due to the rotation of the stars considered.We also discuss the peculiarities of the photometric variability in a number of other white dwarfs. We present the updated “magnetic field–rotation period” diagram for the white dwarfs.     

Anisotropic cosmological models with bulk viscosity for variable <Emphasis Type="Italic">G</Emphasis> and Λ

In this paper we have considered axially symmetric Bianchi-I, Kantowski Sachs and Bianchi-III space-time models with bulk viscosity, where the gravitational constant G and the cosmological term Λ vary with time. In Einstein equations this variation in G and Λ are taken in such a way as to preserve the energy momentum tensor. Solutions are obtained with the cosmological term varying inversely with square of time.     

Seeable universe and its accelerated expansion: an observational test

From the equivalence principle, one gets the strength of the gravitational effect of a mass M on the metric at position r from it. It is proportional to the dimensionless parameter β ²=2GM/rc ², which normally is ?1. Here G is the gravitational constant, M the mass of the gravitating body, r the position of the metric from the gravitating body and c the speed of light. The seeable universe is the sphere, with center at the observer, having a size such that it shall contain all light emitted within it. For this to occur one can impose that the gravitational effect on the velocity of light at r is zero for the radial component, and non zero for the tangential one. Light is then trapped. The condition is given by the equality R _g =2GM/c ², where R _g represents the radius of the seeable universe. It is the gravitational radius of the mass M. The result has been presented elsewhere as the condition for the universe to be treated as a black hole. According to present observations, for the case of our universe taken as flat (k=0), and the equation of state as p=?ρc ², we prove here from the Einstein’s cosmological equations that the universe is expanding in an accelerated way as t ², a constant acceleration as has been observed. This implies that the gravitational radius of the universe (at the event horizon) expands as t ². Taking c as constant, observing the galaxies deep in space this means deep in time as ct, linear. Then, far away galaxies from the observer that we see today will disappear in time as they get out of the distance ct that is <R _g . The accelerated expanding vacuum will drag them out of sight. This may be a valid test for the present ideas in cosmology. Previous calculations are here halved by our results.     

Evolution of Intergalactic Gas in the Neighborhood of Dwarf Galaxies and Its Manifestations in the HI 21 cm Line

Low-mass galaxies are known to have played the crucial role in the hydrogen reionization in the Universe. In this paper we investigate the contribution of soft x-ray radiation (E ~ 0.1–1 keV) from dwarf galaxies to hydrogen ionization during the initial reionization stages. The only possible sources of this radiation in the process of star formation in dwarf galaxies during the epochs preceding the hydrogen reionization epoch are hot intermediate-mass stars (M ~ 5–8 M_⊙) that entered the asymptotic giant branch (AGB) stage and massive x-ray binaries. We analyze the evolution of the intergalactic gas in the neighborhood of a dwarf galaxy with a total mass of 6 × 10⁸M_⊙ formed at the redshift of z ~ 15 and having constant star-formation rate of 0.01–0.1 M_⊙ yr^?1 over a starburst with a duration of up to 100 Myr. We show that the radiation from AGB stars heats intergalactic gas to above 100 K and ensures its ionization x_e ? 0.03 within about 4–10 kpc from the galaxy in the case of a star-formation rate of star formation 0.03–0.1 M_⊙ yr^?1, and that after the end of the starburst this region remains quasi-stationary over the following 200–300 Myr, i.e., until z ~ 7.5. Formation of x-ray binaries form in dwarf galaxies at z ~ 15 results in a 2–3 and 5–6 times greater size of the ionized and heated region compared to the case where ionization is produced by AGB stars exclusively, if computed with the “x-ray luminosity–star-formation rate” dependence (L_X ~ f_XSFR) factor f_X = 0.1 and f_X ~ 1, respectively. For f_X ? 0.03 the effect of x-ray binaries is smaller that that of AGB star population. Lyα emission, heating, and ionization of the intergalactic gas in the neighborhood of dwarf galaxies result in the excitation of the 21 cm HI line. We found that during the period of the starburst end at z ~11.5–12.5 the brightness temperature in the neighborhood of galaxies is 15–25 mK and the region where the brightness temperature remains close to its maximum has a size of about 12–30 kpc. Hence the epoch of the starburst end is most favorable for 21 cm HI line observations of dwarf galaxies, because at that time the size of the region of maximum brightness temperature is the greatest over the entire evolution of the dwarf galaxy. In the case of the sizes corresponding to almost 0.’1 for z ~ 12 regions with maximum emission can be detected with the Square Kilometre Array, which is currently under construction.     

Gravitational Radiation of Relativistic <Emphasis Type="Italic">n</Emphasis> = 1 Polytropes

The gravitational radiation of n = 1 polytropes undergoing quasiradial pulsations is examined. The intensity of the gravitational radiation and the gravitational wave amplitudes are calculated for polytropic models of white dwarfs and neutron stars when the energy of rotation of the object serves as the source of the radiated energy. Calculations of h₀ show that objects with a polytropic equation of state can describe the expected gravitational radiation from white dwarfs and neutron stars. The gravitational radiation of polytropic models of galactic nuclei and quasars is also examined. These objects can create a high enough background of gravitational radiation at frequencies of 10^-8–10^-11 Hz for gravitational wave detectors operating in this frequency range. __________ Translated from Astrofizika, Vol. 48, No. 4, pp. 603–612 (November 2005).     

Mass—luminosity relation and pulsational properties of Wolf—Rayet stars

The evolution of Population I stars with initial masses 70M _⊙ ≤ M _ZAMS ≤ 130M _⊙ is considered. The computations were performed under various assumptions about the mass loss rate and were terminated at the phase of gravitational contraction after core helium exhaustion. The mass loss rate at the helium burning phase, ?_3α , is shown to be the main parameter that determines the coefficients of the mass—luminosity relation for Wolf—Rayet stars. Several more accurate mass—luminosity relations for mass loss rates ? = f _3α ?_3α , where 0.5 ≤ f _3α ≤ 3, are suggested, along with the mass—luminosity relation that combines all of the evolutionary sequences considered. The results of the stellar evolution computations were used as initial conditions in solving the hydrodynamic equations describing the spherically symmetric motions of a self-gravitating gas. The outer layers of massive Population I stars are unstable against radial oscillations throughout the helium burning phase. The oscillation amplitude is largest at enhanced carbon and oxygen abundances in the outer stellar layers, i.e., at a lower initial stellar mass M _ZAMS or a lower mass loss rate during the entire preceding evolution. In the course of evolution, the radial oscillation amplitude decreases and the small nonlinearity of the oscillations at M < 10M _⊙ allow the integral of mechanical work W done by an elementary spherical layer of gas in a closed thermodynamic cycle to be calculated with the necessary accuracy. The maximum of the radial dependence of W is shown to be located in layers with a gas temperature T ～ 2 × 10⁵ K, where the oscillations are excited by the iron Z-bump κ-mechanism. Comparison of the radial dependences of the integral of mechanical work W and the amplitude of the radiative flux variations suggests that the nonlinear radial oscillations of more massive Wolf—Rayet stars are also excited by the κ-mechanism.     

Aperiodic X-ray flux variability of EX Hya and the area of the base of the accretion column at the white dwarf surface

The goal of this paper is to determine the characteristic cooling time of the accretion flowmatter near the surface of the magnetic white dwarf in the binary system EX Hya. Most of the X-ray photons in such binary systems are produced in an optically thin hot plasma with a temperature above 10 keV heated when the matter passes through the shock near the white dwarf surface. The total X-ray luminosity is determined by the matter accumulated below the shock in its cooling time. Thus, the X-ray luminosity variability related to the variations in the accretion rate onto the white dwarf surface must be suppressed at frequencies higher than the inverse cooling time. If the optically thin plasma radiation dominates in the rate of energy losses by the heated matter, which is true for white dwarfs with moderately strong magnetic fields, less than 1–10 MG, then the matter cooling time can give an estimate of the matter density in the accretion column. Given the accretion rate and the matter density in the accretion column at the white dwarf surface, the area of the accretion channel can be estimated. We have analyzed all of the currently available observational data for one of the brightest intermediate polars in the X-ray sky, EX Hya, from the RXTE and XMM-Newton observatories. The power spectra of its aperiodic variability have given an upper limit on the cooling time of the hot plasma: <1.5–2 s. For the observed accretion rate, ×10¹⁵ g s^?1, this corresponds to a matter density below the shock surface ?10¹⁶ cm^?3 and an area of the base of the accretion channel no more than <4.6 × 10¹⁵ cm². Using the information about the maximum geometrical size of the accretion channel obtained by analyzing X-ray eclipses in the binary system EX Hya, we have derived an upper limit on the thickness of the flow over the surface of the magnetosphere near the white dwarf surface, ?3 × 10⁶ cm, and the plasma penetration depth at the magnetospheric boundary, Δr/r ? 6 × 10^?3.     

The number of information bits related to the minimum quantum and gravitational masses in a vacuum dominated universe

Wesson obtained a limit on quantum and gravitational mass in the universe by combining the cosmological constant Λ, Planck’s constant ?, the speed of light c, and also the gravitational constant G. The corresponding masses are 2.0×10^?62 kg and 2.3×10⁵⁴ kg respectively, and in general can be obtained with the help of a generic dimensional analysis, or from an analysis where the cosmological constant appears in a four dimensional space-time and as a result of a higher dimensional reduction. In this paper our goal is to establish a relation for both quantum and gravitational mass as function of the information number bit N. For this reason, we first derive an expression for the cosmological constant as a function of information bit, since both masses depend on it, and then various resulting relations are explored, in relation to information number of bits N. Fractional information bits imply no information extraction is possible. We see, that the order of magnitude of the various parameters as well as their ratios involve the large number 10¹²², that is produced naturally from the fundamental parameters of modern cosmology. Finally, we propose that in a complete quantum gravity theory the idea of information the might have to be included, with the quantum bits of information (q-bits) as one of its fundamental parameters, resulting thus to a more complete understanding of the universe, its laws, and its evolution.     

Implications of a cosmological term coupled to matter

The possibility that the cosmological term is proportional toGU, whereG is the gravitational coupling andU is the mass density of the universe is proposed and discussed. WithG = constant, a cosmological model is obtained, which avoids the flatness and horizon problems and does not affect the well known predictions on the cosmic helium abundance which come from standard big bang cosmology. In such model, the deceleration parameter is a null constant, there is matter creation process throughout the universe at the rate 10^–47 g cm^–3 s^–1 and the cosmological term varies asH ² =t ^–2, whereH is the Hubble constant andt is the cosmic time.The possibility of a time-dependentG is then considered. The main consequence of this is that there is a mass creation process on the local scale; the rate of mass creation inside a body of massM is dM/dt =M H. In Section 6 it is suggested that the new matter might be in the form of neutrinos. This suggestion leads to an interesting consequence in celestial mechanics: the radius of a binary system should depend on time according to the nature of the components (the radius of a binary star should decrease, the radius of a planet-moon system should expand, and the orbital radius of a planet should stay constant).