Mass radii

H. L. Shipman. Masses, radii and model atmospheres for cool white-dwarf stars. Astrophys. J., 213 138, 1977. 

Hence maximum indentation of the two spheres and maximum contact force depend on the material properties, masses, radii, and velocities of the two spheres right before impact. The permanent indentation is evaluated from the initial approach velocities of the spheres and a known coefficient of restitution between the spheres. 

Table 25-IV begins this survey with a comparison of mass, radius, and density of the planets and the sun. These data are probably the most reliable facts known about the planets since...

Basic definitions Let X be some property of a polymer chain such as the degree of polymerization, molar mass, radius of gyration, or comonomer content of a copolymer, etc. In general, the polymer is heterogeneous with respect to X, which can assume discrete values X,. We now define for molecules with X = X,-. 

Fig. 1. The color-magnitude diagram of NGC 188 from [2] with the location of the detached eclipsing binary V12 [3] overplotted. From radial-velocity measurements we find (assuming an inclination of 90 degrees since we do not yet have photometry of the eclipses) that the masses of the two components are 1.06 and 1.08 solar masses. We estimate that we will be able to reach a precision of 1% in the mass estimate. We are in the process of acquiring eclipse photometry such that the radii and orbital inclination can be determined. Since both components are very close to the cluster turnoff their masses and radii can be directly used to give a very accurate age estimate for the cluster by comparing to isochrones in the (mass, radius) plane and requiring that they both lie on the same isochrone.

Star Constellation Type of star Distance (light- years) Mass Radius Surface Intrinsic Magni-(solar (solar temperature brightness tude masses) radius) (Fahrenheit) (relative to sun) ... 

Note Mass, radius and brightness are given in solar units. For example, Sirius A has 2.3 solar masses, is 2.5 times the size of the Sun and is intrinsically 35 times brighter than the Sun. 1 Solar mass = 2 x 1030 kg = 330000 Earth masses 1 solar radius = 700000 km = 110 Earth radii. 

This leads to a mass-density or mass-radius relation... 

Figure 9. Mass-radius relation for pure strange quark matter stars (left) and hybrid stars (right). GO - G4 models of hybrid stars corresponding to different parameters of the model. H pure hadron star, QC star has a quark core, MC star has a mixed core, from Thoma et al. (2003).

Figure 10. Mass-radius relation for a quark star with A//u = 1.6 and A/n = 1. The weak-coupling results for the same choice of renormalization scales are shown as dashed lines, from Andersen and Strickland (2002).

Initial spin period, magnetic field, spatial velocity, mass, radius etc. (plus possible correlations between them). 

Abstract We discuss the high-density nuclear equation of state within the Brueckner-Hartree-Fock approach. Particular attention is paid to the effects of nucleonic three-body forces, the presence of hyperons, and the joining with an eventual quark matter phase. The resulting properties of neutron stars, in particular the mass-radius relation, are determined. It turns out that stars heavier than 1.3 solar masses contain necessarily quark matter. 

In order to calculate the mass-radius relation, one has then to solve the well-known Tolman-Oppenheimer-Volkov equations [23],... 

The consequences for the structure of the neutron stars are illustrated in Fig. 9, where we display the resulting neutron star mass-radius curves, comparing now results obtained with different nucleonic TBF, in analogy to Fig. 5. One notes that while in Fig. 5 the different TBF still yield quite different maximum masses, the presence of hyperons equalizes the results, leading now to a maximum mass of less than 1.3 solar masses for all the nuclear TBF. 

The final result for the structure of hybrid neutron stars is shown in Fig. 11, displaying mass-radius and mass-central density relations. It is evident that the most striking effect of the inclusion of quark matter is the increase of the maximum mass, now reaching about 1.5 M . At the same time, the typical neutron star radius is reduced by about 3 km to typically 9 km. Hybrid neutron stars are thus more compact than purely hadronic ones and their central energy density is larger. For completeness, the figure shows besides static neutron star... 

For a strong first order transition, as suggested in [21], a new branch in the mass-radius diagram could exist, see Figure 9. 

Figure 8. The mass-radius relation, for the linear equation of state (12), of a static quark star.

Figure 9. For a strong first order transition, a new stable branch can exist in the mass-radius relation (Figure from [17]).

A color superconducting phase is a reasonable candidate for the state of strongly interacting matter for very large quark chemical potential [16-20], Many properties of such a state have been investigated for two and three flavor QCD. In some cases these results rely heavily on perturbation theory, which is applicable for very large chemical potentials. Some initial applications to supemovae explosions and gamma ray bursts can be found in [21] and [22] respectively, see also [27], The interested reader can find a discussion of the effects of color superconductivity on the mass-radius relationship of compact stars in [45]... 

Figure 6. Mass-radius relation of different compact star configurations. The left panels correspond to calculations with parameter set RKH for the quark matter phase and the right panels to parameter set HK, respectively. From the upper panel downwards the hadronic phase is described by a BHF calculation without hyperons [55], a relativistic mean field calculation [57] and a chiral SU(3) model [53].

Figure 3. Mass-radius relation for a pure HS described within the GM1 model and that of the HyS or SS configurations for several values of the Bag constant and ms = 150 MeV and as = 0. The configuration marked with an asterisk represents in all cases the HS for which the central pressure is equal to I The conversion process of the HS, with a gravitational mass equal to Mcr, into a final HyS or SS is denoted by the full circles connected by an arrow. In all the panels a is taken equal to 30 MeV/fm2. The dashed lines show the gravitational red shift deduced for the X-ray compact sources EXO 0748-676 (z = 0.35) and IE 1207.4-5209 (z = 0.12-0.23).

In Fig. 6, we report the radius an the mass of the compact star RX J1856.5-3754 inferred by Walter Lattimer (2002) (see also Kaplan et al. 2002) from the fit of the full spectral energy distribution for this isolated radio-quite neutron star , after a revised parallax determination (Kaplan et al. 2002) which implies a distance to the source of 117 12 pc. Comparing the mass-radius box for RX J1856.5-3754 reported in Fig. 6 with the theoretical determination of the MR relation for different equations of state, one concludes that RX J1856.5-3754 could be (see e.g. Fig. 2 in Walter Lattimer, 2002) either an hadronic star or an hybrid or strange star (see also Drake et al. 2002). 

Figure 9. Mass - radius relations for compact star configurations with different EoS purely hadronic star with HHJ EoS (long-dashed), stable hybrid stars for HHJ - INCQM EoS with 2SC (solid) and without 2SC phase (dash-dotted) for the Gaussian formfactor. We show the influence of a tiny variation of the coupling constant Gi by the filled grey band. The difference between the models 2SC and 2SC corresponds to a shift in the bag function (see Fig. 8) 3 MeV/fm3. For comparison, observational constraints on the compactness are given from the "small compact star RX J1856.5-3754 and from the high surface redshift object EXO 0748-676 which can both be obeyed by our hybrid star EoS.

Figure 10. The Mass - radius relation and mass - central energy density dependencies for rotating (dashed lines) and nonrotating (solid lines) hybrid star configurations...

We used a mass-radius relation for white dwarfs. Based on (1) — (4) we found that the probability for a nova not to occur in 50 years in these systems is 0.78-0.84, with the range resulting from differences in mass radius relations. We therefore find, that the fact that a nova outburst has not been observed in these systems is not surprising and does not imply that DN and CN systems are different. Incidentally, the systems found most likely to undergo a CN outburst in the near future were ... 

E°ads — saturation concentration of cellulase/gram of cellulose K = no. adsorption sites/gram of cellulose rj = no. adsorption sites/unit surface area R = mean mass radius/cellulose particle n — no. cellulose particles/gram of cellulose... 

Element Atomic mass Radius (nm) Pauling (1940) Bondi (1968) et al. (1970)... 

As a consequence, particles on the acoustic axis converge during a number of oyoles and eventually oollide. Equations for the dynamios of partioles under standing waves that include the mass, radius and velocity of the partioles, the kinetio and dynamio velooities of the fluid and its velocity due to the aooustio wave have been derived [53],... 

Zapolsky H. S. and Salpeter E. E. (1969) The mass-radius relation for cold spheres of low mass. Astrophys. J. 158, 809-813. 

Lupo M. J. and Lewis J. S. (1979) Mass-radius relationships in icy satellites. Icarus 40, 157-170. 

For estimation of thermodynamic properties of dissolved species, one can use the Entropy Correspondence Principle ( ), where the entropy of an ion at a given temperature is regarded as a function of the charge, the dielectric constant, mass, radius, and other variables. The function depends mainly upon the choice of the standard state, solvent, and temperature. The temperature dependency of entropy was derived based on the above principles and experimental data. By conducting the a square regression on Criss-Cobble s data ( ), we obtained the following eqtiation for calculating the entropies of species in aqueous solution. 

The thermal time scale mass transfer is initiated by the shrinkage of the orbit as a consequence of angular momentum conservation, as long as Md > Mwd- the Roche lobe filling main sequence star is squeezed into a continuously shrinking volume. The stellar radius can fall much below the zero-age main sequence mass-radius relation. Consequently, orbital periods can be achieved which are smaller by a factor / than periods corresponding to unperturbed main sequence stars filling their Roche lobes. For conser-... 

Electrical conductivity of molten alkali and earth alkali metal halides increase by 2 orders at melting. The electrical conductivity of these melts is purely ionic and their electrolysis follows Faraday s law. Deviations from this law are caused by secondary processes during the electrolysis, as for example dissolution or the back reaction of the electrolysis products. Electrical conductivity and thus also the mobility of ions is, in general, given by quantities like ionic charge, ionic mass, radius, polarizability, and the coordination number. 

The notion of an internal coordinate is very general and can include particle mass, radius, geometry, angular momentum, etc. As we describe later, modeling how the internal coordinates change due to the microscale physics is usually nontrivial. 

---