Central temperature

Figure 1.6 The lime-scales of the various processes of element synthesis in. stars. The curve gives the central temperature as a function of lime for a star of about one solar mass. TTie curve is schematic. ...

Fig. 1. Luminosity evolution of tracks of 0.5 M0 starting at different central temperature, labelled in the figure. At the bottom we sketch the energy liberated per gram due to the deuterium fusion with protons. The tracks of logT > 6.0 start in the middle of D-burning.

The central temperature can be deduced from the central pressure and density ... 

The more massive a star is, the higher the central temperatures and pressures are in the later stages. When the helium is consumed, the star fuses the carbon and oxygen into heavier atoms of neon, magnesium, silicon and even silver and gold. In this way, all the elements of the earth... 

Fig. 5.4. Schematic evolution of the internal structure of a star with 25 times the mass of the Sun. The figure shows the various combustion phases (shaded) and their main products. Between two combustion phases, the stellar core contracts and the central temperature rises. Combustion phases grow ever shorter. Before the explosion, the star has assumed a shell-like structure. The centre is occupied by iron and the outer layer by hydrogen, whilst intermediate elements are located between them. CoUapse followed by rebound from the core generates a shock wave that reignites nuclear reactions in the depths and propels the layers it traverses out into space. The collapsed core cools by neutrino emission to become a neutron star or even a black hole. Most of the gravitational energy liberated by implosion of the core (some 10 erg) is released in about 10 seconds in the form of neutrinos. (Courtesy of Marcel Amould, Universite Libre, Brussels.)...

Let us revisit the helium core, for this is the heart of the matter. As a result of the relentless contraction, the density continues to increase, and with it the temperature, but at a slower rate now. When the central temperature reaches 100 million K, the density is 10000 times greater than the density of water. Subject to an imperious quantum principle that forbids them any freedom to overlap one another, the electrons make a final stand against compression and confusion. Eor this reason, they assume an increasing contribution to the pressure. 

The maximum value possible for the temperature gradient must be the difference between the central temperature, Tc, and the surface temperature, Ts,... 

Figure 6. Calculated results from the similarity solution plotted as a function of time. Here R0 is the characteristic radius for energy deposition, A is the nonlinear amplitude of the temperature and density functions, T (R = 0) is the central temperature, and I is the induction parameter. The indicates the predicted induction time r0 = 1.0 x 10 4 sec E0 — 4.0 X 10 ergs R0 = 0.1 cm.

Combining Eq. (12-25) with the central-temperature-gradient relation gives... 

Normal oral temperature is 37°C (98.6°F), plus or minus one degree. Central temperature follows a circadian rhythm. The temperature falls by about 0.8 degrees during sleep and then rises just before waking and the highest temperature is attained at approximately 6 pm. Peripheral temperatures are about 0.5 degrees lower than central temperatures. 

Graph of estimated response versus pH at the central temperature of the design in Table 2.6... 

Figure 3 Evolution of the central temperature and density in stars of 15M and 25M from birth as hydrogen burning stars until iron core collapse (Table 1). In general, the trajectories follow a line of p oc but with some deviation downward (towards higher p at a given T) due to the decreasing entropy of the core. Nonmonotonic behavior is observed when nuclear fuels are ignited and this is exacerbated in the 5Mq model by partial degeneracy of the gas (source Woosley et al., 2002).

Figure 1 Some known giant planets and brown dwarfs, illustrated with a limited azimuthal slice (pie slice) to correct scale. The interiors are color coded according to the principal materials in each zone. Ice and rock refer to elements common in materials that are icy or rocky at normal pressures. Metallic hydrogen indicates ionization primarily through pressure effects. Modeled central temperature in K, and pressure in 10 bar, is shown. The radii of all but G1229b are known directly for G1229b, modeling of the brightness versus wavelength must be used to derive the radius. From left to right, the masses (expressed relative to the mass of Jupiter) are 45,1, 0.7, 0.3, and 0.05...

The temperature profiles within Jupiter and Saturn are thought to be essentially adiabatic, reflecting the high central temperatures and the dominant role of convection below the observable atmosphere where radiative processes become important. There may be deeper layers restricted in radial extent where the temperature profile becomes subadiabatic, due to a decrease in the total opacity, or by virtue of the behavior of the equation of state of hydrogen and helium. The same may hold for Uranus and Neptune, although with less certainty, because of the possibility that stable compositional gradients could exist and dominate the heat flow regime. In particular, Uranus small heat flow, if primordial and not a function of seasonal insolation, could be the result of a stable compositional stratification and hence subadiabatic temperature profile in the interior (Podolak et al., 1991). 

Figure I. Time scale for the various element syntheses in stars. The highly schematic curve gives the central temperature as a function of time fora star of about one. solar mass Degens, I9H9).

Classically, scattering may be pictured as an elastic collision between two particles. If hv breaks down for stars with very high central temperatures). Although a photon is not absorbed, scattering slows down the rate of energy escape by continually changing the photon direction. 

We have found expressions for the central temperature of a star. This must be greater than the ignition temperature for the hydrogen fusion if a star is to result. For a perfect gas, the pressure given by Eq. (93) must be the same as that given by the equation of state. Equating the pressures gives... 

The sun and other main-sequence stars (burning hydrogen in their core quiescently) evolve very slowly by adjusting their central temperature such that... 

The sun with a central temperature of 15.7 million degrees, (Tg0 = 15.7) burns by p-p chains. Slightly more massive star (with central temperature Tq > 20) burns H by the CNO cycle also. Davis et al.s solar neutrino experiment [23], which in 1968 had only an upper limit of the neutrino flux, itself put a limit of less than 9% of the sun s energy is produced by the carbon-nitrogen cycle (the more recent upper limit [36] is 7.3%, from an analysis of several solar neutrino experiments, including the Kamland measurements). Note however that for the standard solar model, the actual contribution of CNO cycle to solar luminosity is 1.5% [15]). In CNO cycle, nuclei such as C, N, O serve as catalysts do in a chemical reaction. The pp-chain and the CNO cycle reaction sequences are illustrated in Figs. 4 and 10. 

Fig. 9. Comparison of the temperature dependence of the p-p chain and the CNO cycle energy production. The points marked for the solar central temperature 7 = Tq = 15.7 are shown on both graphs. The CNO cycle generation dominates over the pp-chain at temperatures higher than Te = 20, so that for sun like stars, the pp-chain dominates. For more massive stars, the CNO cycle dominates as long as one of the catalysts C, N, or O have initial mass concentration at least 1%. Note the logarithmic scales of the graph and how both rates drop sharply with decreasing temperature, with that of CNO cycle even more drastic due to higher Coulomb barriers...

Inertial confinement is a pulsed system in which small pellets of D2 and T2 are irradiated by intense beams of photons or electrons. The surface of the pellet rapidly vaporizes, causing a temperature-pressure wave to move through the pellet, increasing its central temperature to greater than 10 K and causing fusion. If a fusion rate of approximately 100 pellets per second can be achieved, the result is a power output between 1 and 10 gigawatts. 

Figure 2. The base ten logarithm of the central temperature in Kelvins as a function of the base ten logarithm of the central density in grams per cubic centimeter in a model of the evolution of a star with mass twenty-five times greater than that of our Sun (7). The labels H, He, C, Ne, and O indicate the ignition of those burning phases in the star s life.

The external mechanisms by which heat is transferred to the environment are well known. These include convection, conduction, radiation, and evaporation. Although physiologists have been inclined to measure their own heat and mass transfer coefficients on living men and on mannikins, there is no reason to believe that standard engineering correlations are inadequate for this purpose. We will concentrate on the way in which the human body exploits various external heat transfer mechanisms in maintaining a constant central temperature. 

If the mean surface temperature is independent of metabolic rate, a heat exchange mechanism other than convection or radiation must be responsible for removing excess heat generated when a subject exercises. One s attention turns immediately to sweating, a phenomenon which has been studied in many laboratories. For subjects in equilibrium with the environment, the sweat rate is directly proportional to the total heat production rate. Perhaps this is expected because regulating central temperature by controlling the blood flow rate to skin, and, thus, the skin temperature is relatively ineffective. As the skin temperature rises, the difference between the central temperature and the skin temperature... 

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