Abstract. We review the present status of the null cone approach to numerical evolution being developed by the Pittsburgh group. We describe the simplicity of the underlying algorithm as it applies to the global description of general relativistic spacetimes. We also demonstrate its effectiveness in revealing asymptotic physical properties of black hole formation in the gravitational collapse of a scalar field.

INTRODUCTION

We report here on a powerful new approach for relating gravitational radiation to its matter sources based upon the null cone initial value problem (NCIVP), which has been developed at the University of Pittsburgh. We are grateful to the many graduate students and colleagues who have made important contributions: Joel Welling (Pittsburgh Supercomputing Center), Richard Isaacson (National Science Foundation), Paul Reilly, William Fette (Pennsylvania State University at McKeesport) and Philipos Papadopoulous.

As will be detailed, the NCIVP has several major advantages for numerical implementation, (i) There are no constraint equations. This eliminates need for the time consuming iterative methods needed to solve the elliptic constraint equations of the canonical formalism, (ii) No second time derivatives appear so that the number of basic variables is half the number for the Cauchy problem. In fact, the evolution equations reduce to one complex equation for one complex variable. The remaining metric variables (2 real and 1 complex) are obtained by a simple radial integration along the characteristics.

Abstract. We present three-dimensional Newtonian and post-Newtonian codes, including the gravitational radiation damping effect, using a finite difference method. We follow the emission of gravitational radiation using the quadrupole approximation. Using these codes we calculate the coalescence of a neutron star binary. For Newtonian calculations the initial configuration is given as a hydrostatic equilibrium model of a close neutron-star binary. Calculations were performed for neutron stars of different masses as well as of the same masses. In order to evaluate general relativistic effects, we compare the results of the calculation of the coalescence of a binary comprising two spherical neutron stars using the post-Newtonian code with results using the Newtonian code.

INTRODUCTION

The most promising sources for laser-interferometric gravitational-wave detectors are catastrophic events such as the gravitational collapse of a star or the coalescence of a black-hole or neutron-star binary. We need to know the characteristics of the waves for design of detectors. It requires general relativistic calculations of stellar collapse and binary coalescence. In the last decade, 2 dimensional (2D) calculations were successfully performed for a head-on collision of two black holes (Smarr 1979) and axisymmetric collapse of a rotating star (Stark and Piran 1986). They found that the efficiency of gravitational wave emission (the ratio of the energy emitted in gravitational radiation to the total rest mass) is less than 0.1%. Nakamura, Oohara and Kojima (1987), on the other hand, pointed out that the efficiency may be much greater in non-axisymmetric black-hole collision.

Abstract. Our project was inspired by the prospect that a new non-electromagnetic astronomy will develop by the end of the century. Projects like Virgo and Ligo will lead to detectors able to detect extra-Galactic and Galactic sources of gravitational radiation. New generation of neutrino detectors like Superkamiokande will be able to detect various Galactic neutrino sources. All these considerations motivated us to study in detail potential Galactic sources of bursts of the gravitational radiation and neutrinos. In this paper, our projects are described in some detail. The advantages and the drawbacks of the numerical technique used in our computer simulations (pseudospectral methods) are discussed. Possible applications of the numerical methods are illustrated by some examples of astrophysical interest: coalescence of two neutron stars, mini-collapse of a neutron star (phase transition) and formation of a black hole due to the collapse of a neutron star.

INTRODUCTION

The main idea, which motivated our project, is that massive stellar cores, involved in supernovae of type II, are not the only collapsing Galactic objects generating bursts of gravitational waves that could be detected by the next generation of gravitational wave detectors. It is quite likely that SNI and SNII are only an optically detectable subset of a larger class of collapse events, which are less spectacular (as far as electromagnetic radiation is concerned) but perhaps quite frequent, and which are able to radiate a conspicuous amount of gravitational radiation.

Abstract. We investigate initial data for localized gravitational waves in space-times with a cosmological constant Λ. By choosing the appropriate extrinsic curvature, we find that the Hamiltonian and momentum constraints turn out to be the same as those of the time-symmetric initial value problem for vacuum space-times without Λ. As initial data, we consider Brill waves and discuss the cosmological apparent horizon. Just as with Brill waves in asymptotically flat space-time, the gravitational “mass” of these waves is positive. Waves with large gravitational mass cause a strong cosmic expansion. Hence, the large amount of gravitational waves do not seem to be an obstacle to the cosmic no-hair conjecture.

INTRODUCTION

The present isotropy and homogeneity of our universe is something of a mystery within the framework of the standard big bang scenario. The inflationary universe scenario, however, is one of the favourable models which may explain the so-called homogeneity problem [1]. In this scenario, when a phase transition of the vacuum occurs due to an inflaton scalar field and supercooling results, the vacuum energy of the scalar field plays the role of a cosmological constant and the space-time behaves like the de Sitter one with a rapid cosmic expansion. This phenomenon is called inflation. As a result, all inhomogeneities go outside the horizon by rapid cosmic expansion. After inflation, the vacuum energy of the scalar field decays into radiation and the standard big bang scenario is recovered. However, there still remains a question in the above scenario.

Solar models

Here we describe only a few representative main sequence stellar models. One is for the zero age sun, that is, the sun as it was when it had just reached the main sequence and started to burn hydrogen. We also reproduce a model of the present sun, a star with spectral type G2 V, i.e. B − V ∼ 0.63 and Teff ∼ 5800 K, after it has burned hydrogen for about 4.5 × 109 years. In the next section we discuss the internal structures of a B0 star with Teff ∼ 30 000 K and an A0 type main sequence star with Teff ∼ 10 800 K. There are several basic differences between these stars. For the sun the nuclear energy production is due to the proton–proton chain, which approximately depends only on the fourth power of temperature and is therefore not strongly concentrated towards the center. We do not have a convective core in the sun, but we do have an outer hydrogen convection zone in the region where hydrogen and helium are partially ionized. The opacity in the central regions of the sun is due mainly to bound-free and free-free transitions, though at the base of the outer convection zone many strong lines of the heavy elements like C, N, O and Fe also increase the opacity.

In Table 13.1 we reproduce the temperature and pressure stratifications of the zero age sun. In Table 13.2 we give the values for the present sun as given by Bahcall and Ulrich (1987). The central temperature of the sun was around 13 million degrees when it first arrived on the main sequence;…

Available energy sources

So far we have only derived that, because of the observed thermal equilibrium, the energy transport through the star must be independent of depth as long as there is no energy generation. The energy ultimately has to be generated somewhere in the star in order to keep up with the energy loss at the surface and to prevent the star from further contraction. The energy source ultimately determines the radius of the star.

Making use of the condition of hydrostatic equilibrium we estimated the internal temperature but we do not yet know what keeps the temperature at this level. In this chapter we will describe our present knowledge about the energy generation which prevents the star from shrinking further.

First we will see which energy sources are possible candidates. In Chapter 2 we talked about the gravitational energy which is released when the stars contract. We saw that the stars must lose half of the energy liberated by contraction before they can continue to contract. We might therefore suspect that this could be the energy source for the stars.

The first question we have to ask is how much energy is actually needed to keep the stars shining. Each second the sun loses an amount of energy which is given by its luminosity, L = 3.96 × 1033 ergs−1, as we discussed in Chapter 1. From the radioactive decay of uranium in meteorites we can find that the age of these meteorites is about 4.5 × 109 years. We also find signs that the solar wind has been present for about the same time.

In Volume 3 of Introduction to Stellar Astrophysics we will discuss the internal structure and the evolution of stars.

Many astronomers feel that stellar structure and evolution is now completely understood and that further studies will not contribute essential knowledge. It is felt that much more is to be gained by the study of extragalactic objects, particularly the study of cosmology. So why write this series of textbooks on stellar astrophysics?

We would like to emphasize that 97 per cent of the luminous matter in our Galaxy and in most other galaxies is in stars. Unless we understand thoroughly the light emission of the stars, as well as their evolution and their contribution to the chemical evolution of the galaxies, we cannot correctly interpret the light we receive from external galaxies. Without this knowledge our cosmological derivations will be without a solid foundation and might well be wrong. The ages currently derived for globular clusters are larger than the age of the universe derived from cosmological expansion. Which is wrong, the Hubble constant or the ages of the globular clusters? We only want to point out that there are still open problems which might well indicate that we are still missing some important physical processes in our stellar evolution theory. It is important to emphasize these problems so that we keep thinking about them instead of ignoring them. We might waste a lot of effort and money if we build a cosmological structure on uncertain foundations.

The light we receive from external galaxies has contributions from stars of all ages and masses and possibly very different chemical abundances.

Period–density relation

In Volume 1 we saw that there is a group of stars which periodically change their size and luminosities. They are actually pulsating (the pulsars are not). When Leavitt (1912) studied such pulsating stars, also called Cepheids, in the Large Magellanic Cloud she discovered that the brighter the stars, the lpnger their periods, independently of their amplitude of pulsation. In Volume 1 we discussed briefly how this can be understood. The pulsation frequencies are eigenfrequencies of the stars. They are similar to the eigenfrequencies of a rope of length 2l, which is fastened at both ends but free to oscillate in the center (see Fig. 18. la). If you pull the rope periodically down in the center, first slowly and then more rapidly, you find that for a given frequency ν0 a standing wave is generated in the rope. For this frequency you need to put in only a very small amount of energy, much less than for the other frequencies, for which running waves are generated which interfere with each other and are therefore damped rapidly. The frequency ν0, which generates the standing wave, is an eigenfrequency of the rope. If you increase the amplitude of the wave you still find the same eigenfrequency ν0. If you increase the frequency further you again find running waves until you reach another frequency ν2, three times as large as ν0, for which another standing wave is generated. This wave has two nodes and a wavelength which is a third of the wavelength for the eigenfrequency ν0 (Fig. 18.1c).

Evolution along the subgiant branch

Solar mass stars

From previous discussions we know that solar mass stars last about 1010 years on the main sequence. Lower mass stars last longer. Since the age of globular clusters seems to be around 1.2 × 1010 to 1.7 × 1010 years and the age of the universe does not seem to be much greater, we cannot expect stars with masses much smaller than that of the Sun to have evolved off the main sequence yet. We therefore restrict our discussion to stars with masses greater than about 0.8 solar masses, which we observe for globular cluster stars.

We discussed in Section 10.2 that for a homogeneous increase in μ through an entire star (due to an increase in helium abundance and complete mixing), the star would shrink, become hotter and more luminous. It would evolve to the left of the hydrogen star main sequence towards the main sequence position for stars with increasing helium abundance. In fact, we do not observe star clusters with stars along sequences consistent with such an evolution (except perhaps for the socalled blue stragglers seen in some globular clusters which are now believed to be binaries or merged binaries). Nor do we know any mechanism which would keep an entire star well mixed. We therefore expect that stars become helium rich only in their interiors, remaining hydrogen rich in their envelopes. Since nuclear fusion is most efficient in the center where the temperature is highest, hydrogen depletion proceeds fastest in the center. Hydrogen will therefore be exhausted first in the center.

Color magnitude diagrams for globular clusters

The best way to check stellar evolution calculations is, of course, to compare calculated and observed evolutionary tracks. Unfortunately we cannot follow the evolution of one star through its lifetime, because our lifetime is too short – not even the lifetime of scientifically interested humanity is long enough. Only in rare cases may we observe changes in the appearance of one star, for instance when it becomes a supernova. Another example occurred some decades ago when FG Sagittae suddenly became far bluer, a rare example of stellar changes which are too fast to fit into our present understanding of stellar evolution.

Generally evolutionary changes of stars are expected to take place over times of at least 104 years (except perhaps for stars on the Hayashi track, where massive stars may evolve somewhat faster). How then can we compare evolutionary tracks? Fortunately there are star clusters which contain up to 105 stars all of which are nearly the same age but of different masses. In such very populous clusters there are a large number of stars which have nearly the same masses.

In Fig. 17.1 we show schematically evolutionary tracks of stars with about one solar mass. They all originate near spectral types G0 or G2 on the main sequence. Their lifetime, t, on the main sequence is about 1010 years. The evolution to the red giant branch takes about 107 years.

Definition and consequences of thermal equilibrium

As we discussed in Chapter 2, we cannot directly see the stellar interior. We see only photons which are emitted very close to the surface of the star and which therefore can tell us only about the surface layers. But the mere fact that we see the star tells us that the star is losing energy by means of radiation. On the other hand, we also see that apparent magnitude, color, Teff, etc., of stars generally do not change in time. This tells us that, in spite of losing energy at the surface, the stars do not cool off. The stars must be in so-called thermal equilibrium. If you have a cup of coffee which loses energy by radiation, it cools unless you keep heating it. If the star's temperature does not change in time, the surface layers must be heated from below, which means that the same amount of energy must be supplied to the surface layer each second as is taken out each second by radiation.

If this were not the case, how soon would we expect to see any changes? Could we expect to observe it? In other words, how fast would the stellar atmosphere cool?

From the sun we receive photons emitted from a layer of about 100 km thickness (see Volume 2). The gas pressure Pg in this layer is about 0.1 of the pressure in the Earth's atmosphere, namely, Pg = nkT=105 dyn cm−2, where k = 1.38 × 10−16 erg deg−1 is the Boltzmann constant, T the temperature and n the number of particles per cm3.