Notes 1

Atomic Nuclei

The atomic nucleus is quantum system composed of protons, neutron, and electrons who interact together to form a bound system. The forces acting on this system, strong force, weak force, the Coulomb force, and gravity, act on the system over various length scales that effect the overall distributions of the particle wavefunctions composing the nucleus. The strong force acts on length scales of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 10^{-15}} m and below, primarily affecting the quark components of the previously mentioned nucleons. The weak force acts on a scale of $10^{-18}$ m, acting on the range of the nucleons as it is the affect of boson exchange. The electric, i.e. Coulomb, force acting on the nuclei also acts at all ranges, modeling the interaction of the charged nucleon present in the system with one another. Finally, the gravitational force acts also acts at all ranges, and is the weakest of the four nuclear forces.

Within a given nucleus, denoted by Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle _Z^A X_N} , there are multiple levels of organizations. The protons and neutrons, closely bound by the nuclear forces, form a system whose energy levels are quantized, forming a shell structure hierarchy, for both the protons and neutron, whose energy levels often interact between one another. These nucleons are each composed of three individual particles known as quarks, bound together by the strong force.

The characteristics of the nuclides follow general trend lines based on both isomer, and isotope number, as seen by the nuclide chart posted here. A region of stability is observed, beyond which increases in proton or neutron number outside of the dripline (seen here in black) causes rapid decay of the nucleus in question.

Definitions for Abundances

To facilitate an understanding regarding the elements present in a given cosmological event or object, a framework must be established to properly measure the distribution of particles present. To this end, the nuclear abundance is presented as a measure of the number of a given isotope present in a quantity of measured element. This particle abundance is defined as follows:

$X_{i}={\frac {n_{i}}{\sum _{j}n_{j}}}\ ,$

where the sum is counted over all isotopes present in the data and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n_i} is the particle's number density. This is often set logarithmically and normalized to the amount of hydrogen present, creating what is known as the relative particle abundance:

$\epsilon _{i}=\log _{10}X_{i}+12\ ,$

Furthermore, the mass fraction is defined to be the fraction of total mass of a sample that is composed of the particular nucleus in question, i.e.

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X_i = \frac{m_i}{m_{tot}} = \frac{m_i n_i}{\rho} \approx \frac{A_i n_i}{\rho N_A} \ . }

Denoting the mass per baryon as

$Y_{i}\equiv {\frac {X_{i}}{A_{i}}}\ ,$

the particles number density can be defined by this baryon fraction and the density of the sample

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n_i = Y_i \rho N_A \ . }

The mean molecular weight is determined by

Other important quantities for consideration would be the mean molecular weight, $\mu _{i}={\frac {\sum _{i}{A_{i}Y_{i}}}{\sum _{i}Y_{i}}}={\frac {\sum _{i}X_{i}}{\sum _{i}Y_{i}}}={\frac {1}{\sum _{i}Y_{i}}}\,$, the electron abundance,Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Y_e = \sum_i Z_i Y_i \ ,} , and the electron number density, $n_{e}=Y_{e}\rho N_{A}\ .$

Solar System Abundances

Solar abundances, i.e. the quantity of various species of nuclei present in the solar system, tell us a great deal about how the solar system was formed and provide insight into galactic evolution as a whole. To this end, the nature of studying solar abundances can be broken into three disciplines studying the various classes of data present.

Earth materials

To this end, various materials on earth are examined to see what sort of composition of elements were present during the formation of the solar system. The problem that is encountered, however, is the chemical fractionation strongly hinders this process, changing the apparent concentrations. Isotopic compositions, however, do not exhibit this feature however and are ideal candidates for study.

Solar Spectra

Since the sun formed directly from presolar nebula, examination of the spectra observed from its photosphere, i.e. non-fusion process layer, sheds light on the early composition of the solar system prior to structure formation.

Meteorites

Finally, meteorites found on earth, which have not been exposed to extreme temperatures and pressures that would cause the chemical fractionation seen in earthen materials, can be used to sample presolar nebula composition. These meteorites are broken into three classes, stones(93%), stony irons(1.5%), and irons(5.5%). Stones, which are subdivided into chondrites(86%) and achondrites(7%) are by far the largest in abundance, with the chondrites providing some of the best, unfractionated samples of presolar nuclear composition.

Quantum Mechanics

In the late 19th and early 20th centuries, new physical theories were stretching the limits of classical mechanics as well as classical theories of electricity and magnetism. Physicists needed new theoretical tools for describing quantum systems that would reduce to that of macroscopic systems at the proper boundaries.

The famed double slit experiment offered new, unexplained experimental data which led to the to the understanding of particle-wave duality. This is to say that any particle has wavelike properties and thus can be represented as a wave, and visa versa. This is one of the most important theories in all of quantum mechanics. This data was obtained by firing a beam of electrons through 2 slits, and instead of seeing a bright band on the detector screen an interference pattern, similar to that of light waves, was observed.

Another example of the limitations of classical mechanics is in regard to the stability of the atom. Classical theory would suggest that electrons orbiting a nucleus would lose energy by emitting photons and subsequently crash into the nucleus. Niels Bohr proposed an atomic model in which electrons had discrete energy levels. This theory of quantized energy states was the foundation for modern day quantum mechanics. Once experimental data confirming this theory, or a theory similar, was obtained by James Franck and Gustav Hertz, research into quantum theory greatly increased.

The Schrödinger Equation

Possibly the most important equation in all of quantum mechanics is Schrödinger's wave equation. This partial differential equation represents the change of quantum states over time. The wavefunction represents the probability of a particular quantum state being occupied.

The most general form of the equation is as follows, where E is the energy operator, and H is the Hamiltonian operator.

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E\Psi = \hat H \Psi}

The equation can be rewritten in a time dependent form and a time independent form.

The time dependent equation: $i\hbar {\partial \over \partial t}\Psi (x,\,t)={\hat {H}}\Psi (x,\,t)$

The time independent equation: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {E}\psi(r) = - {\hbar^2 \over 2m} \nabla^2 \psi(r) + V(r) \psi(r)}

The time independent equation represents a situation in which the energy of the system is constant in time.

Thermodynamics

In order to build a proper mathematical framework for the understanding of astrophysical processes, a basic background in thermodynamics is needed. Included in this background would be pressure, number density, particle density, state density, total number of states, the equation of state, and various other principle thermodynamic variables.

Thermodynamical Variables

To begin, we look at the various key variables that define a statistical system. First we define entropy, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S\!} , as the energy of a system that is not available to do work. In general, entropy is defined as $S=k_{b}\ln {\Omega }\!$, where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Omega\!} is the number of states in the system.

The internal energy of the system, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle U\!} , is the energy necessary to create the system, defined by $dU=Tds-PdV+\sum _{i}\mu _{i}dN_{i}\!$, where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mu\!} is the chemical potential.

The chemical potential, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mu\!} , of a system is the change in a characteristic thermodynamic state function per change in the number of molecules.

The Helmholtz free energy, $A\!$ is a measure the work obtainable from a closed thermodynamic system at a constant temperature and volume, defined by Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A = U - TS\!} and the change in Helmholtz free energy as $dA=-TdS-PdV+\sum _{i}\mu _{i}dN_{i}\!$.

The Gibbs free energy of the system, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G\!} , measures the useful work obtained from the from an isothermal, isobaric system, and is defined by Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G = U - TS + PV\!} , where the change in Gibbs free energy is said to be $dG=dU-SdT+VdP\!$.

Heat, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Q\!} , is defined to be the energy transferred between two thermodynamical systems via a thermal contact and is defined by $Q=qV\!$ with a change in heat given by Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle dQ = T dS \!} .

Finally, enthalpy, $H\!$, is defined as a measure of the total energy of a thermodynamical system, given by the expression Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H = U + PV\!} and the change in enthalpy as $dH=TdS+PdV+\sum _{i}\mu _{i}dN_{i}\!$.

Thermodynamical Quantities

Now we move to discuss the various equations that define a thermodynamical system outside of its principle variables. Defining the occupational probability function as Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(p)\!} and the state density function as Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \omega\!} , then we can express the following quantities in compact form:

Pressure

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P = \frac{1}{3}\int^{\infty}_{0}pv\omega(p)f(p)dp }

Energy Density

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle u = \frac{U}{V} = \int^{\infty}_{0}E\omega(p)f(p)dp }

Particle Density

$n={\frac {N}{V}}=\int _{0}^{\infty }\omega (p)f(p)dp$