### Astrophysics I

Luminosity: $L_s=\int_o^{\infty} L_{\lambda} d\lambda = 4\pi R_s^2 \int_0^{\infty} F_{\lambda} d\lambda$

Fλ is the radiative flux at the stellar surface. Energy may be lost due to neutrinos or direct mass loss.

Flux: At the Earth’s surface, observed flux is $f_{\lambda}=(\frac{R_S}{D})^2 F_{\lambda}$

Stellar flux $f=\frac{L}{4\pi D^2}$

Apparent magnitude, m, is based on the flux received at the Earth’s surface, fν (flux at frequency ν).

$m=-2.5\log f_{\nu}+const$

Fainter star has larger magnitude.

Absolute magnitude, M, is defined using the flux we would see from a star if it was 10 parsecs distant, Fν. This is a measure of the star’s brightness.

$M=-2.5\log F_{\nu}+const$

Bolometric magnitude is calculated using the total flux f integrated over all frequencies.

$m_{bol}=-2.5\log f_{bol}+const$ apparent

$M_{bol}=-2.5\log F_{bol}+const$ absolute

Distance modulus, d: by taking the difference between the apparent and absolute magnitudes, a measure of the distance to a star can be found.

$m-M=5\log d-5$

Distance measurements using trigonometric parallax: ‘close’ stars appear to move against the background of fixed stars, such that their positions appear different to an observer on Earth when observed at the two extremes of the Earth’s orbit.

Small angle $d=\frac{b}{\theta}$, where b is one astronomical unit (a.u.), i.e. the distance between the Earth and the Sun.

Distances in binary systems: for this we need the ‘deprojected extent’. i.e. the angle in the plane of the sky. Once we know this, everything should be easier.

d=distance to binary

a=semimajor axis of binary

Kepler’s Laws

1. The orbit of each planet is an ellipse with the Sun at one of the foci.
2. A line joining the planet and the Sun sweeps out equal areas in equal times.
3. The square of the orbital period is proportional to the cube of its average distance from the Sun.

$p^2 \propto R^3$

$p^2=\frac{4\pi^2}{G(m_1+m_2)}a^3$

Effective temperature is the equivalent black-body temperature corresponding to a star’s luminosity.

$L_S=4\pi R_S^2\sigma T_{eff}^4$

It ranges between 2000K and 100,000K.

Mass-luminosity relationship $L_S \propto M_S^{\nu}$ 3<ν<5

Lifetime of a star: for now, we’ll only be looking at the star during its lifetime as a star- formation of stars and their endpoints will be covered in their own topics later.

1. Main sequence: A star spends most of its lifetime on the main sequence ($\tau_{ms}\simeq 10^{10}yr (\frac{L_{T_0}}{L_O})^{-\frac{3}{4}})$. The star burns hydrogen in its core, and there is a simple relationship between luminosity and temperature.
2. Main sequence turnoff: core hydrogen is now exhausted; an isothermal helium core develops.

Note: some stars remain on the main sequence for longer than expected; these are known as blue stragglers and may be members of binary systems which have accreted additional mass, or fuel-rich stars powered by the merger of two stars.

1. Subgiant phase: when the core reaches 10-15% of the star’s total mass, it cannot support itself. The core collapses, whilst the envelope expands and cools. The core is now degenerate, and hydrogen burning moves to a shell outside the core. As this phase progresses, the effective temperature decreases and the convection zone gets deeper. When the convection zone hits the burning zone, luminosity increases, and the star enters the
2. Red giant phase: the stellar envelope is now large and the star evolves along a Hayashi track (~constant Teff). The region to the right of the Hayashi track on the H-R diagram is forbidden for highly convective stars. The core mass grows and luminosity increases.
3. Helium flash: the core mass reaches ~0.48 Mo (the subscript refers to our own Sun) and helium burning is ignited- thermonuclear runaway occurs until the core becomes non-degenerate. The core expands and luminosity decreases rapidly as the hydrogen shell source is weakened.
4. Horizontal branch: the core temperature is now ~108 K. The dominant energy source is hydrogen burning in the shell, although there is also helium burning in the core. Horizontal branch lifetime is about 10% of the main sequence lifetime. Once the central helium is exhausted, the core contracts and heats up again, becoming degenerate.
5. Asymptotic branch: Hydrogen and helium burning take place alternately in shells. The core is now isothermal carbon. The enevelope is eventually lost during the superwind phase, exposing the carbon core.
6. White dwarf: the endpoint for stars of M≤1.4Mo; supported by electron degeneracy. Endpoints of more massive stars will be discussed later.

Galactic (open) clusters: loose clusters of 10-1000 stars, not concentrated towards the centre of the cluster. They are found only in the disc of the galaxy.

Globular clusters: massive spherical associations containing 105 or more stars spherically distributed about the centre of the galaxy, many at great distances from the galactic plane.

Hertzsprung-Russell (H-R) diagrams are plots of luminosity versus effective temperature. By looking at H-R diagrams for different cluster types, we can further discuss their properties. Remember, all stars in a cluster are formed at the same time, but more massive stars evolve faster (you might think they’d last longer as they have more fuel to start with, but bear in mind that they’re also larger nuclear reactors that will be burning that fuel more quickly).

HR diagrams for star clusters

1. Young Galactic cluster
2. or

Stars spend most of their lifetime on the main sequence. More massive stars are more luminous and evolve more rapidly ($L \propto M^4$ approx). In young clusters, a few of the most massive stars may have evolved to giants. This evolution proceeds very rapidly, so there is a Hertzsprung gap between the top of the main sequence and the giants. These clusters are not old enough for low-mass stars to have evolved beyond the main sequence stage so no white dwarfs are present.

3. Old Galactic cluster: Short main sequence at faint luminosity (low mass) and well developed giant branch (GB). The more massive stars have evolved to the tip of the giant branch and beyond. Low mass stars evolve slowly, so no Hertzsprung gap; white dwarfs will be present but may be too faint to observe. Helium ignition occurs at the tip of the giant branch.
4.

5. Globular cluster: similar to old galactic cluster but there is also a horizontal branch (HB) containing stars which have evolved beyond the point of He ignition. Only clusters of low (sub-solar) metallicity show an HB. Globular clusters pre-date the formation of the Galactic disc and have metallicities of 0.01 -> 0.1 times solar metallicity. White dwarfs must be present.

Age determination of star clusters: all stars in a cluster are the same age- the thing that differentiates them is their mass.

Starting assumptions

i) Cluster must be a galactic cluster (high luminosity stars on the main sequence), hence the composition is roughly solar and the mass fraction of hydrogen XH~0.7

ii) Stars leave the main sequence when ~15% of their initial mass has been burnt to helium.

iii) $L_S \propto M_S^4$ for stars somewhat brighter than the sun

Given $L_S=500L_O$, $\frac{M_S}{M_O} \simeq \frac{L_S}{L_O}^{\frac{1}{4}} \simeq (500)^ {\frac{1}{4}}=4.73$

Total energy radiated per hydrogen atom consumed is

$(\frac{4\times 1.0078-4.0026}{4})m_Hc^2=7.15\times 10^{-3}m_Hc^2$

Total number of hydrogen atoms consumed is $\frac{fX_HM_S}{m_H}$ f~0.15 XH~0.7

Lifetime of cluster is ~ total energy radiated up to MS turnoff ÷ luminosity at turn-off point

~ total energy per hydrogen atom x no. H atoms consumed ÷ luminosity

More accurate age determination: use theoretical evolutionary tracks for stars of solar composition and various masses. Construct isochromes (lines of equal times) and fit them to observed HR diagrams. The best fit isochrome determines the age of the cluster.

Uncertainties in this method include:

• t=0 is taken to be the time when all stars in the cluster reach the main sequence, but of course they may not all reach it at the same time.
• Errors may occur in determining which stars actually belong to the clusters.
• Errors also occur in converting observing colours such as violet and blue to luminosities and effective temperatures for comparison with theory.