# Massimo Capaccioli » 13.Cosmic distance scale - Part III

### Globular clusters as distance indicators

Globular clusters are spherical systems of coeval Population II stars (from 104 to 106) orbiting about galaxies cores either in the halo or in the bulge. The luminosity function of globular clusters (GCs) is empirically found to have a roughly Gaussian shape:

$\displaystyle{N(M)\propto\exp{\left( -\frac{(M-M_0)^2}{2\sigma^2}\right) }},$

where $\sigma$ is the dispersion about the peak value $M_0.$

Galactic globular cluster M80 = NGC 6093 in Scorpius, pictured by HST. Credit: NASA (HST).

### Globular clusters as distance indicators

If this peak value is a universal constant (e.g., $M_0$ varies by less than $0.2\ mag$ among Virgo clusters ellipticals) or if it is established to vary in a predictable way (as, for instance, is the mean magnitude of Cepheids), than $M_0$can be used as a distance indicator. While a promising technique, the GCLF has several problems, connected with the still poor understanding of this class of objects. For instance, there are:

1. zero point differences with other indicators (PNLF);
2. differences between the Milky Way GCLF and that of M31;
3. systematic differences among populations of GCs with different metallicity; those of elliptical galaxies (older) are brighter than more metal poor in spiral galaxies.

Luminosity function of GCs in the bright Virgo elliptical M87 (Whitmore et al., Ap.J.Lett., 454, L73, 1995).

### Novae

A nova is the result of the accretion of hydrogen-rich material by a white dwarf bound in a close binary system. The abrupt and fast increase of the brightness of this cataclysmic variable is due to thermonuclear (non destructive) reactions on the surface of the degenerated star which liberate some 1045 ergs of energy over a period of some weeks.

Consequently novae are powerful transient phenomena, easy to identify (if seen) and detectable up to distances of several Mpc.

Unfortunately, the peak brightness of these explosions is far from being constant. Nevertheless, by studying the novae of M31, the magnitude at maximum, $m{\rm (max)}$, has been found to define a narrow empirical relation with the rate of decline, $v_d$, that is with the average rate in mag/days at which the nova drops to two magnitudes below maximum (see figures): $m_{pg}({\rm max})=16.88-0.81\times\atan\left[ \frac{\log(100\times v_d)-0.98}{0.19} \right].$

Visual light curve of Nova Herculis 1991, sketching the way the rate of decline is estimated.

### Novae: maximum magnitude vs. rate of decline

Using this relation, Capaccioli et al. were able to calibrate the distance modulus of the Virgo cluster of galaxies with the M31 novae and that of the Coma cluster with the Virgo SNe, deriving a value of the Hubble constant of $H_0=70\ \it km/s/Mpc$, very close to the modern figure. Novae are good distance indicators but:

1. they require time-consuming survey campaigns to be discovered;
2. the rapid rise of the light curve makes it hard to observe $m{\rm (max)}$;
3. it is not certain that the $m({\rm max})$ vs. $v_d$ relation is universal.

Capaccioli et al., Ap.J., 350, 110, 1990.

### Supernovae types: type II

Supernovae (SNe) are extremely energetic stellar explosions with which the lifetime of some stars ends. There are essentially two conditions under which such a dramatic phenomenon occurs.

Type II SN: results from rapid collapse and violent explosion of a massive star ($M/M_{\odot} > 8$). When the mass of the inert core exceeds the Chandrasekar limit of about $1.4 M_{\odot}$, electron degeneracy alone is no longer sufficient to counter gravity and maintain stellar equilibrium. The structure collapses in free fall until the neutron degeneracy installs, which causes the outer layers of the star to bounce away. This is the type of SN called of type II: a funny name if you observe that the progenitors belong to Pop I. In fact, massive stars are required, which exist in a very young population only. So, type II SNe are found in spiral galaxies but not early types (E and S0). A unique characteristic of these core-collapse SNe is the presence of hydrogen lines in their spectra.

Binding energy curve. Note the peak at A = 56÷58, which are the mass numbers for stable iron.

### Supernovae types: type Ia

Type Ia SN: it occurs when a small mass white dwarf ($M<1.44\ M_\odot$) approaches the Chandrasekhar limit because of a mass transfer from a companion red giant which has filled its Roche lobe (figure). The situation is analogous to that of core-collapse SNe, but in this case the entire star is involved in the collapse. The explosion is boosted by the sudden start of the carbon (and, partly, oxygen) fusion, which injects an uncontrolled amount of energy in the degenerate (thus insensitive to energy supplies) core.

It is apparent that these supernovae imply old (Pop II) stars, so it is not surprising that they are found in all types of galaxies. The fact that the explosion involves always the same mass, $\sim1.44\ M_\odot$, is the (debated) reason why the luminosity of the type Ia SNe is pretty constant: a property which promotes these stars to good standard candles.

The figure shows the Roche lobs for a red giant and a white dwarf. The giant star throws material onto the WD through the Lagrangian point L1.

The Roche lobs for a red giant and a white dwarf.

### Supernovae types: type Ia

The same mechanism of mass accretion on a white dwarf and subsequent deflagration describes novae too. In the latter cases, the mass transfer is much slower than for SNe Ia, and the explosion, by which the object gets rid of the mass excess, is not destructive. Note that the classification of SNe contemplates also the types Ib and Ic, which have in fact progenitors of the same physical family of type II. Their classification as type I is historically based on the observation that all these SNe lack hydrogen lines in their spectra.

Z Camelopardalis, a recurrent nova. Credit: NASA/JPL-Caltech.

### Type I Sne and their light curves

Light curves of SNe Ia have roughly the same shape, and a magnitude at maximum, $\Delta M_{pk}\sim\pm 0.5\ mag$ (top figure).

In the early 1990’s an empirical correlation was found between $M_{pk}$ and the rate of decline of the luminosity with time (after peak), $dm/dt$:

$M_{pk}=A+B\left(dm/dt \right).$

Since intrinsically brighter SNe Ia decline more slowly, it is $B > 0$$A$ is instead fixed by assuming a value of $H_0$.

Light curves of several type Ia Sne. Credit: IPAC-Caltech.

Top panel mean light curve for the SNe Ia, after the correction mentioned in the text. Credit: IPAC-Caltech.

### Type I Sne and their light curves

Once corrected for this correlation, SNe Ia turn into good standard candles, well suitable to gauge rather accurate distances even when the discovery occurs past the peak (in which case one needs the colors to extrapolate the value of $M_{pk}$). The peak brightness of these stars is bright enough (for instance, $(M_{pk})_V=-19.6$) to let us see objects out to $z\sim 1.4$ (i.e. at luminosity distances of the order of $10\ Gpc$, when the universe was $4.5\ Gyr$ old).

There is the question, though, of the time evolution of SNe phenomena which is still open. It is clear that $M_{pk}$ varies with the environmental properties (SFR, metallicity). However, it seems that the correction for the decline rate might account for this effect also, allowing us to conclude that corrected SNe Ia peak absolute luminosities do not vary with distance (that is, with cosmic time).

### Supernovae, distances, and cosmology

Supernovae are rare phenomena (1 SN every  $50\ \it yr$ expected in our galaxy) and difficult to discover. Continuous monitoring of galaxy clusters is requested to pick up these stellar detonations.

Extensive survey projects have been undertaken in the last couple of decades (including those with HST) to find out SNe and to measure their light curves and spectra.

A specific space observatory (SNAP) is planned by NASA for year 2020 to search for and measure SNe out to large $z$.

Type Ia Sne help us to select to falsify cosmological models (from E.L. Wrigh).

### Supernovae, distances, and cosmology

The most striking result of this effort, based on type Ia SNe, is possibly the discovery of an accelerated expansion phase in the lifetime of our universe. This finding has placed the yet unsolved question about the energy source for such an acceleration.

The figure plots average data points (luminosity distance vs. redshift) relative to the Union compilation of $>300$ type Ia SNe (Kowalski et al. 2008) against various cosmological models. Red, black and green curves are for closed (density parameter $\Omega=\rho/\rho_{crit}=2$ [what is that?]), flat ($\Omega=1$), and empty ($\Omega=0$) universe. The first two are clearly ruled out.

The data suggest two regimes for the acceleration: the universe is decelerating at high $z$, but it is accelerating at intermediate-low distances. This is consistent with a cosmological constant and a fair amount of Dark Matter.

Type Ia Sne help us to select to falsify cosmological models (from E.L. Wrigh).

### Expansion parallax

Let’s assume that, at the time $t_1$, we observe the explosion of a star at a distance $D$ and that the event produces a spherical shell which expands isotropically with a constant velocity $V$.

If, at the time $t_2$ we measure an angular diameter $\theta$ for the shell, then: $2V\times\left(t_2-t_1 \right)=2R=\theta\times D$.

This method has been successfully applied to planetary nebulae, novae and supernovae. It works well as long as the assumptions work.

### Distances by Surface Brightness Fluctuations

This direct method of gauging distance is based on the well known property of the loss of resolution with distance; a fact you experience every time you look at the landscape from an airplane climbing to route altitude.Since loss of resolution implies a smoothing, the expectation is that one same galaxy looks less and less noisy as it is seen farther and farther out. Let’s compute the effect.

Consider a galaxy at distance $D$, made of identical stars of luminosity $L$, distributed in such a way that there are “n” stars per unit area on the projected image of the galaxy.

Let’s assume that our angular resolution is $\partial\theta$. Every element $\partial\theta\times\partial\theta$ contains $\overline{N}=n\left(D\partial\theta \right)^2$ unresolved stars, each with a flux:

$\displaystyle{ f=\frac{L}{4\pi D^2}}$.

The mean flux is then:

$\displaystyle{ \overline{F}=f\overline{N}= \frac{L}{4\pi D^2}n\left( D\partial\theta\right)^2=\frac{nL(\partial\theta)^2}{4\pi}}$,

which, as expected for a surface brightness, does not depend on distance. However, the different elements of the image are only statistically identical; they contain the same average number of stars $\overline{N}$ within the statistical error.

### Distances by Surface Brightness Fluctuations

Adopting Poisson’s statistics, the dispersion about $\overline{N}$ is: $(\overline{N})^{1/2}$, and the dispersion about $\overline{F}$: $\displaystyle{\sigma_F=f (\overline{N})^{1/2}= \frac{nL(\partial\theta)^2}{4\pi}\left[ n\left( D\partial \theta\right)^2\right]^{1/2}=\frac{n^{1/2}\partial \theta L}{4\pi} D^{-1}}$.

Thus the surface brightness fluctuation (SBF) scales with $D^{-1}$ and can be used to measure the distance. Note that the relative fluctuation is: $\displaystyle{\frac{\sigma_F^2}{F}=\left(\frac{n(\partial \theta)^2 L^2}{(4\pi)^2} D^{-2} \right)=\frac{L}{4\pi D^2}=f},$,

which is the apparent luminosity of a single star. In conclusion, by comparing the relative fluctuations of two galaxies with identical stars we shall obtain the ratio of their squared distances. If the distance of one such galaxy is known (calibration), the other is also found.

### Distances by Surface Brightness Fluctuations

More realistically, let’s imagine that a galaxy consists of a finite number $k$ of star classes. In each class ($i=1,k$) the stars have the same luminosity $L_i$ and a flux:

$\displaystyle{ f_i=\frac{L_i}{4\pi D^2}}$.

In this case the relative fluctuation is:

$\displaystyle{ \frac{\sigma_F^2}{F} =\frac{\sum_{i=1}^k\overline{N}_i f _i^2}{{\sum_{i=1}^k\overline{N}_i f_i}}=\displaystyle{\frac{\langle L\rangle}{4\pi D^2}}}$,

where the weighted mean of the stellar luminosities is:

$\displaystyle{ \langle L\rangle= \frac{\sum_{i=1}^k\overline{N}_i L _i^2}{{\sum_{i=1}^k\overline{N}_i L_i}}$.

The problem is to find $\langle L\rangle$, which is done, in the various photometric bands, using galaxies with known distance or building synthetic galaxies.

### Distances by Surface Brightness Fluctuations

The SBF method, introduced by the seminal paper of Tonry and Schneider (1988, A.J., 96, 807), is hard to use since it requires: the removal of all the source of noise which are not just Poissonian (foreground stars, globular clusters, cosmic rays detections etc.); the identity of the stellar populations of the target and template galaxies. The second condition, i.e. the dependence of the SBF on the stellar population, may be turn into an advantage if we compare galaxies at the same distance (cluster members). The different SBF’s will document different stellar population properties. Let’s see why.

The fluctuations of a SSP depend almost completely on the number $N_3$ of stars within 3 mag from the tip of the Red Giant Branch (RGB). The reason is that these stars are very bright and much less numerous than the others (the smaller the number, the higher the fluctuation). Since an increasing in metallicity reduces the slope of the RGB, it also increases $N_3$ and consequently decreases the SBF.

### Effects of Surface Brightness Fluctuations

Distance effects on the SBFs of two identical galaxies, one 3 times more distant than the other. Credit: J. Tonry.

### Sunyaev-Zel’dovich effect

Galaxies in clusters are embedded in a hot gas which produces an X-ray emission and, at the same time, interacts with the cosmic microwave background (CMB) radiation [what is that?], modifying its spectrum. This information may be used to appraise the linear size of the cluster (in addition to the density of the hot gas), which is then compared with the measured angular size of the projected image of the cluster to derive its distance.

The method, that we are going to sketch, has been devised by the Russian theoretical physicists Rashid Sunyaev and Yakov Zel’dovich. The X-ray photons are generated by the scattering of electrons with ions (bremsstrahlung process). Their rate depends on the temperature $T_e$ and density $\rho_e$ of the hot gas electrons. $T_e$ is estimated by the shape of the X-ray continuum; the electron density and the line-of-sight diameter of the cluster $\Delta D$ are given by the X-ray flux as:

$f\propto\int \rho_e^2 dl=\rho_e^2 \Delta D$

under simplified assumptions on the homogeneity of the gas cloud. With the additional assumption of spherical symmetry, the distance would be readily known if one could disentangle $\Delta D$ from $\rho_e$.

### Sunyaev-Zel’dovich effect

To this purpose we use the distortions on the CMBR spectrum (see next page) caused by the interaction of the cold ($T=2.73\ K$) photons with the hot electrons (inverse Compton effect).

The effect depends on $T_e$ (already known by the X-ray spectrum), and again on $\rho_e$ and $\Delta D$. This is enough to disentangle the value of the linear diameter of the cluster and to obtain the distance to the galaxy cluster by the measure of the apparent diameter [check the assumptions].

While very simple in principle and independent of the parallax measurements and standard candle-based indicators, the Sunyaev-Zeldovich (SZ) effect is of difficult use for both theoretical and practical reasons. The simplified model of a uniform sphere is not realistic. Moreover high resolution X-ray telescopes and improved calibrations for both the X-ray and CMBR detectors are requested. While so far the SZ method has not given robust enough results, it is expected to be a powerful tool to gauge very large distances in the universe as the technology of X-ray and microwave measurements improves.

### Sunyaev-Zel’dovich effect

The interaction between a cold gas of photons (CMBR at $\sim3\ K$) with a very hot gas of electrons ($10^6\div10^7 K$) warms up the former at the expenses of the latter. The phenomenon is described by the so-called inverse Compton effect. The black-body spectrum of the CMBR is reshaped and globally shifted to the blue (higher frequencies; see figure at top). The difference between the original CMBR spectrum and that modified by the SZ effect, with the characteristic sine shape, is shown in the figure at the bottom of the page as a solid line. This is the so called thermal SZ effect. There is another effect, though, which alters the CMBR spectrum, named kinetic SZ effect. It is due to the energy released to the CMB photons by the overall peculiar motion of the cluster with respect to the cosmic background. For a complete treatment of this effects, see: M. Birkinshaw, Physics Reports, 310, 97, 1999; also available in the Web. The figure at the top is the CMBR black-body spectrum (dashed) modified by the SZ effect (solid line). At the bottom the sine wave shape of the SZ thermal effect (solid) and the kinetic SZ (dashed). The dotted line is the rescaled original black-body spectrum of CMBR.

Carlstrom et al., 2002, ARAA , 40, 1.

Carlstrom et al., 2002, ARAA, 40, 1.

### Dynamical indicators

There are indicators which are based on the fact that that, under some assumptions holding for photometric families of galaxies, the total luminosity of an object, $L$, is proportional to some power of the average specific kinetic energy, that is of the rotational velocity $V$ for spirals and of the velocity dispersion $\sigma$ for ellipticals (see lecture 8):

Tully-Fisher relation: $L\propto V^\alpha$, with $3<\alpha<4$; depending on the observing band,

Faber-Jackson relation: $L\propto \sigma^\beta$, with $\beta\simeq 3.1$ in the $V$-band.

The strategy is as follows: assume that the above relations have been calibrated, so that, by measuring the kinematical term ($V$ or $\sigma$), which are not affected by distance [why?], one derives the intrinsic luminosity $L$. This latter can be compared with the apparent luminosity to derive a distance modulus.

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