# Massimo Capaccioli » 12.Cosmic distance scale - Part II

### Distance indicators

Distance indicators are classes of objects and/or of phenomena which allow us to measure the distances of celestial bodies through other observables such as luminosity or size. For this reason they are also called standard ladders.

Let’s consider the example of a photometric indicator, which is based on the fact that flux fades according to $D^{-2}$[always?]. At the distance $D,$ a source of corrected absolute magnitude $M_0$ appears with a magnitude:

$m = M_0 + 5\log D -5 + A + K,$

where $A$ is the total absorption [positive. Why?], and $K$ is a term due to the effect of the cosmic expansion [see ahead]. Then the distance: $\displaystyle{D = dex\left(\frac{m - M_0-A-K}{5}+1\right)},$

becomes known provided that one measures $m,$ estimates $A$ and $K,$ and knows a priori the value of the absolute magnitude $M_0.$ If the latter condition is satisfied, this makes the source a good candidate to become a photometric distance indicator. But it is not the only clause.

### Towards the construction of the cosmic scale

The sequence of cosmic distance indicators, with their ranges of applicability and their overlaps. Notice that each successive indicator is calibrated against the previous one, with an inevitable error propagation.

### Photometric distance indicators

Let’s define this important class of distance indicators properly. Assume that there exists a family of objects with the following properties:

1. standard candles: they all have the same absolute magnitude $M_0$ (or this latter is related to another observable as the period $P$ in a variable star);
2. easy to identify: they can be unambiguously recognized among the other celestial bodies;
3. well visible: they are intrinsically very bright, so that $m$ is measurable at large distances;

If $M_0$ is know, a measure of the apparent magnitude $m$ readily provides the uncorrected (observed) distance modulus: $\left(m-M_0\right)_{obs} = 5\log D_{obs}-5.$ The latter turns into the true distance modulus whether:

1. the apparent magnitude $m$ is corrected for all causes of absorption and for all other phenomena affecting the shape and the position of the observing band;
2. systematic evolutionary effects depending on distance (cosmic time) are accounted for;
3. cosmological effects are accounted for (when the case).

Writing down explicitly the dependence on the observing photometric band: $m_\lambda - M_{0,\lambda} = 5\log D -5 + A_\lambda + {\rm evolution}_\lambda.$

### Photometric distance indicators

In summary, photometric distance indicators are “good” if they are:

1. stable and easy to calibrate;
2. luminous;
3. rather common;
4. scarcely polluted;
5. evolving slowly or in a predictable way.

The question is: how can we calibrate a photometric indicator?

The answer is: usually by comparison with another, already calibrated indicator. This is clearly possible only if there is at least one indicator which can be calibrated (at least indirectly) with trigonometric parallaxes.

### The Period-Luminosity relation: discovery

In 1912 Henrietta S. Leavitt, in collaboration with her director, E.C. Pickering, published an Harvard College Obs. Circ. (vol. 173, pp.1-3) on the Periods of 25 Variable Stars in the Small Magellanic Cloud, containing a fundamental correlation between the average luminosity and the period of the light curve (PL relation) for the family of variable stars known as Cepheids after the prototype, δ Ceph. The discovery was made possible by the fact that the stars belonged to two nearby (nonetheless distant) galaxies, the Magellanic Clouds, so they all were, for each galaxy, approximately at the same distance from the observer.

Henrietta Swan Leavitt and Charles E. Pickering.

Cepheus constellation.

### The Period-Luminosity relation

The relation: $m=a\log P + b,$ where $P$ is the period of the light curve [see ahead], has a slope $a$ and a zero point $b.$ It has been proven that $a$ is more or less constant going from galaxy to galaxy (nearby). The zero point $b$ has to be calibrated in such a way that $m$ turns into an absolute magnitude. Then the PL relation becomes a powerful distance indicator. In fact, you measure the apparent magnitude and the period of a Cepheid at an unknown distance $D,$ via the PL relation compute the absolute magnitude $M,$ and you readily have the distance modulus $(m-M),$ to be further corrected for absorption. It is by this way that in the early 1920 Edwin Hubble proved that the distance of NGC 6822 and (later) M31 was far in excess of the size of the Milky Way, thus proving that this nebula had the same rank of our own galaxy. It was the proof that Kant’s hypothesis on white nebulae (now spiral galaxies) as “island universes” was the right one.

Miss Leavitt's original Period-Luminosity relation for the Large and the Small Magellanic Clouds.

Same as above, with P in logarithmic scale.

### The variable stars named Cepheids

Cepheids are rather luminous $(-2 < M_V < -6)$ and massive $(2 Population I (young) variable stars.

The light curve (figure) is periodic $(5 < P < 100 \;d)$, with a tooth-saw trend $(\phi_{desc}=0.7\pm0.1).$ The average amplitude of the light curve decreases with increasing wavelenght of the observing band: for instance, $\langle M_B\rangle=1.2$ and $\langle M_V\rangle=0.9$.

The average color index is $\langle\Delta(B-V)\rangle=0.4$.

The variablity is due to an oscillation mechanism (“Eddington valve”, resting on ionization/recombination of Hell), which exhibits a radial motion with a velocity amplitude $\langle\Delta_v\rangle=50\;Km/s.$

Typical tooth-saw light curve of a Cepheid.

### The variable stars named Cepheids

Cepheids, being Pop I stars, stay close to the Galactic plane, within a thin layer with an average thickness of $\Delta z =180\ pc.$ They have a small velocity relative to the Sun, $\sim 15\ km/s.$

Cepheids’ luminosities have been classically calibrated by the RR Lyrae stars. There were doubts about their effectiveness in measuring distances, mainly for the effects of metallicity on slope and zero point of the PL relation, and for the absorption corrections (Pop I stars are often embedded in gas/dust clouds). Modern PL relations include a color term to account for metallicity differences: $M=a\log P + b + c\times\left({\rm color\ index}\right).$

The Galactic disk, rich in dust. Credit: ESO.

### The calibration of Cepheids

It is with Cepheids that, in the early 1920’s, Edwin Hubble proved that the distance of NGC 6822 and (later) of M31 was far in excess of the size of the Milky Way, i.e. that this nebula had the same rank of our own galaxy and that Kant’s hypothesis on “white nebulae” (now spiral galaxies) as “island universes” was the right one.

Hubble adopted a calibration for Cepheids provided by Harlow Shapley. It was based on the assumption that RR Lyrae, pulsating variables with $P<1~d$ and constant luminosity, which he had found in some globular clusters of known distance, would represent the faint tail of the PL relation holding for Cepheids.

This was indeed true, but for a another type of variables, akin to Cepheids but of Population II, named W Virginis, which are in fact 2 mag fainter than classical Pop I Cepheids on average (figure).

Recently, using HST, an international team has measured the parallaxes of 10 such stars with an unprecedented accuracy ($\pm0.001\ arcsec$), thus providing an exceptionally accurate zero-point calibration. Within the Hubble Key Project (W. Freedman et al.) many new Cepheids were discovered and studied in 18 nearby galaxies. The final results of the Key project, published in 2001, gave a value of  $H_0 = 72\pm8\ km/s/Mpc.$

The PL relation for pulsating stars: classical Cepheids, RR Lyrae and W Virginis.

### Period-Luminosity-Color relation

The figure shows the Period-Luminosity-Color relation for Cepheids of the SMC from OGLE-III data (Soszynski et al., 2010), with the parallel relation (red) for the first overtone mode. The inset reports the Period-Luminosity relation available in 1960’s (from Shapley’s Galaxies, Harvard Univ. Press, Cambridge, MA, 1961). Notice the much larger dispersion compared to the modern data.

Improvements in the PL relation. From Soszynski et al., 2010, Acta Astron., 60, 17.

### De-reddened distance modulus

The figure shows a method used to correct for extinction the observed monochromatic moduli and derive the de-reddened true modulus. It is based on the extrapolation to infinite wavelength (i.e. to zero effect) of an extinction model (cf., for instance, that of Cardelli et al., Ap.J., 345, 245, 1989) rescaled to the uncorrected values of the distance modulus derived for some colors (4 in the case shown). Notice the perfect fit. Data are relative to the small S/Irr NGC 3109.

Adapted from Musella, Piotto, Capaccioli, 1997, Ap.J., 114, 1976.

Proposed by W. Baade in1926 (Astron.Nacht., 228, 359) and developed by A.J. Wesselink in 1946 (B.A.N., 10, 91), the method applies to distances of pulsating variables as RR Lyrae and Cepheids and rests on measurements of magnitudes, colors, and radial velocities along the oscillation cycle. While of difficult application, it has the merit of being independent of other distance indicators [understand the advantages].

To begin with, let’s assume that the star is a spherical radiator which, at a given epoch $t$, has a radius $R$ and an effective temperature $T_e$ (both vary with time). Were the absolute bolometric magnitude $M_{bol}$ known, the distance $D$ could be computed from: $m_{obs}-M_{bol}=5\log D+ \Delta m_{bol}+A$,

by measuring the apparent magnitude, $m_{obs}$ (in a given photometric band) and by estimating the bolometric correction $\Delta m_{bol}$ (through some empirical relation) and the interstellar absorption $A$ (e.g., through the color excess). Using the Stefan-Boltzmann law, the needed total bolometric luminosity writes as: $L_{bol}=4\pi R^2\sigma T_e^4,$ or:

$M_{bol}=-10\log T_e-5\log R -2.5\log\left(4\pi \sigma\right).$

Thus, the problem of the distance is solved if one knows the simultaneous values of the radius and of the effective temperature. If the object is a black body, its temperature is provided by the color.

In order to measure the radius $R,$ we will assume that, owing to a pulsational mechanism, the star exhibits a a periodic variation of $R$ (and consequently of $T_e.$) The variation of $M_{bol}$ between two successive epochs $t_1$ and $t_2$ (identical to that of $m_{bol}$ since the distance does not change) is:

$\displaystyle{\left(M_{bol}\right)_2 - \left(M_{bol}\right)_1 = \left(m_{bol}\right)_2 - \left(m_{bol}\right)_1 = -10\log\frac{\left(T_e\right)_2}{\left(T_e\right)_1}-5\log\frac{R_1+\Delta R}{R_1}}.$

The above expression gets simpler if we choose two epochs in which the temperature is the same: $\left(T_e\right)_1=\left(T_e\right)_2,$ which happens when the star exhibits the same color:

$\displaystyle{ \left(M_{bol}\right)_2 - \left(M_{bol}\right)_1 = \left(m_{bol}\right)_2 - \left(m_{bol}\right)_1 = -5\log\frac{R_1+\Delta R}{R_1}},$

from where:

$\displaystyle{ R_1=\frac{\Delta R} {{\rm dex} \left( 0.2 \left[ (m_{bol})_1 - (m_{bol})_2\right] \right)-1 }},$

which means that $R_1$ is known if we are able to estimate $\Delta R.$

To this purpose, we assume that the photospheric oscillation is characterized by a purely radial and isotropic velocity $v_r.$ Its mean value, weighted on the luminosity $I(\theta),$ i.e. the radial velocity that we observe, is:

$v_{obs}= \frac{ \displaystyle{ \int_0^{\pi/2} v_r \cos \theta I(\theta)\cos\theta\sin\theta d\theta}} {\displaystyle{\int_0^{\pi/2} I(\theta)\cos\theta\sin\theta d\theta}} = v_r\frac{\displaystyle{\int_0^{\pi/2} I(\theta)\cos^2\theta\sin\theta d\theta}} {\displaystyle{\int_0^{\pi/2} I(\theta)\cos\theta\sin\theta d\theta}}=p^{-1}v_r$.

Neglecting the limb effect [what is it?], $dI(\theta)/d\theta=0;$ and $p =v_r/v_{obs}=1.5$ otherwise, a “reasonable” model gives: $p=24/17\simeq1.41.$ Since $dR=v_r dt=pv_{obs}dt,$ then:

$\displaystyle{\Delta R= p\int_{t_1}^{t_2}v_{obs}(t) dt}.$

In practice, one assumes a biunivocal correspondence between color and effective temperature, and the two epochs are chosen so that the colors are the same. Having measured the apparent magnitudes, and the colors to correct for extinction, one will have the value to compare with the absolute magnitude to find the distance.

### Planetary Nebulae

A planetary nebula (PN) is an emission nebula [how many kinds of nebulae do you know?] where a hot and luminous central star powers an expanding shell of ionized gas. The shell is expelled during the Asymptotic Giant Branch (AGB) phase via pulsations and strong stellar winds, unblanketing a hot core radiating UV radiation. PNe are short-lived  ($10^4 yr$) late phases in the evolution of intermediate- to low-mass stars, which eventually evolve into white dwarfs (WDs). They play a critical role in the chemical enrichment of galaxies (C, N, O, Ca).

NGC 6543, the Cat's eye pictured by HST. Credit: NASA (HST).

Typical spectrum of a Planetary Nebula.

### Planetary Nebulae Luminosity Functions

The emission spectrum offers us the way to disentangle PNe from the diffuse light background of a galaxy: it is enough to compare an image of the host galaxy taken through a very narrow filter centered on [OIII]$\alpha$ galaxies which present a similar behavior and must be flagged out. PNe are quite useful as dynamical tracers as it is relatively easy to measure their radial velocities. It is found that the distribution $N(M)$ of planetary nebula [OIII] brightness $M$ in a galaxy follows the so called planetary nebula luminosity function (PNLF): $N(M) \simeq e^{0.307M}\left(1-e^{3(M^*-M)} \right),$ where $M^*$ ($\simeq-4.48$) is the cutoff magnitude (no PN brighter than this). The PNLF, which is marginally affected by changes in metallicity, has been considered a valid distance indicator.

Luminosity function for M31 PNe (from Van de Steene et al., 2006, A&A, 455, 891).

### Finding PNe in galaxies

On-off technique to identify PNe by comparing a narrow-band image centered on the Oxygen line at 5007 \AA with a broad-band image at the nearby continuum.

### Spectroscopic parallax

Assume that you are able to measure spectroscopically the effective temperature of a star. You may then use the diagram relating the absolute temperature to the absolute magnitude, i.e. an HR diagram calibrated with clusters of stars of known distance, to derive the absolute magnitude of the wanted star, of which you have measured the apparent magnitude. [Evaluate the difficulties and uncertainties of the method.]

A way to derive the absolute magnitude of a star from the color (or, better, the effective temperature).

### I materiali di supporto della lezione

G. Bertin, Dynamics of Galaxies. Cambridge Univ. Press, 2000.

J. Binney & S. Tremaine, Galactic Dynamics, Capter 6, Cambridge Univ. Press, 1987.

J. Binney & S. Tremaine, Galactic Astronomy, Capter 10, Cambridge Univ. Press, 1998.

C. C. Lin & F. H. Shu 1964, Ap.J., 140, 646, 1964.

A. Toomre & J. Toomre, Ap.J., 178, 623, 1972.

W. Freedman, "The Measure of Cosmological Parameters", stro-ph/0202006.

R. Sanders, "Observational Cosmology" astro-ph/0402065.

J. Jensen, J. Tonry and J. Blakeslee, "The Extragalactic Distance Scale", astro-ph/0304427.

S. van den Bergh, "The cosmic distance scale", Astron. Astrophys. Rev, vol. 1, no. 2, 111, 1989.

D. Alloin and W. Gieren, "Stellar candles for cosmic distance scale", Lecture Notes in Physics, vol. 635, Springer, 2003.

"The extragalactic distance scale", proceedings of the Space Telescope Science Institute (U.S.), eds. M.Livio, M.Donahue, N.Panagia, Cambridge Univ. Press, vol. 10, 1997.

D. Hogg, "Distance measures in cosmology", astro-ph/9905116 (see also Hogg et al., "The K-correction", astro-ph/0210394.

G. Jacoby et al., "A Critical Review of Selected Techniques for Measuring Extragalactic Distances", PASP, 104, 599, 1992.

W. Freedman et al., "Final results from the Hubble Space Telescope Key Project to measure the Hubble constant", Ap.J., 553, 47, 2001.

Progetto "Campus Virtuale" dell'Università degli Studi di Napoli Federico II, realizzato con il cofinanziamento dell'Unione europea. Asse V - Società dell'informazione - Obiettivo Operativo 5.1 e-Government ed e-Inclusion