Lecture G3: Galaxies: Distances

Introduction to Astronomy Lecturer: Peder Norberg Galaxies - Lecture G3: Galaxies: Distances -

1 Aims

To discuss redshifts and the Hubble Law, the various methods used to derive absolute distances for galaxies and the evidence for the expansion of the Universe.

2 Objectives

With this lecture, we will:

– Understand some of the basic spectral properties of galaxies and active galactic nuclei (AGN).

– Be able to derive a redshift for a galaxy given the observed and restframe wavelengths of a spectral feature (emission or absorption line).

– Derive a recessional velocity for a galaxy from its redshift, using both the approximation appropriate for small redshifts and the full relativistic Doppler relation.

– Describe the Hubble Law and its origin in the expansion of the Universe. Derive the expansion age of the Universe from the Hubble constant.

– Understand the use of parallax and standard candle methods to determine astronomical distances, including the diﬃculties posed by peculiar velocities.

3 Summary of key concepts

Galaxy spectra show a range of emission (from hot gas) and absorption (from stellar atmospheres) features. Main emission lines: Lyman-α 121.6 nm, [OII] 372.7 nm, [OIII] 500.7 nm, Hβ 486.1 nm, Hα 656.3 nm. Absorption lines: Ca H&K 395.0 nm, Mg I 517.4 nm. AGN also show higher ionisation lines and broader emission lines, which arise due to the inﬂuence of massive black holes. Redshift The wavelengths of these lines are Doppler shifted due to the motion of the source. Hence, if one can identify the species responsible for an emission/absorption line then one can use the observed and restframe wavelengths to derive the redshift of the source. Almost all galaxies are receeding from the Milky Way (their spectra are redshifted). Blueshifts (e.g. Andromeda) arise from peculiar velocities. The Hubble Law Hubble showed that there is a relationship between the recessional velocity of a galaxy (v) and its distance (d) from us: Hubble’s Law v = H0d. H0 is called the Hubble constant −1 and has a value of about H0 = 70 km s per Mpc. THubble = 1/H0 is the time taken by the receeding galaxies to reach their observed distances (assuming no additional acceleration). The Cosmic Distance Ladder We can use a series of methods to obtain distances to higher and higher redshift galaxies. Each method is calibrated using the techniques lower down the ladder (i.e. close to us). The distance ladder includes the following.

– Stellar parallax: the apparent motion of stars due to the orbit of the Earth around the Sun allows us to measure distances out to 1kpc. The ongoing GAIA mission will improve this 1 Introduction to Astronomy Lecturer: Peder Norberg

over the next years measuring distances across the entire Milky Way disk for about a billion stars. – Spectroscopic parallax: use a colour-magnitude diagram calibrated with stars with measured parallaxes to determine the absolute luminosity of a star of a given colour and hence its distance. – Cepheids and RR Lyraes: variable stars which exhibit regular pulsations. Cepheids (Pop I) show a period-luminosity relation which allows us to measure their luminosity once we know the period of their pulsations. RR Lyraes (Pop II) have almost constant luminosity but they are fainter. Cepheids can be used in galaxies out to ∼ 30 Mpc. – Tully-Fisher (TF) relation: spiral galaxy luminosities are proportional to their circular velocity to the 4th power. If one measures the apparent brightness of a spiral galaxy and its rotation speed, one can use it as a standard candle. TF relation is calibrated using Cepheids. It can be applied to high redshifts, but it assumes no intrinsic evolution of the galaxies. – Faber-Jackson (FJ) relation: it is similar to the TF but for ellipticals, and it is based on the velocity dispersion of the stars in the galaxy. The calibration of the FJ is harder as ellipticals have Pop II stellar populations (no Cepheids). This method can be applied to high redshifts, but again it assumes no intrinsic evolution of the galaxies. – Supernovae: bright stellar explosions caused by the collapse of a star. Type Ia are formed by mass accretion onto a white dwarf, and are believed to have uniform peak luminosity 9 (∼ 2 × 10 L ). Thus they can be used as a standard candle out to very high redshifts (again assuming no evolution!).

The most distant objects Among the most distant conﬁrmed objects, there are quasars with redshifts z > 7 and Gamma-ray Burst (GRB) with redshifts z > 9.4, corresponding to an apparent recessional velocity of v > 0.98c and distance of more than 4 Gpc. A few galaxies are known with proposed higher redshifts, e.g. z∼ 11.9, but measurements are harder. We are waiting for the James Webb Space Telescope to be launched in 2021!

4 Readings

Astronomy: The Evolving Universe, Zeilik; Chapter 18

5 Useful equations

5.1 Redshifts

The deﬁnition of redshift, z, for a line with a rest wavelength of λ0 that is observed at wavelength λobs is: λ 1 + z = obs . λ0 A receding object has a redshift due to the Doppler shift: s λ 1 + v/c 1 + z = obs = . λ0 1 − v/c The latter formula is called the relativistic Doppler shift formula and applies for any speed up to that of light, c. For velocites that are small compared to that of the light the approximation z ' v/c works well. 2 Introduction to Astronomy Lecturer: Peder Norberg

5.2 Hubble law

The Hubble law is v = H0d, where H0 is Hubble’s constant, and d is the distance. The Hubble time is deﬁned as 1/H0 and is approximately 14 Gyrs.

5.3 Standard Candles

An object with an absolute magnitude M and apparent magnitude m, and a distance d obeys the following magnitude-distance relation: d m − M = 5 log 10 10 pc The diﬀerence m − M is called the distance modulus. M can be inferred for standard candles (below), hence the distance can be measured from observations of m. Two examples of standard candles are Faber-Jackson relation (ellipticals): L ∝ σ4 4 Tully-Fisher relation (sprials): L ∝ vc where L is the luminosity of the galaxy, σ is the line-of-sight velocity dispersion of the stars and vc is the circular velocity (evaluated on the ﬂat part of the rotation curve).

6 Additional examples

(a) When observed edge-on, the spectrum of a spiral galaxy with a ﬂat rotation curve shows an absorption line which lies at 418.3 nm at the galaxy centre and 418.5 nm at an outer edge. If this is the Hδ line observed at rest wavelength of 410.1 nm in the laboratory, ﬁnd the redshift, z, of the galaxy.

(b) What is the circular velocity at the outer edge?

(d) How far away is the galaxy in (c)?

10 (e) If the galaxy in (c) has a luminosity of 10 L and the Sun has an absolute magnitude V magnitude of 4.8, estimate the apparent magnitude of the galaxy.

Contributions to this material come from: I. Smail, A. Jenkins, M. Fumagalli, P. Norberg, et al.