Measuring Distances

Introduction Trigonometric Parallax Spectroscopic Parallax Cepheid Variables Type Ia Supernovae Tully-Fisher Relationship Hubble’s ‘Law’

Trigonometric Parallax While fundamental, this method is very limited in terms of how far away one can use it. For large numbers of star, ground- based measurements of parallax do not get beyond about 20 pc, although on individual objects this can be extended to about 100 pc (with much effort). The HIPPARCOS satellite extended reliable measurements out to 100 pc for many thousands of stars, but this still compares badly with the 8000 pc to the center of our Galaxy. Clearly we need other methods if we are to be able to measure distances to most objects in our Galaxy let alone to other galaxies.

Improving the Accuracy

Spectroscopic Parallax Spectroscopic parallax has nothing to do with parallax –It makes the assumption that stars of exactly the same type (e.g. have nearly the same spectra) will have the same brightness –If we can measure the distance to one star of this type, and can measure the brightness of both stars, then we can compute the distance to the second star b 1 /b 2 = d 2 2 /d 1 2 The distance values are squared because brightness falls off as an inverse square We try to compare Main Sequence Stars

Spectroscopic Parallax The key to this method is the identification of `identical stars'. This is done by a careful comparison of their spectra, checking for the same features. This is one of the reasons for the classification of stars into `spectral types'. Spectroscopic parallax is a very useful method, but the values it gives are rather approximate in general, and very substantial errors are possible if a normal star is mistaken for a giant or vice versa. However, it can give us distances almost anywhere within our own Galaxy and to nearby galaxies

Cepheid Variables Cepheid variables are high mass stars in a late phase of their evolution during which they become unstable, and start to pulsate. During the pulsations, they expand and contract with oscillation periods of order a few days, and this is visible as a change in their brightness The remarkably useful feature of Cepheid variables is that their luminosity is fixed by their pulsation periods

Cepheid Variables Light curves of some Cepheid variables in the galaxy NGC Their brightness indicates a distance of for this galaxy of 1.4 Mpc. The plot shows B-magnitude versus fraction of a pulsation period.

Cepheid Variables The period-luminosity relation for Cepheids. Visual luminosity or power is measured by absolute magnitude M V The period-luminosity relation was first calibrated by Henrietta Levitt in 1912

Cepheid Variables Cepheids from the ground can be seen to about 1 Mpc (i.e. out to M31 at 750 Kpc). HST has extended this out to 20 Mpc to measure the distance of galaxies in the Virgo cluster. The discovery of Cepheids in M31 by Hubble was one of the key discoveries in establishing the nature of galaxies which many had thought until that point to be part of our own Galaxy. Cepheids continue to be one of the cornerstones of astronomy today.

Type Ia Supernovae A certain type of exploding star called a Type Ia supernova appears to follow a fairly consistent light-curve, peaking at an absolute magnitude of about M v  -19. This makes them 23.8 magnitudes more luminous than the Sun, equivalent to a factor of /2.5 = 3.3 x These are now playing an important role in modern cosmology as there are large projects dedicated to finding such supernovae in very distant galaxies, of order away. Again the principle is the identification of an object that does not change its nature over large distances. These are often called standard candles. Note that both Cepheids and Type Ia supernovae have to be corrected for extinction just as for spectroscopic parallaxes

SN 1987

Tully-Fisher Relation Developed in the 1970's and 1980's by Tully and Fisher. The method has since been improved by several workers and is now one of the more accurate secondary distance indicators for the Universe. The method relies on the fact that there is a relationship between the rate at which a spiral galaxy spins and its intrinsic luminosity. The sense of the relation is that the faster a galaxy spins, the more luminous is the galaxy.

Tully-Fisher Relation Due to the spin of the galaxy, an observer will see part of the galaxy approaching them and part of the galaxy running away. This causes the emission from the galaxy to show redshifted, blueshifted, and no-shifted emission. The motion will thus cause a narrow line, e.g., a line due to some element like hydrogen, to be smeared out and to appear broad to the external observer. The broader the line, the faster the galaxy must be spinning. The gas and stars are in orbit in the galaxy, so from Kepler’s Law, the more mass, the higher the spin rate Since the observable light we see is produced by stars, it does not take a stretch of the imagination to infer that the more massive a galaxy is, the brighter it is likely to be. This is borne out by the Tully-Fisher relation, the empirically deduced relation between spin-rate and luminosity.

Hubble’s Law The dominant motion in the universe is the smooth expansion known as Hubble's Law. Recessional Velocity = Hubble's constant times distance V = H o D where, V is the observed velocity of the galaxy away from us, usually in km/sec; H is Hubble's "constant", in km/sec/Mpc and D is the distance to the galaxy in Mpc. In 1929, Hubble estimated the value of the expansion factor, now called the Hubble constant, to be about 500 km/sec/Mpc. Today the value is still rather uncertain, but is generally believed to be in the range of km/sec/Mpc

Hubble’s Law While in general galaxies follow the smooth expansion, the more distant ones moving faster away from us, other motions cause slight deviations from the line predicted by Hubble's Law. Few of the points fall exactly on the line. This is because all galaxies have some additional residual motion in addition to the pure expansion. –This is referred to as the "cosmic velocity dispersion" or "cosmic scatter" and is probably due to the fact that the gas clouds that formed the galaxies all had some small additional motion of their own. The recessional velocity of a galaxy at a particular distance inferred from Hubble's law is called the "Hubble velocity". This diagram shows a typical plot of distance versus recessional velocity, with each point showing the relationship for an individual galaxy.

About in the middle of the diagram, there are a bunch of galaxies that appear to be at about the same distance but are spread out a lot in the velocity direction. This feature suggests the presence of a large cluster of galaxies, like the Virgo Cluster. In addition to their "Hubble velocities", these galaxies have an extra velocity caused by their orbital motion around the center of the cluster. –Because clusters of galaxies are very massive, this orbital velocity can be very large, more than 1000 km/s. Therefore in the vicinity of nearby clusters of galaxies, we cannot use Hubble's law to determine accurately the distance to the galaxy. Hubble’s Law

The Distance Ladder