Lecture 3:

Handouts:

3. Writing Assignment 1 (due 1/29)

We introduced the nongeometric, or luminosity methods for measuring cosmological distances. All essentially rely on the relationship between an objects luminosity $L$ and flux or apparent brightness $\phi$ as seen from earth. We reviewed how, given objects of the same luminosity, distance scales as the inverse square root of flux. Thus, given objects of standard $L$ (``standard candles''), we may measure distance using flux, or brightness, as our ruler.

Before establishing candidates as standard candles for such a method, we entertained another digression into astronomer jargon, noting that astronomers don't work with the quantities luminosity and flux, but with quantities logarithmically related to them called absolute and apparent magnitude. We defined these quantities, noting that brighter stars have lower magnitudes by convention (think of the magnitude as being -- very loosely -- a required brightness magnification to make the star visible).

We listed some candidates as standard candles for such a scheme, in different distance regimes: main sequence stars, Cepheid variables, novae, brightest galaxies in clusters. We discussed the physics of the two closest standard candles: main sequence stars, and Cepheid variables. We then mentioned other more distant ones discussed in your readings: novae, supernovae, dynamics of galaxies, brightest galaxies in clusters.

We qualitatively described the physics of main sequence stars. Such stars are approximate blackbody sources, with ions, atoms, and molecules in excited states. Occupancy numbers of each state depend on their equilibrium values at the given temperature; that is, the values at which transitions into the state exactly balance transitions out. The intensity of the star's spectral emission and absorption lines depend on the likelihood of the associated transitions -- determined both by the occupancy number of the initial state and by the probability of transition out of that state. Both have complicated temperature dependence. The upshot was that a given spectral line tends to become stronger as the temperature increases and increasingly populates its initial excited state, reaches a maximum, and then weakens as the temperature becomes so high that the relevant atoms are usually found, not in the initial state, but in much more excited or even ionized states. At any given temperature the different spectral lines occur in a unique proportion; that is, a high temperature star may have strong He I, H and Si III lines; while a low temperature star may show predominantly Ca II and Fe I and II lines. Astronomers classify the blends of spectral lines received by spectral type (labeled OBAFGKMNS).

The relevance of this astrophysics to cosmology is that early twentieth century observers noticed -- within nearby clusters whose known distances allowed calculation of intrinsic luminosities -- that each spectral type tended to be observed with a single luminosity. The Hertzsprung-Russell diagram plots the observed absolute magnitudes (logarithmically related to luminosities) versus spectral type (or log T); most stars fall along a line called the main sequence, with some other oddities defining a second branch. Thus identifying a main sequence star of known spectral type allows inference of the star's absolute luminosity. In practice, this is done statistically, by plotting the apparent magnitudes of stars within a distant cluster, keeping those which seem to fall on a main sequence line, then shifting that line by $-(m-M)$ to translate it onto the reference main sequence line, established by the Hyades cluster. $m-M$ obtained is the distance modulus, encoding the distance to the observed cluster.

The next distant standard candles, also used by Hubble, are Cepheid variable stars. These stars, which brighten and dim with a characteristic period, were noticed (again within known distances) to have a peak luminosity directly proportional to their period. Thus measuring the period established a standard candle of known luminosity. However, these were used as standard candles long before their physics was understood; with the unfortunate result that all Cepheids were assumed to have the same constant of proportionality between luminosity and distance. Ultimately two distinct types of Cepheids were found, with different constants of proportionality; Hubble's inclusion of Cepheids of the second type with incorrect constant of proportionality threw off many of his measured distances by a factor of two.

We quickly mentioned novae and supernovae as standard candles, whose peak intensity is related to relaxation timescales after their initial flare; as well as the dynamical properties of galaxies or brightest galaxies in clusters.

After this interlude on the actual process of measuring dynamical properties of the universe -- and how it becomes increasingly indirect, with rich and complicated intervening astrophysics -- we returned to consequences of Hubble's discovery of the expansion.

We first try to accommodate Hubble's law within a quasi-Newtonian picture. We consider an infinite uniform universe obeying Newtonian gravity, but with Hubble's law for the expansion replacing our earlier static assumption. We will use energy conservation for a galaxy in such a universe to explore -- next time -- the dynamics of such a Newtonian cosmological expansion. We will find, qualitatively, that a quasiNewtonian cosmology can accommodate the Hubble expansion, but that expansion always either decelerates or continues at a constant rate. The universe we observe now, instead, has accelerating expansion.

--

KB