Lecture 2:

Handouts:

2: Homework 1 (due 1/22)

We recalled the conceptual puzzles of last time, and discussed Newton's rationalizations of the gravitational collapse problem: claiming that the act of picking an origin violated the intrinsic symmetry of the problem, and that perhaps -- weighting all choices equally -- each star remains stationary because it wants to collapse toward each origin and cannot pick a particular one (all collapse forces cancel). We noted why this is not a scientific solution -- it is a metaphysical answer and not an answer calculated within any physical theory. It does however give some insight into how a new theory might resolve the problem. In the ultimate solution, general relativity, by allowing the gravitational attraction of matter to deform the spatial grid on which particles live, we can reproduce the conjectured tendency for every particle to collapse toward all of its neighbors.

After the conceptual puzzles of last time, we began discussing the actual observational contradiction that killed the static Newtonian cosmology: the Hubble expansion observed in 1929. Hubble measured that -- instead of having random velocities due to local gravitational interactions -- galaxies on average recede from us, with recession velocities proportional to distance. Before discussing the consequences of this fact, we will spend some time in an astrophysical interlude, considering how we might make such measurements of the dynamics of our universe.

We first discussed the measurement of velocities of astronomical objects, using 1) the known frequencies of spectral lines emitted and absorbed by such objects; and 2) the nonrelativistic Doppler shift of those lines when the sources move away from or toward us.

We then focused on measuring distances, noting that the extrapolation from actual measurements to physical distances becomes increasingly indirect as we consider more and more distant objects. This is still an active problem in cosmology, and is the primary reason for our current uncertainty about the value of Hubble's constant $H_o$ (which relates distances to recession velocities in Hubble's law).

This topic is called ``the Cosmological Distance Ladder,'' because it consists of establishing methods for measuring distances at a particular scale; then using those methods to calibrate more indirect methods, which can be extended out to greater distances. It is like having an extension ladder where each piece covers its own particular distance range, with overlap between the pieces establishing consistency and continuity of the entire scheme.

The methods for measuring distances on the closest scales are the most naive and geometric, using essentially no physics beyond definitions of distances and velocities. They do exploit an increasing knowledge of trigonometry and perspective, though.

We derived three geometric methods:

Radar Ranging

or measuring the travel time of a bounced radar signal to and from the object. This works for our neighbors within the solar system, out to a distance of about 10 AU, where we defined an AU as the distance between the earth and sun.

Parallax

This relied on the angular shift of a foreground object against a distant background, when viewed from different observing positions. We related the angular shift $2p$ and observer separation $2B$ to the distance $d$ between observer and object. We then noted the astronomical convention of taking as our observer separation 2 AU, the diameter of the earth's orbit, by comparing angular measurements taken 6 months apart. We labeled $p$ for such a baseline the parallax angle, and defined the units arcsecond, parsec, and light-year. We noted that the nearest stars have $p \le 1''$, corresponding to $d \ge 1 {\rm pc} = 2 \times 10^5 {\rm AU}$. The method fails for earth-based telescopes around 30 pc, and for satellite-based telescopes around 100 pc. Thus we have extended our distance range by a factor of $10^6$, but reached only the nearest stars.

Moving Cluster (Convergent Point)

This relied on two facts: 1) the relationship between transverse and angular velocities with respect to an observer at the origin; and 2) the fact that parallel paths of a group of objects tend to converge to a point at infinity (with that point being in their direction of travel, relative to the observer). Stellar clusters provide such a parallel-moving group, since their stars are gravitationally bound and all move with a single cluster velocity. Thus observing the cluster's motion over time gives its direction of travel, which allows inference of a tranverse velocity from a measured radial velocity (using Doppler shifts as before). Three measurements -- direction of motion, angular velocity, and recessional velocity -- are thus enough to give the distance to such a cluster. This method works for distance scales between 30 and 200 Mpc, far enough to provide a calibration overlap with the nongeometric methods required for greater distance scales.

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