Large cosmic distances require photometric distance measurements.

The apparent brightness is measured of a star of known luminosity (L). Brightness is proportional to L, and inversely proportional to the distance squared. Brightness = L/d2, so d = squareroot(L/B).

The different photometric methods involve different ways of finding the luminosity of very luminous objects.

Expanding Photosphere Method:  Used for Type II supernovae, which have absorption lines. As the near side of the exploding photosphere expands, its lines are blue-shifted and broadened as illustrated below:

 

The top material moves very fast, 08 light-speed for SN1987a. Lower layers, which become the photosphere when the top layer becomes transparent, move less fast. The size of the photosphere grows to an enormous radius, calculated by multiplying the blue-shift (Doppler) speed for each interval of time in the expansion. The greatest radius is several AU at a temperature of around 10,000 K! ThatÕs really luminous, visible to hundreds of millions of parsecs, and the luminosity is known from the radius and temperature.

The geometry of the expansion:

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 


LetÕs analyze an observation. We watch a SN for a week, measuring the approach velocity of its photosphere, its brightness, and its temperature.

  1. Luminosity depends on temperature and radius2.
  2. Distance stays the same, so the ratio of brightness, start and finish, is the ratio of luminosity.
  3. Temperature is known for both times, so we know the ratio of radius2, start and finish.
  4. The square root of the ratio of radius2 is the ratio of the radius, start and finish.
  5. We have measured the approach velocity for the week, so we can calculate the increase in radius, start to finish.
  6. Now we know both the difference and the ratio of the radius over the week. We can find both radii from this. Your high school algebra will work, but here I will use a graph instead.

 

Data: a, Ratio (From 4 above):The radius is 7 times larger at weekÕs end. b. Increase (From 5 above): The radius is 600 million km greater at weekÕs end. (Yes, supernovas are really big.)

On the graph below, the unit is 1 million km. We plot potential values of r1, starting at 10 million km, ending when both expressions for r2 give the same value.

 

The result is that, at weekÕs end, the SN has an expanding photosphere 700 million km radius, 10,000K temperature, about 4 million solar luminosities, visible with modern telescopes and detectors to about 2.5 billion light-years. Actually, at that distance the required spectrum would be hard to measure, but the method does work for very large distances.