## Apparent Magnitude, Absolute Magnitude, and Distance

A simple equation relates apparent magnitude, absolute magnitude, and distance.  Know any two, and you can calculate the third.

Known: Apparent Magnitude (m), Absolute Magnitude (M)
Unknown: Distance (d), in parsecs

Known: Apparent Magnitude (m), Distance (d)
Unknown: Absolute Magnitude (M)

Known: Distance (d), Absolute Magnitude (M)
Unknown: Apparent Magnitude (m)

If this were a perfect universe, the known quantities could always be measured as precisely as one desires.  But, of course, that isn’t the case.

Apparent Magnitude – if the observations are made from the surface of the Earth, atmospheric reddening and extinction (atmospheric r/e) must be taken into account to determine the apparent magnitude above the Earth’s atmosphere.  Even above the Earth’s atmosphere, cosmic reddening and extinction (cosmic r/e) must also be quantified.  Both atmospheric r/e and cosmic r/e1 cause the observed apparent magnitude to appear fainter than it otherwise would be, and bluer wavelengths are more severely affected than redder wavelengths.  The net result is to make objects appear fainter and redder than they would be if there were a perfect vacuum between source and observer.

Absolute Magnitude – is a measure of the intrinsic brightness of a celestial object, and this can only be measured indirectly for objects outside of our solar system.

Distance – is difficult to measure for objects outside of our solar system.  Trigonometric parallax gives the most accurate results for nearby stars, but uncertainty increases rapidly with increasing distance.

Apparent magnitude is the only one of these quantities that is a direct instrumental measurement: absolute magnitude and distance are determined indirectly and thus are subject to greater uncertainty.

`1Atmospheric reddening and extinction (atmospheric r/e) is traditionally called atmospheric extinction, and cosmic reddening and extinction (cosmic r/e) is traditionally called interstellar reddening.  Since in both cases light is both reddened and diminished in intensity, and because "cosmic" encompasses both interstellar and intergalactic matter between source and observer, I suggest here that atmospheric r/e and cosmic r/e might be an improvement in terminology.`

## Distant Supernovae Evince Accelerating Expansion of our Universe

In 1998, it was discovered by two independent research teams through the study of distant Type Ia supernovae that our expanding universe has an expansion rate that is accelerating.  This was a completely unexpected result.

A Type Ia supernova occurs in a close binary star system where mass from one star accretes onto a white dwarf until it reaches a critical mass and a supernova explosion ensues.  Many of these events, chosen carefully, can be used as “standard candles” for distance determination.  The intrinsic peak luminosity of a typical Type Ia supernova is a function of the light curve decay time.  Type Ia supernovae whose luminosity curves rise and fall more rapidly are less intrinsically luminous at maximum brightness.  Type Ia supernovae whose luminosity curves rise and fall more slowly are more intrinsically luminous at maximum brightness.

If we know the intrinsic luminosity of an object (the absolute magnitude) and can measure the apparent luminosity of that object (the apparent magnitude), we can calculate its distance.  Type Ia supernovae are on the order of a million times brighter than Cepheid variables, and are in fact the brightest of all “normal” supernovae.  They can thus be used to measure the distance to extremely distant objects.

The evidence for an accelerating universe is that these distant supernovae appear fainter than they should be at their measured cosmological redshift, indicating that they are farther away than expected.  A number of possible explanations for the faint supernova phenomenon had to be eliminated before the conclusion that the universe’s expansion is accelerating could be arrived at, including

(1) Do distant supernovae (and therefore supernovae that occurred many billions of years ago) have the same intrinsic brightness as comparable nearby supernovae that occurred in the recent past?

(2) Are the distant supernovae being dimmed by galactic and intergalactic extinction due to dust and gas along our line of sight to the supernova?

As described above, the shape of the supernova light curve indicates the supernova’s intrinsic brightness, analogous in a way to the period of a Cepheid indicating its intrinsic brightness.  Though there is evidence that ancient supernovae may have been a little different than those today because of lower metallicity, the effect is small and doesn’t change the overall conclusion of an accelerating universe.  However, properly characterizing the influence of metallicity will result in less uncertainly in distance and therefore less uncertainty in the acceleration rate of the universe.

Extinction is worse at bluer wavelengths, but how the apparent magnitude changes as a function of distance is independent of wavelength, so the two effects can be disentangled.  2011 Nobel physics laureate Adam Riess in his award-winning 1996 Ph.D. thesis developed a “Multicolor Light Curve Shape Method” to analyze the light curves of a large ensemble of type Ia supernovae, both near and far, allowing him to determine their distances more accurately by removing the effects of extinction.

## Cosmologically Distant Objects Appear Magnified

George F. R. Ellis writes in section 2.3.3 of his outstanding survey paper, Issues in the Philosophy of Cosmology:

…there is a minimum apparent size for objects of ﬁxed physical size at some redshift zc = z depending on the density parameter and the cosmological constant.  The past light cone of the observer attains a maximum area at z; the entire universe acts as a gravitational lens for further oﬀ objects, magnifying their apparent size so that very distant objects can appear to have the same angular size as nearby ones.  For the Einstein-de Sitter universe, the minimum angular diameter is at z = 1.25; in low density universes, it occurs at higher redshifts.

Electromagnetic radiation such as visible light that we observe from a source that is in motion relative to us (the observer) experiences a change in wavelength that is given by

This is called redshift and is positive for a source that is moving away from us and negative for a source that is moving towards us.  The higher the relative speed toward or away from us, the greater the magnitude of the redshift.  Superimposed upon the kinematic velocities of individual galaxies relative to our Milky Way galaxy, since 1929 we have known that there is a cosmological redshift (called the Hubble flow) that is always positive and increasing in magnitude with increasing distance between any two galaxies.  In the nearby universe, the redshift (or blueshift) from kinematic velocities (often referred to as “peculiar velocities”) swamp the contribution from the Hubble flow, so some galaxies are actually approaching each other.  A good example of this is M31 and the Milky Way galaxy.  For more distant galaxies, however, the cosmological redshift swamps any contribution from the kinematic velocities.  Thus, redshift becomes a useful proxy for distance at cosmological distances.

From our everyday experience, we know that the further away an object is, the smaller is its angular size.  However, there comes a point where the angular size of an object reaches a minimum, and at even greater distances, its angular size increases!  As George Ellis states above, the entire universe acts as a gravitational lens to magnify distant objects.

Michael Richmond presents an equation for angular size as a function of redshift (based on some classical assumptions about the structure of the universe).  In his equation, the angular size of an object also depends upon the value we choose for H0, the Hubble constant, the matter density parameter, ΩM, and, of course, the physical size of the object of interest.

Let’s work through an example using this equation.  The visible part of the Andromeda Galaxy is estimated to be about 220,000 light years across.  In megaparsecs, that is 0.0675.  This is the value we will use for S.

For the Hubble constant, H0, let use a recent result: 71.9 km/s/Mpc.

And, for the matter density parameter, ΩM, let’s use 1.0.  This indicates that we live in a universe that has just enough matter for the universe to eventually recollapse, were it not for dark energy.  Though Richmond’s equation above only applies to a matter-dominated universe where the dark energy density parameter ΩΛ is zero, as George Ellis indicates above, a minimum angular diameter is still reached in a universe with dark-energy (i.e. low density universe), only this occurs at a higher redshift than that presented here.

I have not been able to find or derive a more general equation for angular size as a function of redshift that will work for a dark-energy-dominated universe (perhaps a knowledgeable reader will post a comment here providing some insight into this issue), but it will be a useful exercise to continue with the calculation assuming the matter-dominated Einstein-de Sitter universe.

Casting Michael Richmond’s equation into the following SAS program, I was able to find that the Andromeda galaxy would reach a minimum angular size of 11.3 arcseconds at z = 1.25, as shown below.

In principle, measuring the angular size of a “standard” object at various redshifts could allow us to determine what kind of universe we live in.  But there’s a problem.  As we look further out into space we are also looking further back in time, so there is no guarantee that a “standard” object in today’s universe (say, a spiral galaxy such as M31) would have looked the same or even existed billions of years ago.

References
Ellis, G. F. R. 2006, Issues in the Philosophy of Cosmology, Philosophy of Physics (Handbook of the Philosophy of Science), Ed. J. Butterfield and J. Earman (Elsevier, 2006), 1183-1285.
[http://arxiv.org/abs/astro-ph/0602280]

Richmond, Michael, Two classic cosmological tests
[http://spiff.rit.edu/classes/phys443/lectures/classic/classic.html]