16: Distance and Magnitude

Distances

We have the invariant distance equation for a homogeneous and isotropic universe (an FRW spacetime):

\[
ds^2 = -c^2 dt^2 + a^2(t)\left[\frac{dr^2}{1-kr^2} +r^2\left(d\theta^2 + \sin^2(\theta) d\phi^2\right)\right].
\]

We now introduce no fewer than five kinds of spatial distances.

Coordinate distance: The coordinate distance between, e.g., \(r,\theta,\phi\) and \(r+dr,\theta,\phi\) is simply \(dr\). The generalization to non-radial distances is not clear, but we don't use this concept much anyway.

Physical distance: At time \(t\), the physical distance between \(r,\theta,\phi\) and \(r+dr,\theta,\phi\) is given by \(a(t) dr/\sqrt{1-kr^2}\). This is the distance you'd measure with a ruler. In general, for pairs of spacetime points at the same time, \(t\), it's given by \(\sqrt{ds^2}\). 

Comoving distance (or comoving length): If we call the physical distance, \(D\), the comoving distance is \(d = D/a\). 

Luminosity distance: By definition of luminosity distance \(d_L\), 

\[
F = \frac{L}{4\pi d_L^2}
\]

which is the relationship we expect in a Euclidean geometry with no expansion, assuming an isotropic emitter. We also calculated the relationship between flux and luminosity in an FRW spacetime and found

\[
F = \frac{L}{4\pi r^2 (1+z)^2}
\]

so we conclude that in an FRW spacetime, \(d_L = r(1+z)\). 

Due to how apparent magnitude \(m\), and absolute magnitude \(M\) are defined, we have

\[
\mu \equiv m-M = 5 \log_{10}\left(\frac{d_L}{10\ {\rm pc}}\right)
\]

where \(\mu\) is called the distance modulus. IF you know the flux of an object (how bright it is) then you can determine apparent magnitude \(m\). If you know the luminosity of an object (its total power output), then 

Comoving angular diameter distance: By definition of comoving angular-diameter distance, \(d_A\), 

\[
\ell = \theta d_A
\]

where \(\theta\) is the angle subtended by an arc of a circle with comoving length \(\ell\). We can calculate the relationship between \(\ell\), \(\theta\) and \(r\) in an FRW spacetime. The comoving length of an arc at coordinate distance \(r\), that subtends, at the origin, an angle \(\theta\) is

\[
\ell = \int ds = \int_0^{\theta} r d\theta = \theta r.
\]

Comparing with the definition of \(d_A\) we also see that \(\ell = \theta d_A\). So we conclude that \(d_A = r\).

Box \(\PageIndex{1}\)

Exercise 16.1.1: In an FRW spacetime, how are \(d_A\) and \(d_L\) related?

Box \(\PageIndex{2}\)

Exercise 16.2.1: What is the comoving distance, \(\ell\), from the origin to some point with radial coordinate value \(r\), along a path of constant \(\theta\) and \(\phi\)?

Curvature integrals: Although we've made use of a first order Taylor expansion to analytically solve the above integral, the exact integral does have an analytic solution. For \(k> 0\), \(\ell = (1/\sqrt{k}) \sin^{-1}(\sqrt{k}r)\). For \(k< 0\), \(\ell = (1/\sqrt{-k}) \sinh^{-1}(\sqrt{-k}r)\). 

To work out how angular diameter distance \(d_A\) is related to the scale factor at the time light was emitted, \(a\), we look at how light travels from coordinate value \(r\) to the origin. Light has \(ds^2 = 0\), and from that we get

\[\begin{equation}
  \begin{aligned}
        \int_0^r \frac{dr}{\sqrt{1-kr^2}} & = \int_t^{t_0} cdt/a = c  \int_a^1 da/(a^2 H), \\ \\ (1/\sqrt{-k}) \sinh^{-1}(\sqrt{-k} r) & = c \int_a^1 da/(a^2 H) \\ \\ r & = (1/\sqrt{-k}) \sinh\left(\sqrt{-k} c\int_a^1 da/(a^2 H)\right) \\ \\ d_A & = (1/\sqrt{-k}) \sinh\left(\sqrt{-k} c\int_a^1 da/(a^2 H)\right)
  \end{aligned}
\end{equation}\]

where, except for the first line, we have assumed \(k < 0\). I leave it to the student to work out the \(k > 0\) case. The \(k=0\) case should also be clear. 

Box \(\PageIndex{3}\)

Exercise 16.3.1: In calculating \(d_A\) vs. \(a\), what are the two different ways curvature makes a difference?

Box \(\PageIndex{4}\)

We have defined the density parameters \(\Omega_x = \rho_{x,0}/\rho_{c,0}\) where \(\rho_c \equiv 3H^2/(8\pi G)\) is the critical density, defined to be the total density for which the curvature, \(k\) is zero.  

Exercise 16.4.1: Using the Friedmann equation, convince yourself that if \(\rho = \rho_c\), then \(k=0\). 

With this notation we can write

\[
H^2(a) = H_0^2\left(\Omega_\Lambda + \Omega_m a^{-3} + \Omega_K a^{-2}\right)
\]

where \(\Omega_K \equiv -k/(a^2 H_0^2)\).

Exercise 16.4.2: Show that this can be derived from the Friedmann equation and the fact that \(\rho_\Lambda \propto a^0\) and \(\rho_m \propto a^{-3}\).

Exercise 16.4.3: Further, show that \(\Omega_\Lambda + \Omega_m + \Omega_K = 1\).

Some Definitions of Astronomical Terms  

Magnitudes are absurd but useful if you want to use data from astronomers.

Luminosity: The luminosity of an object, \(L\), is its power output. Usually its total electromagnetic power output, sometimes referred to as bolometric luminosity. Typical units for luminosity are ergs/sec (\(10^7\) erg = 1 Joule, 1 Watt = 1 Joule/sec) or solar luminosity, \(L_{\rm Sun}\). The Sun, by definition has a luminosity of one solar luminosity and \(L_{\rm Sun} = 3.826 \times 10^{33}\) erg/sec = more than \(10^{24}\) 100 Watt light bulbs. (You can remember this if you remember it's about as luminous as 7 Avogadro's number of 100 Watt light bulbs).  

Flux: The flux, \(F\), from an object is not an intrinsic property of the object, but also depends on the distance to the object. It is the amount of energy passing through a unit area, per unit of time. For an isotropic emitter in a non-expanding, Euclidean three-dimensional space, \(F = L/(4 \pi d^2)\) where \(d\) is the distance between source and observer. This equation just follows from energy conservation; note that the total power flowing through a spherical shell of radius \(d\) completely surrounding the emitter at its center is \(4 \pi d^2 \times F = L\).  

Luminosity Distance: The universe is expanding, and the spatial geometry is non-Euclidean to some degree. We define luminosity distance such that the Euclidean, non-expanding relationship between flux, luminosity, and distance is preserved if one simply swaps out distance for luminosity distance. That is, 

\[F = \frac{L}{4\pi d_L^2}.\]

We showed previously that in an FRW universe \(d_L = d\times(1+z)\) where \(d\) is the coordinate distance to the object from us that we are observing in the current epoch. 

Spectral Flux density: We usually do not measure the total flux from an object, but instead measure the flux in a manner that depends on how the flux is spread out in frequency. Thus a useful concept is the spectral flux density, \(S\), that quantifies how much flux there is per unit frequency.  The units of spectral flux density are erg/s/m\(^2\)/Hz.  Where Hz is the unit of frequency called Hertz, equal to 1/s.  

Apparent magnitude: Astronomers often use apparent magnitude, \(m\), instead of flux. The apparent magnitude has a logarithmic dependence on flux; the reason for this is historical, and is fundamentally due to the logarithmic sensitivity of our eyes to flux. Not only is it logarithmic instead of linear, but brighter objects have smaller magnitudes. This is because the Greeks defined the brightest stars as stars of the first magnitude, and next brightest as stars of the 2nd magnitude, down to the stars we could just barely see at all, which are stars of the 6th magnitude. This ancient system, updated with precise definitions related to flux is still in use today (otherwise I would not bother telling you about it). One way of relating apparent magnitude to flux is the following:

\[m = M_{\rm Sun} - 2.5 \log_{10}\left(\frac{F}{F_{\rm Sun 10}}\right)\]

where \(M_{\rm Sun} = 4.76\) is the absolute magnitude of the Sun (see next definition) and \(F_{\rm Sun 10}\) is the flux we would get from the Sun if it were 10pc away. Since \(L_{\rm Sun} = 3.826 \times 10^{33}\) erg/sec and 1pc = \(3.0856 \times 10^{18}\) cm we get \(F_{\rm Sun 10} = 3.198 \times 10^{-7}\) erg/cm\(^2\)/sec.  

Note that because of the -2.5 factor in front of the \(\log_{10}\), if the flux increases by a factor of 10, the apparent magnitude decreases by -2.5. Conversely, if the magnitude increases by 1, the flux decreases by a factor of \(10^{1/2.5} = 10^{0.4}\).  

Absolute Magnitude: The absolute magnitude of an object, denoted by \(M\), is another way of expressing its luminosity. It is designed so that the apparent magnitude and absolute magnitude of an object are the same, for an object at 10pc. Therefore:

\[M = M_{\rm Sun} - 2.5\log_{10}\left(\frac{L}{L_{\rm Sun}}\right).  \]

Distance Modulus: The distance modulus is defined as \(\mu \equiv m - M\). Note that as a difference between apparent and absolute magnitudes, this is equal to a log of the ratio of flux and luminosity. By plugging in the definitions above of \(m\) and \(M\) one finds

\[ \mu = 5\log_{10}\left(\frac{d_L}{10 {\rm pc}}\right)= 5\log_{10}\left(\frac{d_L}{1 {\rm pc}}\right) - 5\]