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# Monthly Archives: December 2013

# Units of Astronomical Images and Conversions

If you’re like me, you frequently have to re-check the definitions for units and conversions with astronomical data. We all learned this in our intro astronomy courses, but details sometimes get fuzzy after time. It doesn’t help that there is occasional laxity in the use of the term ‘flux’ in the Literature, when they really mean flux density or even brightness (and vice versa)! I have not yet found definitions and conversions for the most common astronomical image units in one resource, so I am venturing to create something like that here.

Some excellent resources I’ve found on the web on this topic, with figures and examples:

http://www.cv.nrao.edu/course/astr534/Brightness.html

http://www.astro.queensu.ca/~courteau/Phys216/sb.html

https://dept.astro.lsa.umich.edu/ugactivities/Labs/brightness/index.html

Beware, though, as even these sources use terms like brightness to mean different things.

To go straight to the quick reference guide, click here.

### The big picture

As observational astronomers, we observe the sky, record the incoming photons, and quantify the emission in order to say something about the physics in the target system. We generally want to determine an object’s luminosity, but what we measure with our instrument is the flux from that object incident on our detector area, within a certain wavelength range, coming from the source which covers a certain solid angle on the sky. It takes a little bit of work to back out the luminosity.

### Common units

There are many different units in use for the same quantities in astronomy, and some of the more historical units are still commonly seen (cgs, magnitudes). Here I am concerned with the units you are likely to encounter in your images, with a slight preference for the radio part of the spectrum. Note that I am sticking to physical units, not instrument units such as ADU. Most books and guides I’ve seen start with the quantities you get in your images (such as brightness) and work their way down to the more fundamental quantities that we are interested in studying (such as luminosity). I will do that later in the quick guide, but for a first look, I find it more intuitive to start with the simple quantities (what we’re trying to determine) and work our way up to what we get in our data products. I will be working in SI units throughout, which is becoming more common in the Literature, but I will also use common units like Janskys where convenient.

Luminosity (L). This is a fundamental characteristic of the source you observe. Dimensionally equivalent to the physics quantity *Power*, it’s simply the amount of energy radiated per unit time. Though it is a measure of how bright something is in the colloquial sense, it is NOT equivalent to the term ‘brightness’ in astronomy. (See below.) The most common units are Watts [W], though you may still occasionally see L in terms of ergs/s. (1Jy = 10^{7} erg) Since we typically assume isotropic radiation for simple cases (definitely not true for many sources you may encounter!), the luminosity is generally taken to be the flux times the area of the sphere at your distance D. (c.f. the Stefan-Boltzmann Law!) That is, L = A_{sphere }F = 4πD^{2}F.

Flux (F). The farther away a source is, the dimmer it will appear to an observer. Think of how the high beams of an oncoming car seem brighter as the car gets closer, and less blinding when they are farther away. Of course, this is because the radiated energy is spread out over a larger area as distance increases. The amount of radiated energy that passes through an area is flux. SI units are W/m^{2}, though you will still see erg/s/cm^{2}. Since we are interested in quantifying the luminosity of an object – that is, the inherent power radiated – for analysis and science goals we are not as much interested in the amount of light that passes through the detector area, but rather the total amount of energy radiated at the source. (However, see caveat below.) To get this, as mentioned above, we assume isotropic radiation (for simple cases) and multiply the flux by the area of the great sphere at the observer’s distance D from the source: A_{sphere} = 4πD^{2}. Notice the D^{2}: flux follows the inverse square law.

HUGE CAVEAT: We don’t live in a conceptually simple textbook universe. (Surprise!) In our analysis of physical systems, we often make assumptions about source geometry, isotrpic emission, and a host of other factors. It’s not that we’re lazy – we often don’t have the information, resolution, or telescope time to do a detailed analysis of every system. Instead, we make reasonable educated guesses. This leads to a subtle but crucial point: our instruments do not give us the properties of a source, but rather they give us a snapshot of the light that our sensors detect when pointed in that direction. We don’t know that we recover all the flux, for starters – this is especially pertinent for interferometers, which act as spatial filters. Flux on large or small scales can be missed, depending on dish configurations, and flux also depends on frequency. Though we are trying to measure the properties of the sources we observe (e.g., luminosity), in reality we are limited to measuring the signal registered in our detectors (e.g., specific intensity or flux density). In other words, the final product you get out depends on the model you use, so it’s always a good idea to report results in terms of observables!

Brightness (B) or Intensity (I). In astronomy, “brightness” is a technical term that describes the power radiated through a surface area, divided by solid angle. SI units are W/m^2/steradian. In physics terminology, this quantity is sometimes referred to as radiance. When flux is divided by the solid angle that the source covers on the sky, the result is a quantity that is independent of distance. CAUTION: There seem to be two camps when it comes to brightness: It is most common to use the terms brightness and intensity to describe the same thing (Power per area per solid angle) – this is related to the B in Planck’s blackbody law or the Rayleigh-Jeans and Wien approximations. [B* _{ν}*]= W/m

^{2}/Hz/sr. Thus, brightness is independent of distance to the source. Some, however, use brightness and

*flux*interchangeably (making it dependent on distance). This makes things confusing, to say the least. Even my favorite intro astronomy textbook (the venerable Astronomy & Astrophysics by Zeilik & Gregory) falls into the latter category, though they do mention that they

*mean*flux when they say brightness, and use the symbol

*f*for it. I assume for the rest of this discussion that brightness is dimensionally equivalent to intensity (NOT flux), to remain consistent with the convention of the Planck law definition and its approximations. The NRAO page linked above has a very good discussion of brightness.

Surface Brightness (S). The term “surface brightness” is used specifically for objects whose projection on the sky is extended (covering many pixels or beam sizes). The units are equivalent to brightness from above: W/m^{2}/steradian. If you’re looking at a point source, you know that all the radiation is coming from that discrete location. But if you’re looking at something like a gas cloud, you really need to specify how spread out the emission is. For example, let’s say you are looking at a movie screen from the back of a theater. You have good eyesight, so you can resolve things down to 1cm from your distance. If someone shines a laser pointer on it before the movie starts, the resulting dot (<~1cm) is essentially a point source, so you can determine the flux easily – it’s all coming from the single tiny spot. However, if you now observe the screen during the movie, you may see quite a bit of illumination from the region – maybe maybe it’s an underwater shot with a gradient of bright blue at the surface to dim blue below. You can only make things out down to about 1cm, so you can’t determine the flux due to every infinitesimally small point, but you can determine the brightness [flux per square degree, say] of any piece of the screen just fine. You can always divide the brightness in a region by its solid angle to get its *average* flux. This scenario is obviously contrived, but it demonstrates the necessity of care when dealing with extended sources.

Optical images are often given in terms of magnitudes, for example mag/square”. It is also common to see magnitudes per physical measure of the solid angle based on the source, for example mag/pc^{2}. Here, the pc^{2} means the source area on the sky, which means that an assumption about the distance to the object is built in.

A quick note on summing: If you are summing the pixel brightnesses (to determine luminosity, for example), the appropriate solid angle to multiply by is the solid angle of **one pixel **(such as the red square in the image above), not the total solid angle of the source (the blue galaxy above). This is because the sum already takes into account the total area. See the tutorial on reprojection for further discussion.

Magnitudes (mag) & surface brightness (mag/arcsec^2). Apparent magnitudes (m) are a measure of how bright celestial objects appear to us. It’s a measure of flux peculiar to astronomy. It’s based on the system developed by Hipparchus (or so the story goes) a few thousand years ago, and we still use the magnitude scale because astronomers love tradition, though the modern one has been modified. Absolute magnitudes (M) are magnitudes that objects would have if they were observed from a standard distance of 10pc. I’ll be sticking to apparent magnitudes, since that’s what you will encounter if your images are in [mag]. Magnitudes are logarithmic, and we determine flux in SI units by comparing to a reference source. The sun is the reference most commonly used, but of course, you can always use another one. Here is the relation between magnitude and flux:

m_{⊙} – m = -2.5 log(F/F_{⊙}).

I should note here that this is true in general, but you will want to take care to use reference values for a specific wavelength/frequency band. The Sun has slightly different flux and magnitude in blue, red, etc. light. So a better way to write this relation might be m_{⊙,λ} – m = -2.5 log(F/F_{⊙,λ}). I am using λ here instead of *ν* to avoid confusion with the typical notation for flux density, F* _{ν}*. (See next section)

Turning the equation around, we get the formula for flux:

F = F_{⊙,λ} × 100^{(m⊙,λ – m)/5}.

Spectral variations:

Each instrument has a finite bandwith over which it can detect light (think: red or blue filters in a camera or telescope). When an instrument is capable of detecting light over a large-enough range of different channels, you will often get images in units that reflect the fact that the detected light is a function of wavelength of frequency. Flux detected over many spectral channels is usually called Flux Density (F* _{ν}*), and comes in units of W/m

^{2}/Hz or, as is common in the radio and FIR: Janskys (1 Jy = 10

^{-26}W/m

^{2}/Hz). Luminosity L [W] becomes spectral luminosity L

*[W/Hz], S becomes S*

_{ν}*(still called surface brightness), and Intensity/Brightness become Specific Intensity/Brightness (I*

_{ν}*, B*

_{ν}*) in W/m*

_{ν}^{2}/Hz/sr or Jy/sr. The most familiar use of spectral brightness / specific intensity / spectral radiance in textbook studies is the Planck law, which quantifies the emission from a blackbody at a given temperature.

It is also common to encounter specific intensity images in units of Jy/beam. “Beam” is the term used ubiquitously in radio (and often in the FIR) to refer to the angular size of the resolution element or point-spread-function, which is usually assumed to be a 2D gaussian. The area under a single gaussian is , and thus the area under a 2D gaussian is computed as . For the beam response, we set the amplitude = 1. σ = FWHM/(2 sqrt(2 ln2)), so the double integral simplifies to ~ 1.133 FWHM_{1}* FWHM_{2}. The FWHM can be given in terms of arcseconds or pixels, depending on what you want. Thus, Jy/beam is a measure of surface brightness.

Conceptually, to get the total Flux, luminosity or brightness over a continuous distribution of emission, you would integrate the function over all observed frequencies. Since our instruments discretize the signal into channels, we can take the approximation of sums.

If the channels are uniform in size (usually the case), then things simplify because the sum changes to ∑ F_{ν,i} × d*ν*_{i} → n_{chan} × d*ν* × ∑ F_{ν,i}. Thus you can sum the flux and multiply by the total bandpass, Δ*ν* = n×d*ν*.

Since the total flux comes from the integration of flux over all channels, the total flux, (and intensity, etc.) is sometimes called the “integrated flux” (“integrated intensity”, etc.). Be careful, though, as this term is also used to describe the total flux in an image resulting from integration of the flux in several pixels.

### Conversions

Don’t just blindly apply dimensional analysis! To start, let’s define some commonly used quantities.

- D = distance to source.
- A
_{beam}= . The FWHM can be in pixels or arcsec. - Δ
*ν*=*ν*_{2}–*ν*_{1}. This is the effective passband of the instrument. - Ω
_{sr}= solid angle of a pixel in the image

Once all the definitions and units from the previous section are sorted out, converting between them is straightforward. To go from intensity or (surface) brightness [W/m^{2}/sr] to flux [W/m^{2}/pixel], simply multiply by the pixel size (in steradians, beams, square arcsec, or whatever units the brightness was in). There is a subtle point to be aware of here: notice that I wrote flux in units of [something *per pixel*]. Total flux from a source in an image depends on how many pixels you include. A single pixel will have units of [Jy/pix] or [W/m^{2}/pix] – this makes the sum of several pixels [Jy] or [W/m^{2}]. e.g., f[Jy/pixel]×n[pixels]=F[Jy]. The tricky part here is that sums of brightness maps still need to include the factor of steradians/pixel, etc. See the examples section below for more discussion on this point.

To go from flux to luminosity, multiply your flux by the area of the sphere at distance D (from source): 4 pi D^2. Easy peasy! Again, luminosity in units of [W] or [erg/s] implies that you have summed over several pixels. If you are making a luminosity map, just be aware that your units will technically be [W/pixel].

For magnitudes, we need to first convert to flux or luminosity. Again, we determine flux from apparent magnitude by comparing to a reference – you can choose any reference of known flux/magnitude that you like, but the easiest to use is the Sun. Since F=L/4πD^{2}, you can work in terms of luminosity if you prefer:

m – m_{⊙} = -2.5 log(L/L_{⊙} × D_{⊙}^{2}/D^{2}). So, if you’re going TO flux or luminosity from magnitudes, you can use

F [W/m^{2}] = F_{⊙,λ} × 100^{(m⊙,λ – m)/5} and

L [W] = L_{⊙,λ} × (D/D_{⊙})^{2} × 100^{(m⊙,λ – m)/5}

For single maps in spectral units (Jy, erg/s/angstrom, etc.): divide by the total bandwidth used in that frame. If it’s an image with several frames covering different frequencies/wavelengths/velocities, etc., then use the bandwidth of the individual channels.

Integrated moment maps (Moment-0, flux over all channels) can be a little tricky. The image started out as a multi-channel data cube [Jy/beam or similar], and the channel information (frequency or wavelength) may have been converted to something more physically useful for analysis, such as recessional velocity v, redshift z, or distance from the observer. When the image was then summed, it may have kept those converted units, giving the resulting data mixed units such as Jy/beam × m/s. Note that this is an intensity (or brightness) – Janskys denote flux density, where 1 Jy = 10^{-26} W/m^{2}/Hz. The “per hertz” bit makes it a measure of flux *per channel*. Likewise, the m/s bit is a measure of the total bandwidth – it’s the recessional velocity range (determined from the redshift equation) that comes from the bandwidth. In effect, the m/s and 1/Hz cancel out (with a leftover factor).

You may be thinking, “Why don’t we just divide by the total velocity range covered and multiply by the frequency passband – that would cancel the units!” For one, if you download an image from NED or elsewhere, sometimes you don’t have all the info on hand, such as the channel widths, number of channels, or total bandwidth summed over. But the real rub is that intensity is technically defined by an integral. On its own, when it depends on a single variable, it’s OK to approximate ∫ I(*ν*)d*ν* ≈ ∑ I(*ν*) Δ*ν*. But now we are dealing with an additional variable: I = I(*ν*,v) So when we want to simplify, we have to put in the differentials: I [W/m^{2}/beam] = I [W/m^{2}/beam/Hz · m/s] × d*ν*/dv. This is NOT the same as I [W/m^{2}/beam/Hz ×m/s] × Δ*ν*/Δv because the differentials are, well, differentials – variables, not static.

To avoid confusion between the notation for velocity v and frequency *ν*, I will switch the notation for frequency from *ν* to *f* for this part. Again, I [W/m^{2}/beam] = I [W/m^{2}/Hz · m/s] × d*f*/dv. The recessional velocity is given by the standard redshift equation, using the rest frequency *f*_{0}:

v(*f*) = *c*×(*f*–*f*_{0})/*f*_{0}

Note, however, that this definition is used mostly by radio astronomers. Optical astronomers use v/*c* = (*f*–*f*_{0})/*f* instead. See here ( http://iram.fr/IRAMFR/ARN/may95/node4.html ) for more discussion on that point.

Now,

∂v/∂*f* = ∂/∂*f* [( 1-*f*/*f*_{0} ) c ] = –*c */*f*_{0}

Therefore,

| ∂*f*/∂v | = | ∂v/∂*f *|^{-1} = *f*_{0}/*c*

So, to convert from Jy/beam · m/s, which is an intensity, we multiply by ∂*f*/∂v = *f*_{0}/*c* , NOT by Δ*f*/Δv.

Here are some other factors that you may (or may not?) come across:

∂v/∂λ = ∂/∂λ [ (λ-λ_{0})/λ_{0}×*c *] = *c*/λ_{0}, so | ∂λ/∂v | = λ_{0}/*c*

*c *= λ*f* → ∂*f*/∂λ = ∂/∂λ( *c*/λ ) = –*c*/λ^{2 } → | ∂*f*/∂λ | = c/λ^{2}, | ∂λ/*∂f *| = *c*/*f*^{2}

## Examples

Coming soon!

## Quick Reference:

Click here for a printout cheat sheet of the units and conversions discussed on this page.