# General Units

Wavelengths in infrared astronomy are commonly expressed in microns = micrometers = µm (or um if you don't have a µ).

• 5000 Å =500 nm =0.5 µm =Visible light
• ~0.9 to 5 µm =Near-infrared (~smoke particles)
• 5 µm to ~30 µm = Mid-infrared (~hair)
• 30 µm to ~350 µm = Far-infrared (~salt grain)

Brightnesses or fluxes are most likely to be given in Janskys (Jy) or mJy (milli Jy) or µJy (micro Jy). 1 Jansky = 10 − 26 Watts/m^2/Hz.

Jy can be converted to magnitudes which have historically been relatively rarely used in the mid- or far-infrared.

Because the unit is named for Karl Jansky, the plural of the unit is really Janskys, not Janskies.

## Aside on fluxes and flux densities

Astronomically, it can be important to understand the difference between luminosity, flux, and flux density. In practice for this stuff, you probably don't need to know the gritty details of this until you are more familiar with the numbers and the jargon.

Colloquially, flux means the rate of something through something else, such as water through a pipe, or traffic on a highway. In physics and astronomy, it means the same thing.

Flux is a measurement of energy per unit area per unit time. Using our analogies above, this would be the number of cars per lane per second that pass under a bridge on a highway (or grams of water through the cross-sectional area of the pipe per second). In measuring energy from celestial objects, the units of flux are Joules per second per meter squared if you like mks (meters-kilograms-seconds) units, or ergs per second per centimeter squared if you like cgs (centimeters-grams-seconds) units.

Luminosity is a measurement of energy per unit of time, such as Joules per second if you like mks units, or ergs per second if you like cgs units. This would be, in our analogy, the total number of cars on the highway passing under the bridge per second. (The flux of cars is the luminosity per lane.)

Flux density is a measurement essentially of energy per unit area per unit time "per photon". In our analogy, this would be the number of RED cars per lane per second that pass under the bridge on the highway. In this analogy, the "per photon" is seen in the red cars. In astronomy, the "per photon" manifests itself as a "per Hz" (unit of frequency) or "per cm" (unit of wavelength). A Jansky is proportional to Watts/m^2/Hz. Recall that Watts are energy per second. So this is energy per second per square meter per Hertz.

Now, just to further confuse things, the units of Spitzer mosaics are not just Janskys, but Janskys per area! To make the numbers easier, they are in MJy/sr, but they could also be in uJy/square arcsecond. Read on for more, including definitions and scale factors!

For completeness, we note here that magnitudes are proportional to the log of the ratio of two fluxes. Most magnitudes with which you are most likely familiar are tied to the magnitude of Vega, so a magnitude of 0 means that the object has the same flux as Vega. There's more on this below, too.

# Units of Spitzer Images

Optical data with which you are familiar may be in counts or photons, or possibly (like Hubble data) calibrated to be energies. That, combined with the exposure time of the image, gives you flux units. Spitzer data comes in flux (density) per unit (pixel) area instead, MegaJanskys per steradian (MJy/sr). 1 MJy = 106 Jy, and a sr is a solid angle.

If you've done photometry before, and expect to do it exactly the same way again here, it won't work, because this matters.

1 square arcsec is $2.3504 \times 10^{-11}$ sr. (1 degree = 60 arcmin = 3600 arcsec.)

If you want to convert the image from MJy/sr to uJy/square arcsec, multiply the image by 23.5045. The units of this number are (uJy/arcsec)/(MJy/sr).

If you want to take a Spitzer image and use your previous routines on it, the most efficient way to do this is probably to take the image in MJy/sr and multiply out the "per sr" part of it so that it is instead in MJy/px. The subtlety in this step is that each Spitzer array has slightly different pixel sizes, and the mosaics that we create have different sizes yet again from the original images. You can make mosaics with whatever size pixels you want, so if you get Spitzer mosaics from more than one astronomer, or more than one Spitzer wavelength, chances are excellent that the pixels will be slightly different sizes. The information on the pixel sizes are in the FITS header of each image.

The following paragraphs are a high-level summary of what to do for any Spitzer image data you may encounter; see below for a cookbook of the process for one mosaic.

Look in the FITS header of the mosaics for the keywords "CDELT1" and "CDELT2". These keywords are set to be the scale of the rows and columns in degrees per pixel. Using the values of these keywords, and the conversions above, you can figure out the number of square degrees per pixel, the number of square arcsec per pixel, and finally the number of steradians per pixel. Multiply the whole image in MJy/sr by the number of sr/px to get MJy/px.

If you are instead working with the individual BCDs (read this as: the individual little images that went into the big mosaic), you should look for keywords "PXSCAL1" and "PXSCAL2". NOTE that these pixels ARE NOT SQUARE, and this is more important for MIPS data. From here, you now have the same information as the "CDELT1" and "CDELT2" above, so you can follow the same procedure.

# Units of Spitzer Photometry

## Introduction

The photometry software that people at the SSC wrote for use with Spitzer data, called APEX, produces flux densities in microJanskys. The final bandmerged catalog you can get has listed flux densities in microJanskys, as well as magnitudes.

Astronomers use magnitudes in color-color or color-magnitude plots. Astronomers use a variant on flux densities in spectral energy distribution (SED) plots.

## Magnitudes

A magnitude is really a flux ratio. It is defined as follows, where M's are magnitudes and F's are fluxes:

$M_1 - M_2 = 2.5 \times \log \left(\frac{F_2}{F_1}\right)$      (eqn 1)


The magnitude system (in the optical) was defined to be referenced to Vega. In other words, Vega is defined to be zero magnitude, and you would then define magnitudes of anything else as follows:

  $M = 2.5 \times \log \left(\frac{F_{\mathrm{Vega}}}{F}\right)$       (eqn 2)


When they looked at Vega with IRAS, they discovered that it did NOT look like they expected, and in fact it has a large infrared excess! Therefore, infrared magnitudes are defined with respect to what Vega would be, if it did not have an excess.

Generally, the zero points (e.g., the "Vega flux") are published for most of the bandpasses you might encounter. They are consolidated on the Central wavelengths and zero points page. Therefore, in order to convert the uJy that apex returns into magnitudes, use the equation 2 above, substituting these so-called "zero-point fluxes" in for "Fvega." Note that the zero-point fluxes are in Janskys and the fluxes returned by APEX are in microJanskys. You can find the zeropoints for many other bands on the web as well, such that you can freely convert between mags and flux densities.

Note that plain magnitudes get fainter (the number gets larger) as the distance of the object increases. This happens because Vega, your reference object, stays the same while such an example object moves farther away. BUT, colors (which are differences in magnitudes) are ratios of fluxes of the same object, and therefore independent of distance. This is powerful when you are studying objects whose distances you don't know, or comparing objects at a variety of distances.

To convert magnitudes back into fluxes (e.g., if you have optical or 2MASS magnitudes and need to get fluxes), you need to invert the equation above. Recall that to invert a logarithm (base 10), you have to raise both sides to the power of 10, e.g., if log x = y then x = 10^y.

### Aside on AB mags

BE CAREFUL to keep track of whether you are working with Vega-based magnitudes or AB mags. Vega magnitudes define things with respect to a Vega spectrum (as above), but some folks (largely extragalactic folks) define things with respect to a flat spectrum source instead, and those are AB mags. Most Sloan folks (even those folks working with stars, and even those working with Sloan filters but not necessarily SDSS archival data) work in AB mags instead. For AB mags, you always use a flat reference spectrum, so the zero point is 3631 Jy for all bands.

## Spectral Energy Distributions (SEDs)

SEDs are energy plotted against some measure of the photon -- frequency or wavelength. The reason astronomers do this is to see how much energy is produced by the object as a function of frequency or wavelength. Now it's really going to get a little hairy! Steel your nerves and plunge onwards... it really all comes down to unit conversion.

The units that are used in Spitzer data are Janskys. 1 Jy = 10 − 23 erg/s/cm^2/Hz (in cgs units rather than mks units, sorry -- and just to be clear, I mean erg / (s*cm^2*Hz), it's just easier to read in the way I wrote it above). A Jansky is technically a unit of "flux density," represented by Fν. We want to plot "energy density", so one way to do this (DON'T DO THIS TO YOUR DATA YET) is to get rid of the "per Hz", e.g., multiply the Jy by the frequency (ν) of the bandpass center, or νFν. You could just stop here, and plot νFν vs. ν to get something that is technically a spectral energy distribution. BUT, now it starts to get a little hairy, because there are some cultural influences here. Do you know off the top of your head what the frequency of the IRAC-1 band is? I don't either, but I do know its wavelength -- 3.6 microns. Astronomers coming from the longer wavelengths (mm, radio, etc.), because they think in units of frequencies, will tend to plot up νFν against ν (nu). The units of Fν really are Janskys. BUT, astronomers coming from the shorter wavelengths (optical, etc.), because they think in units of wavelengths, will tend to plot instead λFλ , where λ (lambda) is the wavelength of the light. This is what we want to do here (because we have been thinking of the wavelengths of the Spitzer bands but not the frequencies). The catch here is that the units of Fλ are NOT Janskys.

$\lambda \times \nu = c$, the speed of light. In order to convert the Janskys into units of Fλ, you need to take into account the differentials (ah-HA, calculus being used here!), e.g., the fact that

   $\frac{dF}{d\lambda} = \frac{dF}{d\nu} \frac{d\nu}{d\lambda}$ and $d\nu = \frac{c}{\lambda^2}d\lambda$


So you need to multiply the Fν by c / λ2 to convert it into Fλ. But we are not done yet! Recalling from above, the units of Fλ are not an energy density. You need to get another factor of λ in there to make the units work out to be energy density: calculate λFλ to get units of ergs/s/cm^2.

SO, IN SUMMARY: Take your Fν measurements that are in Jy. (Ensure they are in Jy! If they're in magnitudes, convert them to Jy first; see 'magnitude' discussion above.) Multiply by 10 − 23 to get them into cgs units. Multiply these Fν values by c / λ2 to get them into Fλ. Multiply them by λ to get them into λFλ. WATCH YOUR UNITS. NB: c = 2.997924d10 cm/sec

In other words, in order to convert our data from photometry to SEDs, we need to do the following:

1. Read in the catalogs you have.
2. Convert any magnitudes (and errors) into flux densities (if necessary).
3. Make SED plots for individual objects, but converting numbers first into the right units:
1. Create an array of the wavelengths of each measurement, keeps a copy of the version in microns, and convert to cm.
2. For any real measurements, convert the flux densities (probably in microJanskys) into cgs units.
3. For any real measurements, convert Fν into Fλ by multiplying the Fν values by the dν / dλ corresponding to the wavelength of each bandpass.
4. For any real measurements, multiply Fλ by the lambda corresponding to the wavelength of each bandpass to get λFλ.
4. For any real measurements, plot the log of the λFλ data points (in cgs units) against the log of the lambda data points (in microns, only because that makes it easier to read). Label the axes (with units)! Plot the error bars on top of the data points (also converted from uJy).

### Notes on plotting

Why are we plotting λFλ vs. λ instead of ν? Well, only because I think in wavelength, not frequency. I don't know off the top of my head the frequencies of the Spitzer bandpasses, but I do know their wavelengths.

Why are we plotting λFλ instead of νFν? Well, only for internal consistency. Since one axis is in wavelength units, it makes sense to have the other axis also in wavelength units.

If you have gotten this far using real data, you will find (if you have done the calculations correctly) that you have numbers that are very small, like 9.77237e-12, 1.99526e-11, etc. Any time you find yourself with these kinds of numbers, you should automatically plot in log space (or log/log space), NOT linear space. You want to actually plot log(λFλ) vs. log(λ).

### The next step

IF you are highly motivated and ready to go on to the next step... It can be useful, in the course of analysis of the Spitzer data, to pretend that the contribution from the star is a blackbody. It's not really, but it's awful close, especially in the infrared. A blackbody's flux density is given by (where T is temperature, and other constants are given below)

  $B_{\lambda} = \left(\frac{2hc^2/\lambda^5}{\exp(hc/\lambda kT)-1)}\right)$       (eqn 3)


but of course we want to plot $\lambda \times B_{\lambda}$:

  $\lambda B_{\lambda} = \left(\frac{2hc^2/\lambda^4}{\exp(hc/\lambda kT)-1)}\right)$       (eqn 4)


Values of these constants all in cgs units:

• h = 6.6260755d-27 erg*sec
• c = 2.997924d10 cm/sec
• k = 1.380658d-16 erg/deg

So, in summary, for the list of things to do above, add this one:

1. For any real measurements, for any star with at least 2 fluxes, fit a model -- one very simple way to do that is to fit a blackbody to the energies derived from the three 2MASS and first 2 IRAC bands. There are two free parameters in this fit -- the temperature of the blackbody and an additive (in the log) offset related to the distance of the object. If we know the temperature of the star (via a spectral type) and the distance to the object, then we know the values for the temperature and the offset.

# Cookbook for image conversion (or calculating the number APT needs): Method One

## Step zero. What do you have and what do you need?

You have an image in MJy/sr(/px). You have the number of degrees per pixel.

You need to convert the image to MJy(/px), a.k.a "get rid of the steradians." This is also the number that APT needs to do the conversion to MJy.

The things that make this hard are:

• The pixel size of the mosaic changes depending on wavelength and where you got the mosaic, so I can't just give you one number to work for all mosaics every time.
• The "pixels" in the above are kind of a funny, hidden unit and the accounting of it works in some unexpected ways, which is why it's in parentheses above (and below).

## Step one. Find out what the size of the pixels are in your images

For the IRAC-1 mosaic I created for you in July 2006, CDELT1=-0.000339 degrees per pixel, and CDELT2=0.000339 degrees per pixel. (ignore the minus sign; it has something to do with a fits convention.) You can find this out for any given mosaic by looking in the fits header.

## Step two. Find out what the size of the pixels are in square degrees per pixel

          degrees             degrees              square degrees
0.000339 ------- *  0.000339 ------- = 1.14921e-7 ---------------
pixel               pixel               (square) pixel


## Step three. Find out what the conversion is between square degrees and sr.

There are 60 arcminutes in a degree. There are 60 arcseconds in an arcminute.

60 arcminutes   60 arcseconds     3600 arcseconds
------------- * -------------- =  ---------------
1 degree        1 arcminutes        1 degree


Square it!

  (1 degree)^2 = 1 square degree = (3600 arcsec)^2 = 1.296e7 square arcsec


We look up that 1 square arcsec is 2.3504x10^(-11) sr.

  1.296e7 square arcsec    2.3504e-11 sr                       sr
------------- * ---------------- = 0.0003046118 ------------
square degree    1 square arcsec                square degree


## Step four. Find out what the size of the pixels are in sr.

           square degrees                        sr                       sr
1.14921e-7 ---------------  *  0.0003046118 ------------ = 3.500629e-11  ----
(square) pixel                   square degree                 px


## Step five. Convert the units of the image.

    MJy                       sr    MJy
---------   *  3.500629e-11 ---- = ---
sr                        px    px


So you or APT needs to multiply this whole image by 3.500629e-11. This is what APT does with the number that you type into the "More settings" window -- it multiplies the result by the number we enter. The number that you need to enter, then is the number that we just calculated. BUT look at the units of that number. The units of the image (and consequently the photometry you get out) are then in MJy (MegaJanskys). None of the sources will be that bright. If you want to get it in Janskys:

 1e6  Jy
----
MJy


so then multiply the image (a.k.a. the number above) by 1e6 to get the image (e.g., the result of the photometry) into Jy (or multiply the number you get above by 1e6 before entering it into APT):

 MJy   1e6  Jy     Jy
--- *     ---- = ----
px         MJy     px


For the specific example we're working through, 3.500629e-11 * 1e6 = 3.500629e-5. If you enter that number into APT, the result of its calculation will be in Jy. But, most sources will be even fainter. MOPEX returns results in MICROJANSKYS. If you want to have your results produced in microJy, there are 1e6 uJy in a Jy, and I'll let you do that one.

# Method 2: Alternative but completely equivalent and possibly more straightforward solution

## Step zero. What do you have and what do you need?

You have an image in MJy/sr(/px). You have the number of degrees per pixel.

You need to convert the image to uJy(/px), a.k.a "get rid of the steradians" AND convert to microJanskys to get the numbers to still be reasonable and not very tiny or very large.

The things that make this hard are:

• The pixel size of the mosaic changes depending on wavelength and where you got the mosaic, so I can't just give you one number to work for all mosaics every time.
• The "pixels" in the above are kind of a funny, hidden unit and the accounting of it works in some unexpected ways.

## Step one. Find out what the size of the pixels are in your images

For the IRAC-1 mosaic I created for you in July 2006, CDELT1=-0.000339 degrees per pixel, and CDELT2=0.000339 degrees per pixel. (ignore the minus sign; it has something to do with a fits convention.) For any given mosaic, you can find out these values by looking in the fits header.

## Step two. Convert your image from MJy/sr to uJy/square arcsec

We look up that there are :

          [uJy/sq. arcsec]
23.5045 -----------------
[MJy/sr]


So the first factor to multiply the image by is 23.5045 to get it into uJy/square arcsec. If you are attempting to get the number for APT, this is the first factor to write down on a piece of scrap paper.

## Step three. Find out what the size of the pixels are in square degrees per pixel

          degrees             degrees              square degrees
0.000339 ------- *  0.000339 ------- = 1.14921e-7 ---------------
pixel               pixel               (square) pixel


## Step four. Find out how many square arcsec there are in a pixel.

There are 60 arcminutes in a degree. There are 60 arcseconds in an arcminute.

60 arcminutes   60 arcseconds     3600 arcseconds
------------- * -------------- =  ---------------
1 degree        1 arcminutes        1 degree


Square it!

  (1 degree)^2 = 1 square degree = (3600 arcsec)^2 = 1.296e7 square arcsec


Convert the pixel size.

            square degrees            square arcsec             square arcsec
1.14921e-7 --------------- * 1.296e7 -------------- = 1.48938 ---------------
(square) pixel            square degree             (square) px


## Step five. Convert the image

     uJy            1.48938  square arcsec      uJy
-----------    *           --------------- = ------
sq arcsec (*px)              (square) px        (px)


So multiply the image by another factor of 1.48938 to get it into uJy/px. (If you are trying to get the number for APT, get the factor from way up above, 23.5045, and multiply it by 1.48938 [for this example] to get the number you want to put into APT so that it can do the multiplication for you.) Note that the output of APT will then be in microJanskys.