# 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 = $\displaystyle 10^{-26}$ 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 pixel! 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!

# 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 = $\displaystyle 10^{6}$ 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 $\displaystyle 2.3504 \times 10^{-11}$ 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 this page 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 use at the SSC, called APEX, produces fluxes in microJanskys. The final bandmerged catalog you can get has listed fluxes in microJanskys, as well as magnitudes.

Astronomers use magnitudes in color-color or color-magnitude plots. Astronomers use a variant on fluxes 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:

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

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

We have published the zero points (e.g., the "Vega flux") for most of our bandpasses. They are (copied from various places on the web):

• IRAC 1 : 280.9 Jy
• IRAC 2 : 179.7 Jy
• IRAC 3 : 115.0 Jy
• IRAC 4 : 64.13 Jy
• MIPS 1 : 7.14 Jy
• MIPS 2 : 0.775 Jy
• MIPS 3 : 0.159 Jy

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 2MASS magnitudes on the web as well:

• J : 1594 Jy
• H : 1024 Jy
• K : 666.7 Jy

Note that plain magnitudes get fainter (the number gets larger) as the distance of the object increases. BUT, colors (differences in magnitudes) are ratios of fluxes, and therefore independent of distance.

## 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.

1Jy = $\displaystyle 10^{-23}$ erg/s/cm^2/Hz (in cgs units rather than mks units, sorry). A Jansky is technically a unit of "flux density." In order to get rid of the "per Hz", you need to multiply the Jy by the frequency of the bandpass center.

Astronomers coming from the longer wavelengths will tend to plot up nu * F(nu) (written as $\displaystyle \nu F_{\nu}$ ) against nu, where "nu" ($\displaystyle \nu$ ) is the frequency. The units of $\displaystyle F_{\nu}$ are Janskys.

Astronomers coming from the shorter wavelengths will tend to plot up lambda * F(lambda) (written as $\displaystyle \lambda F_\lambda$ ), where "lambda" ($\displaystyle \lambda$ ) is the wavelength of the light. The units of $\displaystyle F_{\lambda}$ are NOT Janskys.

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

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



So you need to multiply the $\displaystyle F_{\nu}$ by $\displaystyle c/\lambda^2$ to convert it into $\displaystyle F_{\lambda}$ .

Additionally, to analyze the Spitzer data, it's often useful 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)

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


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

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

In words, in order to analyze our data, we need to have something that does the following:

1. Reads in the fluxes from the files.
2. Converts the Spitzer fluxes (and errors) into magnitudes (if necessary).
3. Converts the 2MASS magnitudes (and errors) into fluxes (if necessary).
4. Makes color-color and color-magnitude plots for stars in our region using magnitudes.
5. Makes SED plots for individual objects, but converting numbers first into the right units:
1. Creates an array of the wavelengths of each measurement, keeps a copy of the version in microns, and converts to cm.
2. For any real measurements, converts the microJanskys into cgs units.
3. For any real measurements, converts $\displaystyle F_{\nu}$ into $\displaystyle F_{\lambda}$ by multiplying the $\displaystyle F_{\nu}$ values by the $\displaystyle d\nu/d\lambda$ corresponding to the wavelength of each bandpass.
4. For any real measurements, multiplies $\displaystyle F_{\lambda}$ by the lambda corresponding to the wavelength of each bandpass to get $\displaystyle \lambda F_{\lambda}$ .
6. For any real measurements, plots the log of the $\displaystyle \lambda F_{\lambda}$ data points (in cgs units) against the log of the lambda data points (in microns, only because that makes it easier to read). Labels the axes (with units)! Plots the error bars on top of the data points (also converted from uJy).
7. For any real measurements, for any star with at least 2 fluxes, fits 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.

Why are we plotting $\displaystyle \lambda F_{\lambda}$ vs. $\displaystyle \lambda$ ? 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 $\displaystyle \lambda F_{\lambda}$ instead of $\displaystyle \nu F_{\nu}$ ? Well, only for internal consistency. Since one axis is in wavelength units, it makes sense to have the other axis also in wavelength units.

# Cookbook for image conversion: 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."

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.)

## 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.

  1.297e7 square arcsec    2.3504e-11 sr                     sr
------------- * ---------------- = 0.000304847 ------------
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 ---------------  *  0.000304847 ------------ = 3.50333e-11 ----
(square) pixel                 square degree                px


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

    MJy                      sr    MJy
---------   *  3.50333e-11 ---- = ---
sr                       px    px


So multiply this whole image by 3.50333e-11. The units of the image (and consequently the photometry you get out) are in MJy (MegaJanskys). If you want to get it in Janskys:

 1e6  Jy
----
MJy


so multiply the image by 1e6 to get the image into Jy:

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


If you want to get it into 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.)

## 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 multiply the image by 23.5045 to get it into uJy/square arcsec

## 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 1.48938 to get it into uJy/px.