# Units

## Contents

# General IR astronomy 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 = **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\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.

# 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.

Magnitudes has a different wiki page.

# Units of Spitzer and Herschel 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 (and some Herschel data) come in *flux (density) per unit (pixel) area* instead, MegaJanskys per steradian (MJy/sr). 1 MJy = **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\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 **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\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 or Herschel 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 (or Herschel) detector has 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(or Herschel) mosaics from more than one astronomer, or more than one Spitzer(or Herschel) wavelength, chances are excellent that the pixels will be slightly different sizes (and for Herschel, maybe slightly different units). The information on the pixel sizes are in the FITS header of each image.

To figure out the pixel size, look in the FITS header of the mosaics for keywords corresponding to pixel size -- they are often "CDELT1" and "CDELT2", but may be "PXSCAL1" and "PXSCAL2" or something else. 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.

=

## 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 = **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 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 **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle F_{\nu}}**
. 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 (**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \nu}**
) of the bandpass center, or **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \nu F_{\nu}}**
. You could just stop here, and plot **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \nu F_{\nu}}**
vs. **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \nu}**
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 **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \nu F_{\nu}}**
against **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \nu}**
(nu). The units of **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle F_{\nu}}**
really are Janskys. BUT, astronomers coming from the shorter wavelengths (optical, etc.), because they think in units of wavelengths, will tend to plot instead **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \lambda F_\lambda}**
, where **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \lambda}**
(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 **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle F_{\lambda}}**
are NOT Janskys.

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \lambda \times \nu = c}**
, the speed of light. In order to convert the Janskys into units of **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle F_{\lambda}}**
, you need to take into account the differentials (ah-HA, calculus being used here!), e.g., the fact that

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \frac{dF}{d\lambda} = \frac{dF}{d\nu} \frac{d\nu}{d\lambda}}andFailed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle d\nu = \frac{c}{\lambda^2}d\lambda}

So you need to multiply the **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle F_{\nu}}**
by **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle c/\lambda^2}**
to convert it into **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle F_{\lambda}}**
. But we are not done yet! Recalling from above, the units of **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \lambda}**
in there to make the units work out to be energy density: calculate **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \lambda F_{\lambda}}**
to get units of ergs/s/cm^2.

**SO, IN SUMMARY**: Take your **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 10^{-23}}**
to get them into cgs units. Multiply these **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle c/\lambda^2}**
to get them into **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \lambda}**
to get them into **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \lambda F_\lambda}**
. 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:

- Read in the catalogs you have.
- Convert any magnitudes (and errors) into flux densities (if necessary).
- Make SED plots for individual objects, but converting numbers first into the right units:
- Create an array of the wavelengths of each measurement, keeps a copy of the version in microns, and convert to cm.
- For any real measurements, convert the flux densities (probably in microJanskys) into cgs units.
- For any real measurements, convert
**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle d\nu/d\lambda}**corresponding to the wavelength of each bandpass. - For any real measurements, multiply
**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \lambda F_{\lambda}}**.

- For any real measurements, plot the log of the
**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\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). 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

Why are we plotting

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(

### 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)

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle B_{\lambda} = \left(\frac{2hc^2/\lambda^5}{\exp(hc/\lambda kT)-1)}\right)}(eqn 3)

but of course we want to plot **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \lambda \times B_{\lambda}}**
:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\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

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

- 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.