Observing a Y Dwarf

Today, let' s look at a type of astrophysical object known as a Y dwarf.
Y dwarfs (n.b. not dwarves) are a recently discovered subclass of brown dwarfs that have an apparent temperature of around 350 K.
In this case we will be looking at one close to a Sun-like star to see how difficult it might be to detect one of these.

Using the blackbody curve for an object at 350 K,
${\lambda }_{\max }=\frac{b}{T}$ Where b = $2.8977685\left(51\right)\times {10}^{-3}$ m·K
∴ ${\lambda }_{\max }=8.28\times {10}^{-6}m=8280\mathrm{nm}$   which is in the far infrared
This is around the wavelength you would want to be looking at to detect this type of object.

Now, we want to think about how many photons we could actually see from the Y dwarf compared to the Sun-like star.
Let’s assume that the star has a radius equal to the Sun’s and that the dwarf has a radius equal to Jupiter’s.  Also, let’s say that these objects are both about 30 light years from where we will detect them.
To figure this out, we will need to use Planck’s Law:
$B_{\lambda }\left(T\right)=\frac{2hc}{{\lambda }^5}\frac{1}{e^{\frac{hc}{\lambda kT}}-1}$    where B is the spectral irradiance in units of ergs per steradian per ${\mathrm{cm}}^2$per wavelength in cgs, h is Planck’s Constant, k is Boltzmann’s Constant, and c is the speed of light
Using this, at ${\lambda }_{\max }$a Y dwarf has a spectral irradiance of
$B_{{\lambda }_{\max }}=$$2.15\times {10}^8$ergs ${\mathrm{sr}}^{-1}{\mathrm{cm}}^{-2}$
To find the number of photons per ${\mathrm{cm}}^2$ per second at a distant of 30 ly we need to multiply by the visible surface of the Y dwarf, then multiply by the solid angle subtended by 1 ${\mathrm{cm}}^2$ at 30 ly, then divide by the energy per photon at this wavelength:
$N_{\mathrm{photons}}=2.15\times {10}^8*\left(\pi {R_{\mathrm{Jupiter}}}^2\right)*\left(\frac{1}{4{\pi \left(30\mathrm{ly}*9.46\times {10}^{17}\frac{\mathrm{cm}}{\mathrm{ly}}\right)}^2}\right)*\left(\frac{1}{\frac{hc}{{\lambda }_{\max }}}\right)=14\mathrm{photons}{\mathrm{cm}}^{-2}{s}^{-1}$
This is clearly not very many photons
Doing the same calculation for the Sun-like object at 5777 K (but still at the same wavelength):
$N_{\mathrm{photons}}=552000\mathrm{photons}{\mathrm{cm}}^{-2}{s}^{-1}$

Looking at these numbers, it is easy to see why it would be very difficult to detect a Y dwarf next close to a Sun-like star (especially when you consider uncertainty in detection and possible error).  The ratio of photons at our detector is 41,000:1; that’s 40,000 photons from the Sun-like star to every photon from the Y dwarf, every second.

This was worked on in class with Cassi, Lauren, and Joanna.

Created with Wolfram Mathematica 8.0