ERRATUM: "ARE THE ULTRA-FAINT DWARF GALAXIES JUST CUSPS?" (2011, ApJ, 727, L14)

Both of the exponential and cusp models tested here have six free parameters. The cusp model's free parameters are: central R.A. and decl., rotation angle and scale, a smoothing scale, and a constant background level. The exponential model's free parameters are: central R.A. and decl., a rotation angle, ellipticity, half-light radius, and a constant background level. For each model we step through the six parameters (Θ) and evaluate the likelihood of the data given each model. The full likelihood, which is a product of the individual star likelihoods, is

$$\begin{eqnarray} &&\mathscr{L}(\mathbf {\Theta })= \prod _{i=1}^{N_{\rm star}} \frac{1}{Z(\mathbf {\Theta })}\Sigma (x_i|\mathbf {\Theta }) \end{eqnarray} \tag{ 1 }$$

$$\begin{eqnarray} &&Z(\mathbf {\Theta })=\frac{1}{N_{{\rm random}}} \sum _{j=1}^{N_{{\rm random}}} \Sigma (x_{\rm random,j}|\mathbf {\Theta }), \end{eqnarray} \tag{ 2 }$$

where Θ is the list of parameters for the cusp or exponential model, Σ(·) is the surface density of the model given the parameters, and Z(Θ) is a normalization constant, estimated using a spatially randomized stellar catalog over the observed sky region. N_star is the total number of stars, and x_i is the two-dimensional position (on the sky) of star i. N_random is the total number of random points, set to equal N_star, and x_random,j is the position of random point j.