We study the bivariate size-luminosity distribution of Lyman Break Galaxies ( LBGs ) selected at redshifts around 4 and 5 in GOODS and the HUDF fields .
We model the size-luminosity distribution as a combination of log-normal distribution ( in size ) and Schechter function ( in luminosity ) , therefore it enables a more detailed study of the selection effects .
We perform extensive simulations to quantify the dropout-selection completenesses and measurement biases and uncertainties in two-dimensional size and magnitude bins , and transform the theoretical size-luminosity distribution to the expected distribution for the observed data .
Using maximum-likelihood estimator ( MLE ) , we find that the Schechter function parameters for B _ { 435 } -dropouts are \alpha = -1.68 ^ { +0.068 } _ { -0.095 } , M ^ { * } = -20.60 ^ { +0.13 } _ { -0.17 } , and \phi ^ { * } = 1.79 ^ { +0.32 } _ { -0.52 } \times 10 ^ { -3 } \mathrm { Mpc } ^ { -3 } .
The log-normal size distribution is characterized by the peak R _ { 0 } = 1.34 ^ { +0.099 } _ { -0.108 } kpc at M _ { 1500 } = -21 mag , width \sigma _ { \ln \mathrm { R } } = 0.83 ^ { +0.046 } _ { -0.044 } , and the slope of the size-luminosity ( RL ) relation \beta = 0.22 ^ { +0.058 } _ { -0.056 } .
Similarly , for V _ { 606 } -dropouts we find \alpha = -1.74 ^ { +0.15 } _ { -0.20 } , M ^ { * } = -20.53 ^ { +0.24 } _ { -0.27 } , \phi ^ { * } = 1.55 ^ { +0.62 } _ { -0.77 } \times 10 ^ { -3 } \mathrm { Mpc } ^ { -3 } , R _ { 0 } = 1.19 ^ { +0.21 } _ { -0.16 } kpc , \sigma _ { \ln \mathrm { R } } = 0.90 ^ { +0.15 } _ { -0.065 } , and \beta = 0.25 ^ { +0.15 } _ { -0.14 } .
The Schechter function parameters are consistent with the values in the literature , while the size distributions are wider than expected from the angular momentum distribution of the underlying dark matter halos .
The slope of the size-luminosity ( RL ) relation is similar to those found for local disk galaxies , but considerably shallower than local early-type galaxies .