The major axis and ellipse-fit intensity profiles of spiral galaxies larger than 0.3 in the Hubble Space Telescope Ultra Deep Field (UDF) are generally exponential, whereas the major axis profiles in irregular disk galaxies, called clump-clusters in our previous studies, are clearly not. Here we show that the deprojected positions of star-forming clumps in both galaxy types are exponential, as are the deprojected luminosity profiles of the total emissions from these clumps. These exponentials are the same for both types when normalized to the outer isophotal radii. The results imply that clumps form or accrete in exponential radial distributions, and when they disperse they form smooth exponential disks. The exponential scale lengths for UDF spirals average 1.5 kpc for a standard cosmology. This length is smaller than the average for local spirals by a factor of 2. Selection effects that may account for this size difference among spirals are discussed. Regardless of these effects, the mere existence of small UDF galaxies with grand-design spiral arms differs significantly from the situation in local fields, where equally small disks are usually dwarf Irregulars that rarely have spiral arms. Spiral arms require a disk mass comparable to the halo mass in the visible region -- something local spirals have but local dwarfs Irregulars do not. Our UDF result then implies that galaxy disks grow from the inside out, starting with a dense halo and dense disk that can form spiral arms, and then adding lower density halo and disk material over time. Bars that form early in such small, dense, gas-rich disks should disperse more quickly than bars that form later in fully developed disks.
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