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## A..4 Bulge+Disk Profile

The surface brightness radial profiles of disk galaxies are usually modelled as the sum of a bulge and a disk component. The resulting bulge+disk profile can in general be written as

 (32)

where quantities subscripted with and refer to the bulge and disk component, respectively. The total brightness predicted by this profile is

 (33)

The relative importance of the bulge and disk component in terms of the brightness they contribute to the overall profile can be quantified by the bulge/disk ratio , which, from Equations 22 and 28, can be expressed in terms of the bulge and disk parameters as

 (34)

where

 (35)

is related to the more frequently used bulge/bulge+disk ratio by the

 (36)

The relation between the radii and and the effective radius of the bulge+disk profile , can be determined using Equations 18 and 19, whose combination yields

 (37)

which, after trivial modifications, becomes

 (38)

When values for the ratios and are assumed, Equation 38 can be solved numerically to obtain the corresponding values of the ratios and . Numerical values of the latter two ratios are given in Table 7 for some values of the former two.

 0.333 0.444 0.555 0.666 0.777 0.3 0.37776903 0.39659331 0.41501285 0.43296121 0.45038387 0.4 0.47738606 0.49489417 0.51165418 0.52767183 0.54296003 0.5 0.57169562 0.58717625 0.60178337 0.61556793 0.62857967 0.6 0.66216968 0.67514232 0.68725805 0.69858888 0.70920016 0.7 0.74977666 0.75987986 0.76924321 0.77794075 0.78603748
 0.333 0.444 0.555 0.666 0.777 0.3 1.2592301 1.3219777 1.3833761 1.4432040 1.5012795 0.4 1.1934651 1.2372354 1.2791355 1.3191796 1.3574001 0.5 1.1433912 1.1743525 1.2035667 1.2311359 1.2571593 0.6 1.1036161 1.1252372 1.1454301 1.1643148 1.1820002 0.7 1.0711095 1.0855427 1.0989189 1.1113440 1.1229107

Now, the general bulge+disk profile can be written as

 (39)

In much the same way as Equation 38, Equation 39 can be numerically solved with respect to the ratios and if values of the ratios and are assumed. Numerical values of the two former ratios are given in Table 8 for the same values of the latter two as in Table 7.

 0.333 0.444 0.555 0.666 0.777 0.3 1.5059000 1.6695642 1.7862309 1.8669555 1.9202392 0.4 0.90250402 1.0169289 1.1059437 1.1750876 1.2286100 0.5 0.61171687 0.69840479 0.76930751 0.82759966 0.87573973 0.6 0.44733235 0.51643686 0.57490397 0.62470279 0.66737834 0.7 0.34431760 0.40137789 0.45089086 0.49413744 0.53213802
 0.333 0.444 0.555 0.666 0.777 0.3 0.57903215 0.51357003 0.45788132 0.41020644 0.36917465 0.4 0.61692617 0.55611501 0.50399454 0.45900376 0.41992150 0.5 0.65336297 0.59676211 0.54778829 0.50511037 0.46768035 0.6 0.68801330 0.63543886 0.58948209 0.54903740 0.51322600 0.7 0.72080756 0.67220776 0.62927470 0.59111205 0.55699899

Tables 7 and 8, combined with Equation 39, allows to write the appropriate bulge+disk profile for different values of and . The total brightness predicted by the law can then be written in terms of the bulge+disk quantities and as

 (40)

and, proceeding like for the pure bulge and disk profiles, can be written as

 (41)

Values of are given in Table 9.

 0.333 0.444 0.555 0.666 0.777 0.3 15.841388 16.675985 17.443019 18.140889 18.770256 0.4 15.161182 15.816591 16.415234 16.959955 17.454163 0.5 14.737528 15.291208 15.795789 16.255446 16.674198 0.6 14.458169 14.948771 15.395577 15.803140 16.175547 0.7 14.268144 14.717733 15.127242 15.501275 15.843825

Next: Bibliography Up: A. Galaxy Surface Brightness Previous: A..3 Disk Profile
Mattia Vaccari 2002-01-31