The Astrophysical Journal, 203:39-51, 1976 January 1 © 1976. the American Astronomical Society

^-CORRECTIONS FOR GALAXIES OF DIFFERENT MORPHOLOGICAL TYPES W. Pence Department of Astronomy, University of Texas, Austin Received 1975 May 6 ABSTRACT Mean energy distributions from 1500 to 8000 Â for different types of galaxies have been derived from satellite ultraviolet filter photometry and from Earth-based spectrum scans. X-corrections for the standard U, B, V filters have been calculated from these energy distributions. The K- corrections are used to calculate the mean redshift, mean observed color, and color distribution of galaxies as a function of apparent magnitude and type in a Euclidean universe assuming no galaxy evolution and no intergalactic extinction. Subject headings: galaxies: photometry — galaxies: redshifts — galaxies: stellar content

I. INTRODUCTION with 0.24 according to van den Bergh (1967) and 0.25 according to Holmberg (1958, 1974). The AB for each The X-correction is the magnitude difference be- galaxy is given in Table 1. The extinction at all other tween a redshifted and non-redshifted spectral energy distribution when observed through a fixed spectral wavelengths was determined from an interstellar interval. The ^-corrections for galaxies of different extinction curve provided by Bless (private communication) based on OAO-2 photometry of O and B stars. morphological types are necessary to interpret the magnitude-redshift relation m(z) and the luminosity Each galaxy energy curve was corrected to zero function N(m) of galaxies. The X-correction as defined galactic extinction before the mean curve for each by Oke and Sandage (1968) has been used in this paper : galactic type was determined. When the K-corrections were calculated from these energy curves, the U, B, V Kiz) = 2.5 log (1 + z) filter functions were multiplied by the galactic trans- mission function scaled to produce 0.23 mag absorp- tion in the B band. + 2.5 log JJ” F{X)Si{X)dX TABLE 1 OBSERVED GALAXIES Wavelength Interval 1500-3500 Â 3500-5500 A 5500-8000 A (1) Galaxy Observer Observer Observer Type NGC t NGC t AB NGC t Ab where F(X) is the energy flux density in the reference 0A0-2 Wells (1972) Oke & Sandage(1968) frame of the galaxy and S^X) is the sensitivity function 4552 -5 0.24 467 -2 0.26 3379 -4 0.28 of detector /. 5866 -1 0.29 4874 -2 0.23 The F(A) used in the integrations should be corrected for the effects of interstellar and intergalactic extinc- 224 (nucleus) 0.49 6703 -2 0.62 tion and galaxy evolution. In this paper, intergalactic 3C-295 -4 0.27 7785 -5 0.28 extinction and galaxy evolution have been ignored, OAO-2 Wells (1972) Oke & Sandage(1968) since there are no reliable data on the amount of 4736 1357 2 0.33 3379 -4 0.28 correction to apply. Galactic extinction was taken into 7177 3 0.48 account although its amount is still in doubt. Because of this uncertainty, two sets of X-corrections are pre- Interpolation Wells (1972) Wells (1972) sented here. Sandage (1973) and Knapp (1975) claim Between Sab & ! 470 4 0.26 Extrapolation that there is essentially zero extinction near the north 1659 4 0.51 pole. Therefore, for the first set of K-corrections, the OAO-2 Interpolation Interpolation observed galaxy energy distributions were not cor- Between Sbc and rected for interstellar extinction. For the second set of 598 6 0.37 Between Sbc and Sdm-Im K-corrections, AB for each galaxy was determined from 5457 6 0.28 Sdm-Im a formula provided by de Vaucouleurs using values of OAO-2 Wells (1972) Wells (1972) Ab = 0.23 and 0.25 mag at the north and south 4449 10 145 8 0.25 Extrapolation galactic poles, derived from a revision of the de 1140 10 0.32 Vaucouleurs and Malik (1969) study, and in agreement

© American Astronomical Society • Provided by the NASA Astrophysics Data System O'!cu 00 00 CMo 40 PENCE Vol. 203 ^0 ft II. PREVIOUS WORK filter. However, to calculate the m(z) and N(m) rela- r- tions at cosmologically significant look-back times, In two previous papers, Schild and Oke (1971) and one needs ^-corrections out to much larger z. There- Whitford (1971) have presented independent calcula- fore in this paper the OAO-2 ultraviolet observations tions of ^-corrections for giant elliptical galaxies out have been added to those of Wells in order to calculate to z = 0.28 with the B filter. Oke (1971) extends these ^-corrections out to z = 1.40 through the B filter. results out to z = 0.52 in the B by using the energy distribution of 3C 295 (z = 0.46) corrected back to the III. METHOD OF CALCULATION rest frame of the galaxy. This gives a usable energy curve down to 2700 Â. The new ^-corrections pre- Table 1 summarizes the observations that were used sented in this paper are in good agreement with the to derive an energy curve for each galaxy type. The Whitford and the Oke and Sandage data in the range energy curves in the wavelength range 1500-3500 Â, in common. with one exception, were determined solely by the Wells (1972) was the first to obtain a standard set of OAO-2 photoelectric photometry (Code et al 1972) integrated galaxy energy-curves for different galaxy of the tabulated galaxies. The OAO-2 was equipped types covering the whole range of the Hubble sequence. with a set of 10 medium-band interference filters with He observed 10 galaxies with the scanning spectro- mean wavelengths of 1430, 1554, 1679, 1913, 2386, photometer attached to the McDonald Observatory 2462, 2945, 2985, 3317, and 4252 Â for a constant 2 m or 2.7 m telescopes. He derived average energy energy curve with half-power bandwidths of 240, 240, curves in the range 3500-5500 Á for galaxies in four 260, 260, 330, 380, 440, 420, 540, and 840 Â, respec- groups of types E-S0, Sa-Sb, Sbc, and Sdm-Im, tively. Ideally, one should use the isophotal wave- corresponding to <¿> = —3, 2, 4, and 9 in the nu- length, Ai? of each filter. However, the published OAO merical sequence introduced by de Vaucouleurs (1962, data do not give enough information to derive 1974). With these energy curves Wells was able to Most of the galaxies were observed only through seven calculate the ^-corrections out to z = 0.28 with the B or eight of these filters. It should be noted that these TABLE Z E - SO AVERAGE ENERGY DISTRIBUTION Observed Corrected for Galactic Extinction X Flux X Flux X Flux Flux Flux X Flux X Flux A. Flux 1900 38 3840 393 4580 1225 5320 1386 1900 40 3840 401 4580 1201 5320 1306 2000 44 3860 403 4600 1243 5340 1402 2000 45 3860 411 4600 1217 5340 1320 2100 51 3880 415 4620 1248 536o 1397 ¿100 51 3880 422 4620 1221 5360 1314 2200 59 3900 423 4640 1242 5380 1375 2200 59 3900 430 4640 1213 5380 1292 2300 68 3920 432 4660 1232 5400 1352 2300 68 3920 439 4660 1201 5400 1269 2400 79 3940 456 4680 1229 5420 1342 2400 80 3940 463 4680 1197 5420 1259 2500 92 3960 510 4700 1242 5440 1347 2500 95 3960 517 4700 1208 5440 1262 2600 108 3980 592 4720 1270 54bo 1378 2600 113 3980 599 4720 1234 5460 1290 2700 127 4000 683 4740 1305 5480 1396 2700 137 4000 691 4740 1267 5480 1305 2800 150 4020 756 4760 1333 5500 14Q5 2800 165 4020 764 4760 1293 5500 1313 2900 17B 4040 797 4780 1343 5600 1430 2900 202 4040 805 4780 1301 5600 1331 3000 215 4060 807 4800 1332 5700 1460 3000 246 4060 814 4800 1289 5700 1352 3100 255 4080 800 4820 1306 5800 1420 3100 305 4080 806 4820 1263 5800 1309 3200 291 4100 786 4840 1276 5900 1440 3200 348 4100 791 4840 1232 5900 1324 3300 345 4120 774 4860 1257 6000 1510 3300 390 4120 778 4860 1213 6000 1381 3400 370 4140 770 4880 1261 6100 1520 3400 400 4140 774 4880 1215 6100 1384 3420 380 4160 776 4900 1286 6200 1500 3420 402 4160 779 4900 1238 6200 1362 3440 384 4180 789 4920 1316 6300 1400 3440 404 4180 791 4920 1265 6300 1266 3460 38 7 *►200 806 4940 1330 6400 1420 3460 406 4200 808 4940 1277 6400 1282 3480 390 4220 820 4960 1319 6500 1480 3480 408 4220 8?1 4960 1265 6500 1332 3500 392 4240 825 498o 1289 6600 1520 3500 408 4240 825 4980 1235 6600 1363 3520 396 4260 822 5000 1258 6700 1490 3520 412 4260 822 5000 1204 6700 1329 3540 393 4280 823 5020 1244 6800 1485 3540 408 4280 822 5020 1189 6800 1322 3560 384 4300 841 5040 1250 6900 1480 3560 398 4300 839 5040 1194 6900 1314 3580 379 4320 876 5060 1265 7000 1480 3580 392 4320 873 5060 1207 7000 1309 3600 38 7 4340 919 5080 1273 7100 1475 3600 400 4340 915 5080 1213 7100 1301 3620 403 4360 956 5100 1265 7200 1340 3620 416 4360 951 5100 1204 7200 1177 3640 424 *►380 983 5120 1242 7300 1480 3640 437 4380 976 5120 1181 7300 1296 3660 442 4400 1007 5140 1211 7400 1500 3660 455 4400 999 5140 1150 7400 1309 3680 457 4420 1036 5160 1189 7500 1500 3680 470 4420 1027 5160 1129 7500 1305 3700 465 4440 1072 5180 1185 7600 1440 3700 478 4440 1061 5180 1124 7600 1246 3720 464 4460 1111 5200 1203 7700 1430 3720 476 4460 1098 5200 1140 7700 1235 3740 455 4480 1140 5220 1236 7800 1450 3740 466 4480 1126 5220 1170 7800 1250 3760 441 4500 1157 5240 1271 7900 1460 3760 452 4500 1141 5240 1202 7900 1257 3780 423 4520 1167 5260 1300 8000 1420 3780 433 4520 1149 5260 1229 8000 1221 3800 405 4540 1182 5280 1327 3800 414 4540 1162 5280 1253 3820 394 4560 1203 5300 1357 3820 402 4560 1181 5300 1280

© American Astronomical Society • Provided by the NASA Astrophysics Data System O'!cu 00 00 CMo No. 1, 1976 ^-CORRECTIONS 41 ^0 ft observations are somewhat uncertain and are based In the range 3500-5500 Â the energy curves given by only on a preliminary analysis of the OAO data. It is r- Wells (1972) were used. These are tabulated at 20 Â possible that they may be significantly revised in the intervals and have 50 Â resolution. future. The OAO observations of E-SO galaxies are In the range 5500-7500 Â the linear extrapolations especially unreliable because of the very low ultraviolet of magnitudes per unit frequency versus frequency counts produced by galaxies of this type. The K- given by Wells were adopted for types Sbc and Sdm. corrections for E-SO galaxies calculated from the For type E-SO, observations of the giant elliptical OAO data alone are excessively large and violate any NGC 3379 by Oke and Sandage (1968) were used for sense of continuity between E-SO galaxies and Sab À > 5500 Â. This curve was also used for the Sab galaxies. Therefore it was decided to use all possible galaxy energy curve which is dominated by the stars in data to synthesize the ultraviolet energy curve for the the spheroidal component since the shape of the curve E-SO type galaxy. The adopted energy curve was in this wavelength region does not vary much with constructed by combining Oke’s 3C 295 observation galaxy type (Johnson 1966). Also this region is not (Oke 1971) between 2700 and 3500 Â (ignoring the critical for calculation of ^-corrections for the U, B, V emission lines) and the OAO observations of late F filters. It only slightly affects Kv for small z. dwarf stars in the range 2000-2700 Â. The latter were In two cases it was necessary to interpolate between suggested by B. Tinsley (private communication) as adjacent types. There are no OAO-2 observations of being a theoretically plausible representation of the an Sbc (t = 4) galaxy, so a linear interpolation of ultraviolet spectral energy distribution of a pre- log F(X) versus t was performed between the Sab dominantly old stellar population. The published (t = 2) and Scd {t = 6) curves. Therefore, log F¿X) = preliminary reductions of the OAO observations 0.5 log F2(A) + 0.5 log F6(A) for A < 3500 À. Simi- suggest that the flux from E-SO galaxies increases larly, Wells did not observe any Scd galaxies but had rapidly below 2000 Â. However, the data are much too data for Sbc (t = 4) and Sdm-Im (t = 9), so log F6(A) uncertain to use here; therefore the E-SO energy curve = 0.4 log F9(A) + 0.6 log F4(A) for A > 3500 Â. was not extended below 2000 Â. To obtain the final energy curve for each galaxy TABLE 3 Sab AVERAGE ENERGY DISTRIBUTION Observed Corrected for Galactic Extinction X Flux X Flux ^ Flux Flux Flux ^ Flux ^ Flux Flux 1400 770 3760 552 4520 1092 5280 1132 1400 1015 3760 568 4520 1072 5280 1056 1500 580 3780 568 4540 1097 5300 1156 1500 772 3780 584 4540 10 75 5300 1077 1600 450 3800 562 4560 1101 5320 1179 1600 610 3800 577 4560 1077 5320 1097 1700 383 3820 544 4580 1109 5340 1185 1700 498 3820 558 4580 1083 5340 1101 1800 333 3840 536 4600 1113 5360 1176 1800 424 3840 549 4600 1085 5360 1092 1900 296 3860 552 4620 1113 5380 1166 1900 378 3860 565 4620 1083 5380 1081 2000 270 3880 579 4640 1106 5400 1156 2000 342 3880 591 4640 1074 5400. 1070 2100 253 3900 589 4660 1097 5420 1148 2100 320 3900 601 4660 1064 5420 1062 2200 240 3920 574 4680 1093 5440 1152 2200 297 3920 585 4680 1058 5440 1064 2300 232 394o 566 4700 1102 5460 1168 2300 282 3940 576 4700 1065 5460 1077 2400 228 3960 604 4720 1124 5480 1185 2400 275 3960 614 4720 1085 5480 1092 2500 230 3980 688 4740 1145 5500 1197 2500 275 3980 698 4740 1104 5500 1102 2600 236 4000 774 4760 1152 5600 1220 2600 282 4000 785 4760 1110 5600 1118 2700 253 4020 823 478o 1148 5700 1240 2700 300 4020 834 4780 1105 5700 1129 2800 274 4040 832 4800 1137 5800 1230 2800 319 4040 842 4800 1093 5800 1114 2900 300 4060 820 4820 1124 5900 1230 2900 338 4060 828 4820 1079 5900 1110 3000 327 4080 808 4840 1114 6000 1280 3000 352 4080 615 4840 1068 6000 1148 3100 347 4100 802 4860 1109 6100 1310 3100 373 4100 808 4860 1061 6100 1169 3200 368 4120 806 4880 1108 6200 1290 3200 393 4120 611 4880 1059 6200 1147 3300 392 4140 818 4900 1107 6300 1190 3300 410 4140 823 4900 1057 6300 1053 3400 443 4160 835 4920 1108 6400 1200 3400 469 4160 839 4920 1056 6400 1059 3420 466 4180 851 4940 1115 6500 1260 3420 492 4180 854 4940 1061 6500 1108 3440 487 4200 857 4960 1125 6600 1300 3440 514 4200 859 4960 1069 6600 1138 3460 510 4220 854 4980 1131 6700 1260 3460 537 4220 855 4980 1074 6700 1096 3480 510 4240 849 5000 1130 6800 1260 3480 536 4240 849 5000 1071 6800 1093 3500 488 4260 838 5020 1123 6900 1260 3500 512 4260 838 5020 1063 6900 1090 3520 470 428o 825 504o 1112 7000 1260 3520 493 4280 824 5040 1051 7000 1085 3540 458 430 n 823 5060 1103 7100 1250 3540 479 4300 821 5060 1041 7100 1073 3560 447 432o 846 5080 1107 7200 1130 3560 467 4320 843 5080 1044 7200 965 3580 439 4340 884 5100 1117 7300 1250 3580 458 4340 879 5100 1052 7300 1063 3600 444 4360 918 5120 1111 7400 1275 3600 462 4360 912 5120 1045 7400 1080 3620 458 438o 942 5140 1083 7500 1275 3620 476 4330 934 5140 1017 7500 1076 3640 472 4400 960 5160 1051 7600 1230 3640 490 4400 951 5160 986 7600 1031 3660 487 4420 978 5180 1038 7700 1210 3660 505 4420 967 5180 973 7700 1012 3680 501 4440 999 5200 1054 7800 1230 3680 518 4440 987 5200 987 7800 1027 3700 511 4460 1026 5220 1083 7900 1250 3700 528 4460 1012 5220 1013 7900 1042 3720 518 4480 1056 5240 1106 8000 1210 3720 535 4480 1040 5240 1034 8000 1006 3740 531 4500 1080 5260 1119 3740 547 4500 1062 5260 1045

© American Astronomical Society • Provided by the NASA Astrophysics Data System O'!cu 00 00 CMo 42 PENCE ^0 ft type, the observations in the different wavelength was no apparent advantage to their use. The B and intervals were normalized at a common wavelength r- V filters produced identical results, but the Azusienis (A = 4250 Â in the blue and A = 5500 Â in the red). and Straizys U filter produced ^-corrections which The adopted energy distributions are tabulated in typically differed by 15 percent from those calculated Tables 2 through 6 (in arbitrary flux units). The left with Matthews and Sandage’s U filter. half of each table gives the observed energy distribu- From an analysis of 16 stars covering the range of tion, i.e., with no correction for interstellar extinction. spectral types O to M, Wells (1972) found that scale The right half gives the corrected energy distribution factors of 0.95 and 1.07 must be applied to the {7 — 1? assuming AB = 0.23 and 0.25 at the north and south and B — V color indices calculated with the one air poles. Figures 1 and 2 illustrate the effect of the ex- mass filters of Matthews and Sandage to bring them tinction correction on the E-SO and Scd energy into agreement with the observed colors. All the distributions. Note that for E-SO galaxies the energy K(U — B)- and K(B — F)-corrections in this paper distributions below 2700 Â are essentially identical for have been multiplied by these factors. the two cases because the nearby F stars that were used have no significant extinction correction. It should be added that the observed galaxies have IV. COMPUTED ^-CORRECTIONS U — B and B — V color indices close to the mean for A computer program was written by the author to their morphological type (de Vaucouleurs and de compute the ^-corrections from the above input filter Vaucouleurs 1972). Thus the ^-corrections presented and flux functions. The program was checked by here should be typical of an average galaxy of each reproducing Wells’s ^-corrections using his input type. data. The calculated ^-corrections for each group of The ¿7,2?, V filter functions for one air mass given by galaxy types and each filter are tabulated in Tables Matthews and Sandage (1963) were used for all cal- 7-9. The K{B) function is shown in Figure 3. The right culations in this paper. Similar functions given by half of each table shows the effect of correcting the Azusienis and Straizys (1966) were tried, but there energy curve of each galaxy for galactic extinction as TABLE 4 Sbc AVERAGE ENERGY DISTRIBUTION Observed Corrected for Galactic Extinction X Flux \ Flux \ Flux X Flux X Flux X Flux X Flux X Flux

1400 1541 3760 719 4520 1074 5280 975 1400 2091 3760 742 4520 1052 5280 903 1500 1175 3780 707 4540 1071 5300 969 1500 1620 3780 729 4540 1047 5300 896 1600 917 3800 711 4560 1065 5320 960 1600 1275 3800 732 4560 1039 5320 887 1700 763 3820 733 4580 1061 5340 949 1700 1036 3820 754 4580 1034 5340 876 1800 641 3840 764 4600 1062 5360 944 1800 857 3840 784 4600 1033 5360 870 1900 559 3860 792 4620 1069 5380 952 1900 738 3860 812 4620 1038 5380 876 2000 504 3880 808 4640 1073 5400 974 2000 659 3880 827 4640 1039 5400 895 2100 470 3900 809 4660 1066 5420 1007 2100 608 3900 8?7 4660 1031 5420 924 2200 441 3920 799 4680 1055 5440 1038 2200 568 3920 816 4680 1018 5440 951 2300 424 3940 791 4700 1051 546q 1051 2300 535 3940 806 4700 1013 5460 962 2400 413 3960 802 4720 1060 5480 996 2400 512 3960 817 4720 1020 5480 910 2500 410 3980 843 4740 1072 5500 968 2500 499 3990 857 4740 1030 5500 884 2600 413 4000 904 4760 1076 5600 977 2600 493 4000 918 4760 1033 5600 887 2700 423 4020 956 4780 1067 5700 955 2700 502 4020 969 4780 1023 5700 862 2800 439 4040 977 4800 1054 5800 946 2800 513 4040 989 4800 1009 5800 849 2900 459 4060 965 4820 1045 5900 934 2900 524 4060 976 4820 999 5900 834 3000 479 4080 937 4840 1042 6000 923 3000 530 4080 946 4840 994 6000 819 3100 496 4100 918 4860 1041 6100 912 3100 544 4100 926 4860 992 6100 805 3200 515 4120 921 4880 1036 6200 901 3200 558 4120 928 4880 986 6200 791 3300 534 4140 941 4900 1028 6300 891 3300 570 4140 947 4900 977 6300 779 3400 543 4160 965 4920 1023 6400 880 3400 583 4160 970 4920 970 6400 767 3420 553 4180 980 4940 1026 6500 870 3420 588 4180 984 4940 972 6500 755 3440 558 4200 989 4960 1036 6600 860 3440 592 4200 992 4960 980 6600 743 3460 563 4220 991 4980 1046 6700 850 3460 596 4220 993 4980 988 6700 729 3480 568 4240 985 5000 1045 6800 940 3480 600 4240 986 5000 985 6800 719 3500 574 4260 972 5020 1031 6900 830 3500 606 4260 971 5020 971 6900 708 3520 600 428o 955 5040 1013 7000 821 3520 632 4280 953 50^0 952 7000 696 3540 616 4300 941 5060 1004 7100 812 3540 648 4300 938 5060 942 7100 687 3560 612 4320 935 5080 1005 7200 802 3560 642 4320 931 5080 942 7200 674 3580 596 4340 941 5100 1008 7300 793 3580 624 4340 936 5100 943 7300 664 3600 582 436o 957 5120 1006 7400 784 3600 608 4360 950 5120 940 7400 653 3620 581 4380 977 5140 997 7500 776 3620 606 4380 968 5140 931 7500 644 3640 593 4400 998 5160 986 7600 767 3640 618 4400 988 5160 919 7600 632 3660 618 4420 1018 5180 980 7700 759 3660 643 4420 1006 5180 913 7700 624 3680 655 4440 1034 5200 985 7800 751 3680 680 4440 1020 5200 916 7800 616 3700 698 4460 1049 5220 992 7900 743 3700 724 4460 1033 5220 922 7900 608 3720 728 4480 1062 5240 993 8000 735 3720 754 4480 1044 5240 922 8000 600 3740 732 4500 1071 5260 985 3740 757 4500 1051 5260 913

TABLE 5 h) ft Sed AVERAGE ENERGY DISTRIBUTION Observed Corrected for Galactic Extinction Flux A. Flux X Flux Flux ^ Flux ^ Flux A. Flux A. Flux 1400 2720 3760 840 4520 1033 5280 916 1400 3800 376o 864 4520 1015 5280 857 1500 2100 3780 774 4540 1037 5300 918 1500 3000 3790 795 4540 1017 5300 858 1600 1650 3800 777 4560 1036 5320 912 1600 2350 3800 797 4560 1014 5320 851 1700 1340 3820 809 4580 1038 5340 900 1700 1900 3820 829 4580 1015 5340 839 1800 1090 3840 842 4600 1045 5360 972 1800 1530 3840 862 4600 1020 5360 812 1900 930 3860 873 4620 1038 5380 875 1900 1270 3860 892 4620 1011 5380 814 2000 830 3880 884 4640 1039 5400 909 2000 1120 3880 902 4640 1011 5400 845 2100 770 3900 883 4660 1044 5420 925 2100 1020 3900 900 4660 1014 5420 858 2200 715 3920 870 4680 1037 5440 939 2200 960 3920 886 4680 1005 5440 870 2300 685 3940 859 4700 1019 5460 969 2300 895 3940 873 4700 987 5460 897 2400 660 3960 864 4720 1021 5480 938 2400 840 3960 878 4720 987 5480 867 2500 645 3980 901 4740 1025 5500 996 2500 800 3980 914 4740 990 5500 828 2600 637 4000 961 4760 1021 5600 878 2600 762 4000 974 4760 985 5600 807 2700 62d 4020 1011 4780 1020 5700 978 2700 741 4020 1023 4780 983 5700 803 2800 620 4040 1023 4800 1011 5800 969 2800 728 4040 1034 4800 973 5800 791 2900 619 4060 1000 4820 1003 5900 956 2900 716 4060 1010 4820 964 5900 776 3000 620 4080 967 4840 1035 6000 842 3000 705 4080 975 4840 994 6000 759 3100 626 4100 943 4860 1074 6100 828 3100 701 4100 950 4860 1030 6100 742 3200 635 4120 939 4880 1053 6200 815 3200 700 4120 945 4880 1008 6200 728 3300 641 4140 965 4900 995 6300 803 3300 699 4140 970 4900 952 6300 714 3^00 656 4160 1000 4920 986 6400 790 3400 699 4150 1004 4920 942 6400 701 3420 659 4180 1012 4940 1028 6500 778 3420 700 4180 1015 4940 981 6500 688 3440 662 4200 1010 4960 1064 6600 766 3440 701 4200 1012 4960 1014 6600 675 3460 665 4220 1010 4980 1105 6700 754 3460 702 4220 1011 4980 1051 6700 660 3480 668 4240 1004 5000 1173 6800 742 3480 703 4240 1004 5000 1115 6800 648 3500 672 4260 987 5020 1156 6900 730 3500 7q4 4260 987 5020 1097 6900 635 3520 669 4280 968 5040 1025 7000 720 3520 700 4290 967 5040 971 7000 624 3540 672 4300 956 5060 943 7100 7q9 3540 702 4300 953 5060 892 7100 613 3560 683 4320 954 5080 940 7200 698 3560 712 4320 950 5080 888 7200 600 3580 669 4340 972 5100 939 7300 688 3580 696 4340 967 5100 886 7300 589 3600 644 4360 985 5120 945 7400 678 3600 669 4360 979 5120 891 7400 578 3620 640 4380 991 5140 932 7500 669 3620 664 4380 983 5140 878 7500 569 3640 653 4400 1003 5160 917 7600 659 3640 677 4400 994 5160 863 7600 557 3660 672 4420 1019 5180 920 7700 650 3660 695 4420 1008 5180 865 7700 548 3680 697 4440 1017 5200 919 7800 641 3680 720 4440 1005 5200 863 7800 539 3700 816 4460 1022 5220 923 7900 632 3700 842 4460 1008 5220 866 7900 531 3720 949 4480 1043 5240 914 8000 623 3720 978 4480 1028 5240 857 8000 522 3740 942 4500 1043 5260 911 3740 970 4500 1026 5260 853

Fig. 1.—Mean E-S0 energy distributions with [^(pole) = 0.23] and without correction for galactic extinction

TABLE 6 h) ft Sdm - Im AVERAGE ENERGY DISTRIBUTION Observed Corrected for Galactic Extinction X Flux X Flux X Flux Flux X Flux X Flux A. Flux A. Flux 1400 5200 3760 1062 4520 975 5280 834 1400 6830 3760 1084 4520 962 5280 794 1500 3940 3780 888 4540 989 5300 346 1500 4980 3780 905 4540 975 5300 805 1600 2900 3800 889 4560 993 5320 844 1600 3480 3800 906 4560 978 5320 803 1700 2200 3820 937 4580 1005 5340 831 1700 2750 3820 954 4580 989 5340 790 1800 1800 3840 973 4600 1020 5300 775 1800 2350 3840 989 4600 1002 5360 736 1900 1570 3860 1010 4620 994 5380 770 1900 2050 3860 1026 4620 975 5380 730 2000 1380 3880 1012 4640 991 5400 320 ¿000 1770 3880 1027 4640 971 5400 777 2100 1210 3900 1007 4660 1013 5420 815 2100 1570 3900 1021 4660 992 5420 772 2200 1110 3920 989 4680 1010 5440 80 7 2200 1420 3920 1002 4680 988 5440 764 2300 1040 3940 973 4700 973 5460 958 2300 1300 3940 985 4700 950 5460 811 2400 990 3960 966 4720 966 5480 857 2400 1220 3960 977 4720 943 5480 810 2500 960 3980 995 4740 958 5500 799 2500 1170 3980 1006 4740 934 5500 754 2600 940 4000 1054 4760 944 5600 748 2600 1102 4000 1064 4760 920 5600 704 2700 930 4020 1099 4780 954 5700 775 2700 1070 4020 1109 4780 929 5700 726 2800 925 4040 1097 4800 950 5800 765 2800 1030 4040 1106 4800 924 5800 714 2900 920 4060 1055 4820 942 5900 750 2900 1000 4060 1063 4820 915 5900 698 3000 910 4080 1013 4840 1025 6000 733 3000 970 4080 1019 4840 995 6000 680 3100 900 4100 983 4860 1125 6100 716 3100 940 4100 988 4860 1091 6100 661 3200 870 4120 966 4880 1078 6200 701 3200 900 4120 971 4880 1045 6200 646 3300 840 4140 1003 4900 948 6300 686 3300 860 4140 1007 4900 918 6300 630 3400 830 4160 1054 4920 932 6400 671 3400 849 4160 1057 4920 901 6400 615 3420 833 4180 1061 4940 1030 6500 657 3420 853 4180 1064 4940 995 6500 601 3440 836 4200 1042 4960 1107 6600 643 3440 858 4200 1044 4960 1069 6600 586 3460 839 4220 1039 4980 1200 6700 629 3460 864 4220 1040 4980 1157 6700 571 3480 842 4240 1034 5000 1394 6800 616 3480 876 4240 1034 5000 1343 6800 558 3500 852 4260 1010 5020 1372 6900 603 3500 881 4260 1010 5020 1321 6900 545 3520 787 4280 989 5040 1043 7000 591 3520 813 4280 988 5040 1003 7000 533 3540 766 4300 978 5060 858 7100 579 3540 791 4300 976 5060 824 7100 521 3560 806 4320 983 5080 851 7200 567 3560 831 4320 980 5080 817 7200 508 3580 796 4340 1020 5100 845 7300 556 3580 820 4340 1016 5100 810 7300 497 3600 750 4360 1028 5120 861 7400 545 3600 771 4360 1023 5120 825 7400 485 3620 739 4380 1013 5140 843 7500 535 3620 759 4380 1007 5140 80 7 7500 475 3640 756 440 0 1010 5160 822 7600 524 3640 776 4400 1003 5160 786 7600 464 3660 762 4420 1020 5160 836' 7700 515 3660 781 4420 1012 5180 799 7700 455 3680 765 4440 991 5200 829 7800 505 3680 783 4440 982 5200 792 7800 445 3700 1032 4460 983 5220 828 7900 496 3700 1056 4460 973 5220 790 7900 437 3720 1413 4480 1014 5240 808 8000 487 3720 1444 4480 1003 5240 771 8000 428 3740 1374 4500 1003 5260 810 3740 1403 4500 991 5260 772

Fig. 2.—Mean Sed energy distributions with |X4ß(pole) = 0.23] and without correction for galactic extinction

Ky for Arpóle) = 0.00 for Aß(pole) = 0.23 Sdm- Sdm- E-S0 Sab Sbc Scd Im 'E-S0 Sab Sbc Scd Im .02 .065 .072 .074 .063 .046 .053 .068 .068 .054 .044 .04 .137 .140 .141 .122 .096 .111 .130 .129 .104 .091 .08 .291 .261 .249 .200 . 134 .235 .242 .223 .163 .124 .12 .468 .396 .341 .260 .151 .382 .368 .300 .202 .133 .16 .671 .530 .423 .311 .165 .565 .487 .367 .234 .132 .20 .885 .649 .502 .355 .182 .779 .587 .428 .262 .130 .24 1.096 .761 .577 .393 .204 1.009 .678 .484 .284 .130 .28 1.301 .869 . 647 .424 .228 1.238 • 7 64 .535 .300 .131 .32 1 .494 .969 .709 .450 .252 1.452 .843 .580 .311 .131 .36 1.671 1.055 .763 .472 .272 1.651 .915 .619 .317 .129 .40 1.837 1.124 .806 .489 .287 1.834 .975 .650 .318 .124 .44 1.991 1.177 .839 .503 .297 2.004 1.022 .671 .314 .116 .48 2.135 1.215 .862 .513 .301 2.161 1.054 .682 .306 .104 .52 1.240 .878 .520 .299 1.072 .686 .296 .089 .60 1.264 .889 .518 .274 1.081 .677 .270 • 046 .68 1.264 .879 .498 .228 1.065 .651 .235 ■•015 .76 1.245 .850 .460 .164 1.035 .612 .186 -.086 .84 1.212 .805 .402 .089 .995 .557 .115 .161 .92 1.166 • 744 .326 .004 .943 .485 .024 .241 1.00 1.110 .671 .234 -.094 .878 .400 .080 .326

discussed previously. This is shown on the figure with and lower envelopes of the observation points used to dotted lines. draw the adopted energy curve. The mean relative In order to get an indication of the size of error on difference {LKjK = |^max - Kmiri\l2(Ky) between the ^-corrections produced by uncertainties of the the ÆT-corrections produced with these upper and lower energy curves in the ultraviolet, two other energy curves and the adopted mean curve is shown in Table curves for the E-S0 type were used and the results 10. The energy curves for the other galaxy types have a were compared. The two curves represent the upper smaller uncertainty in the ultraviolet because the TABLE 8 K CORRECTION FUNCTIONS FOR THE B FILTER

Kß for AB(pole) =0.00 K for A pole) 0.23 Sdm- B B^ Sdm- E-S0 Sab Sbc Scd Im E-S0 Sab Sbc Scd Im .02 .092 .069 .042 .032 .017 .087 .063 .036 .027 .013 .04 .189 .143 .089 .068 .035 .179 .131 .076 .056 .026 .08 .389 .298 .195 .154 .087 .369 .275 .170 .131 .071 .12 .593 .458 .313 .251 .153 .565 .424 .277 .219 .130 .16 .793 .615 .438 .352 .221 .753 .572 .391 .309 .191 .20 .978 .767 .561 .452 .286 .924 .715 .504 .398 .252 .24 1.147 .909 .680 . 546 .346 1.074 .849 .612 .479 .308 .28 1.304 1.040 .790 .633 .397 1.211 .972 .712 .553 .355 .32 1.455 1.162 .892 .711 .440 1.344 1.086 .803 .616 .392 .36 1.599 1.274 .983 .777 .475 1.472 1.188 .882 .667 .419 .40 1.739 1.378 1.064 .833 .504 1.600 1.281 .951 .708 .438 .44 1.884 1.478 1.138 .880 .527 1.737 1.368 1.011 .740 .449 .48 2.037 1.575 1.205 .919 .547 1.884 1.451 1.064 .766 .455 .52 2.191 1.668 1.266 .950 • 564 2.037 1.531 1.112 .784 .456 .60 2.502 1.841 1.375 1.004 .597 2.359 1.676 1.195 .811 .455 .68 2.800 1.981 1.465 1.044 .625 2.684 1.793 1.259 .825 .450 .76 3.073 2.089 1.532 1.073 .645 2.986 1.881 1.303 .825 .440 .84 2.163 1.576 1.090 .652 1.940 1.327 .817 .421 .92 2.208 1.601 1.095 .643 1.971 1.334 .802 .392 1.00 2.229 1.608 1.090 .618 1.980 1.328 .780 .352 1.10 2.230 1.597 1.067 .569 1.969 1.301 .739 .290 1.20 2.210 1.565 1.023 .507 1.940 1.257 .680 .221 1.30 2.175 1.516 .959 .431 1.896 1.196 .600 .147 1.40 2.128 1.452 .876 .339 1.839 1.120 .507 .066

TABLE 9 K CORRECTION FUNCTIONS FOR THE V FILTER h) a Ky for Ag (pole) = 0.00 Ky for Ag (pole) 0.23 Sdm- Sdm- E-S0 Sab Sbc Scd Im E-S0 Sab Sbc Scd Im .0? .039 .036 .010 .002 -.028 .034 .030 .003 -.009 .027 • 04 .075 .069 .019 .005 -.043 • 064 .056 .005 -.017 .052 .08 .139 .128 .037 .003 -.066 .118 .103 .008 -.028 .084 .12 .206 .184 .057 .008 -.067 .174 .145 .015 -.029 .094 .16 .292 .252 .090 .032 -.056 .251 .202 .034 -.017 .091 .20 .406 .341 .135 .067 -.035 .355 .278 .067 .008 .079 .24 .538 .440 .189 .109 -.010 .478 .366 .108 .039 .061 .28 .692 .553 .254 .160 .018 .624 .470 .163 .080 .040 .32 .876 .686 .335 .222 .050 .800 .594 .234 .134 .014 .36 1.068 .829 .430 .292 .082 .985 .728 .319 .196 .012 .40 1.249 .972 .535 .377 .133 1.159 .864 .416 .273 .057 .44 1.413 1.108 • 646 .469 .199 1.318 .991 .518 .358 .118 .48 1.559 1.228 .756 .561 .268 1.455 1.104 .620 .442 . 182 .52 1.684 1.339 .860 .648 .330 1.568 1.208 .716 .519 .242 .60 1.904 1.542 1.041 .794 .425 1.750 1.400 .882 .642 .333 * 68 2.124 1.722 1.187 .903 .479 1.930 1.568 1.010 .727 .378 .76 2.368 1.890 1.308 .985 .509 2.151 1.721 1.112 .784 .394 .84 2.628 2.051 1.417 1.049 .537 2.406 1.857 1.198 .825 .399 .92 2.890 2.203 1.518 1.100 .567 2.679 1.982 1.274 .856 .402 1.00 3.143 2.341 1.606 1.140 .597 2.953 2.095 1.341 .877 .403 1.10 3.438 2.486 1.697 1.179 .631 3.277 2.217 1.408 .890 .400 1.20 3.704 2.591 1.764 1.208 .653 3.571 2.308 1.452 .890 .392 1.30 3.944 2.659 1.806 1.226 .661 3.833 2.363 1.474 .679 .375 1.40 2.698 1.828 1.233 .654 2.388 1.477 .862 • 346 1.50 2.714 1.834 1.228 .630 2.391 1.469 .843 .308 1.60 2.715 1.826 1.212 .589 2.381 1.450 .816 .258 1.70 2.703 1.806 1.185 .539 2.361 1.421 .778 .201 1 .80 2.681 1.775 1.145 .484 2.332 1.381 .726 .141 1.90 2.649 1.731 1.089 .424 2.296 1.329 .658 .081 2.00 2.610 1.678 1.022 .355 2.251 1.267 .579 .020 2.10 2.566 1.618 .946 .274 2.197 1.197 .494 .046 2.18 2.526,1.567 .882 .199 2.147 1.138 .425 .107

Fig. 3.—KB functions for galaxies of various types. Dotted curves were calculated using the energy distributions corrected for galactic extinction.

© American Astronomical Society • Provided by the NASA Astrophysics Data System O'!cu 00 00 CMo «-CORRECTIONS 47 ft TABLE 10 V. APPLICATIONS UD Relative Uncertainties of Kv, Kb, r- and K for E-SO Types In studies of general galaxy counts to different v limiting apparent magnitudes, one does not know the AKuIKu AKbIKb AKv/Kv value of z for any particular galaxy, and it is difficult or impossible to determine the galaxy type with any 0.1... 0.18 0.2... 0.00 0.01 precision since the images are usually small and faint. 0.3... 0.20 0.4... 0.06 0.06 The only known quantity is the apparent magnitude. 0.5... 0.17 0.6... 0.10 0.09 0.8... 0.11 0.09 Therefore one is interested in knowing the mean in- 0.09 correction for each galaxy type and for all galaxy types combined as a function of apparent magnitude. These quantities have been calculated below for a Euclidean universe with q = In these calculations emitted flux is larger. Therefore the uncertainty in the any change in absolute magnitude or color because of i^-corrections for the other galaxy types is smaller galactic evolution or intergalactic extinction has been than the uncertainties given above. neglected. The parameters used in these calculations The agreement between the author’s A-corrections are given in Table 11. The mean absolute B magnitude and those of Schild and Oke (1971), Whitford (1971), of each galaxy type (row 1) and the dispersion o- of the and Oke (1971) for the E-S0 galaxies is very good over Gaussian absolute luminosity distribution (row 2) the range in common. At large z, Oke’s ^-corrections have been provided by de Yaucouleurs based on are systematically smaller than those calculated by statistical studies of bright galaxies. A Gaussian Whitford and by the author. This is because the 3C 295 absolute luminosity function has been adopted because energy curve used by Oke levels off below 2700 Â. the dwarf systems, which are compact or low in surface However, the accidental errors are so large (see Oke brightness and which populate the faint end of the 1971, Fig. 2) that I believe it is better to use a theo- luminosity function, are seldom recorded when faint retically plausible energy curve, such as the curve of an galaxies are counted. The frequency distribution of F dwarf star, instead of trying to use extremely un- galaxies of different types per unit volume of space certain observations. (row 3) was chosen such that it would reproduce the Studies of the color of galaxies as a function of observed apparent frequency distribution of different red shift provide an observational check on the com- galaxy types among the brighter galaxies (row 4) when puted ^-corrections. Three investigators have esti- integrated in conjunction with the above luminosity mated the change in color with redshift for nearby functions. This apparent distribution was obtained bright galaxies. Stebbins and Whitford (1948) found a -1 from the morphological classification of 1500 bright change of ÀQ/A F = 0.0133 mag per 1,000 km s galaxies (de Yaucouleurs 1963). for E-S0 galaxies out to z = 0.13, which reduces to Mattig (1958, 1959) has derived formulae for the 0.0114 in the J? — V system. Holmberg (1958) found a distance modulus and the volume of observed space as value of 0.016 ± 0.010 (reduced to B — V system) for a function of redshift. For q = %> spirals (z < 0.008), and de Yaucouleurs (1961) determined a value of 0.012 ± 0.003 from a sample of m — M = 5 log — 5 log /7 — 45.06 + K(z, t), (2) galaxies of all types (z < 0.013). The author’s K- corrections at small z are in good agreement, predicting 32ttA3 V = 3 3 (3) a value of 0.0113 for E-S0 galaxies and 0.0075 for all 3// [l + A + V(1 + 2^)] ’ other types. A more sensitive test of the ^-corrections for elliptical galaxies is provided by the B — V color and indices of the brightest galaxies in clusters. Figure 2 in ^ = 2z - 2[a/(1 + z) - 1], (4) Sandage (1973) shows the corrected observed colors of giant elliptical galaxies out to z = 0.46. The author’s where m is the observed apparent magnitude. 77, the K(B — V) function (normalized to B — V = 0.97 at Hubble constant, has been assumed to equal 100 km z = 0) fits the observations nearly perfectly. s“1Mpc~1 to be consistent with (Table 18,

TABLE 11 Galaxy Parameters Parameter E-S0 Sab Sbc Scd Sdm-Im -18.40 -18.40 -18.40 -17.00 -17.00 oM (mag) 0.80 0.80 0.80 1.20 1.20 Space frequency 0.215 0.185 0.160 0.275 0.165 Apparent frequency 0.309 0.268 0.232 0.119 0.072 0.426 0.064 - 0.130 - 0.213 - 0.186 <£ - V} 0.916 0.729 0.579 0.478 0.457 A(C/-B)(mag) 0.05 0.09 0.11 0.07 0.06 A(B-K)(mag) 0.09 0.11 0.14 0.08 0.06

row 1). The ^-corrections corrected for galactic extinc- pole. Table 12 tabulates these quantities for the B tion given in Tables 7-9 were used. filter. The asterisks indicate that it was necessary to By performing integrations over the appropriate extrapolate the K(z) function out to larger z than given frequency functions and volume elements one can in Tables 7-9. It is interesting to note that at faint compute the mean ^-correction and mean z at each magnitudes the mean z of the late-type spirals is greater apparent magnitude for galaxies near the Galactic than for the ellipticals. Thus, even though the former TABLE 13

* Uncertain because an extrapolation of Ky(z) (Table 7) was used

TABLE 15 APPARENT RELATIVE FREQUENCY OF DIFFERENT GALAXY TYPES In The U Band In The B Band In The V Band Apparent E- Sab Sbc Scd Sdm - E- Sab Sbc Scd Sdm- E- Sab Sbc Scd Sdm - Mag. SO Im SO Im SO Im 12.0 *182 .261 • ¿93 . 167 .097 .311 .268 .232 .118 .071 .406 .270 .190 .084 .049 12.5 *182 .261 .293 .167 .097 .311 .268 .232 .118 .071 .405 .270 .191 .085 .050 13.0 .184 .261 .291 .166 .097 .311 .268 .232 .118 .071 .399 .269 .194 .087 .051 13.5 .186 .262 .290 ,164 .097 .311 .269 .231 .118 .071 .396 .269 .194 .089 .052 14.0 .190 .259 .284 .168 .098 .303 .265 .236 .122 .074 .389 .265 .200 .091 .055 14.5 .193 .259 .282 .167 .100 .301 .266 .233 .124 .076 .388 .264 .198 .093 .056 15.0 .191 .254 .278 .174 .102 .287 .260 • 242 .131 .080 .384 .262 .202 .095 .058 15.5 .191 .255 . 28U .171 .103 .286 .261 .240 .131 .081 .381 .261 .203 .096 .059 16.0 .189 .253 .280 .175 .104 .279 .258 .245 .134 .084 .375 .259 .206 .098 .061 16.5 . 190 .252 .279 .174 .106 .275 .258 • 246 .136 .086 .368 .258 .209 .101 .063 17.0 .189 .250 .277 .177 .108 .269 .255 • 2*+7 .139 .089 .360 .256 .213 .105 .066 17.5 .188 .246 .275 .180 .112 .263 .252 .249 .143 .093 .350 .254 .218 .109 .069 18.0 .185 .240 .273 .185 .116 .255 .249 .250 .148 .097 .338 .252 .223 .114 .073 18.5 .181 .234 .272 .190 .123 .248 .246 .251 .154 .102 .324 .249 .228 .121 .078 19.0 .174 .227 .271 .197 .130 .240 .242 .250 .160 .107 .307 .245 .234 .130 ,084 19.5 .165 .221 .270 .206 .139 .232 .238 .2*9 .167 .114 .289 .241 .238 . 140 .092 20.0 .153 .214 .268 .216 .149 .225 .233 .247 .173 .121 .270 .235 .241 .152 .102 20.5 .139 .208 .266 .227 .160 • 218 .228 .244 .181 .129 .252 .229 .242 .166 .112 21.0 .124 .200 .264 .240 .172 .211 .222 .241 .189 .138 .235 .221 .239 .180 .125 21.5 .107 .191 .262 .254 .186 .203 .215 .237 .197 .148 .221 .214 .234 .193 .137 22.0 .091 .182 .263 .267 .197 .193 .207 .234 .206 .160 .212 .207 .228 .204 . 149 22.5 .076 .175 .267 .276 .206 .180 .199 .232 .216 .173 .205 .200 .222 .212 .161 23.0 .063 .172 .273 .283 .209 • 163 .189 .229 .229 .189 .197 .192 .218 .219 .174 23.5 . 144 .179 .229 .243 .205 .184 .183 .216 .228 . 189 24.0 • 122 .169 .231 .258 .220 .165 .173 .215 .240 .207 24.5 .140 .162 .219 .256 .223 25.0 .114 .153 .227 .269 .238

© American Astronomical Society • Provided by the NASA Astrophysics Data System 50 PENCE Vol. 203 are intrinsically fainter, they can be seen to a greater distance. This is because of the much larger K- corrections for early-type galaxies which causes them to fade out with distance more rapidly than later types. The apparent relative frequency of each galaxy type as seen through the B filter at successively fainter apparent magnitudes is tabulated in Table 15. At bright magnitudes the distribution matches the observed distribution among 1500 nearby galaxies (de Vaucouleurs 1963). As expected, the table shows that the late-type galaxies become relatively more abundant at faint magnitudes because of their smaller K- corrections. Identical calculations have been performed for the U and V filter bands. The only difference in the calculations is that the mean absolute magnitude of each type changes when viewed through the different filters. The difference is given by the mean U — B and B — V colors of each type listed in Table 11. Tables 13 through 15 give the numerical results which are qualitatively similar to the B filter calculations. Figure 4 shows the predicted mean color-color relation of each galaxy type at successively fainter apparent B magnitudes. The mean color-color relation for galaxies of all morphological types together is also shown. Unfortunately there are very few U, B, V observations of randomly selected faint galaxies to check these predictions. The dotted circle on the figure shows the mean colors ( = 18.0 near the north galactic Fig. 4.—Predicted mean color-color relation for different pole measured by de Vaucouleurs and Brown (Brown galaxy types as a function of B magnitude. The relation for all galaxy types combined is shown by the dashed curve. The mean 1974). Obviously, the agreement with the predicted observed color indices of eight faint «.ß) = 18.0) field galaxies color (B — V = 0.86, U — B = 0.03) is not good. More near the north galactic pole are shown by the dashed circle. observations are needed to see if this discrepancy is real.

Fig. 5.—Calculated frequency distribution of the B — V color index for E-S0 and Sdm-Im type galaxies and for all galaxy types combined at successively fainter B magnitudes.

© American Astronomical Society • Provided by the NASA Astrophysics Data System No. 1, 1976 ^-CORRECTIONS 51 To carry these calculations one step further, the Figure 5 should only be taken qualitatively because expected frequency distribution of colors of galaxies at there are too many uncertain parameters in the model each apparent magnitude was computed. Studies of (#, H9 , M distribution, (B — V), intrinsic R — V nearby galaxies have shown that the colors of galaxies distribution) for the results to be exact. of a given type have an intrinsic spread along the color- It should be stressed again that in all these calcula- color sequence. Line 7 in Table 11 gives a parameter tions the effects of galactic evolution and intergalactic (equivalent to a probable error) which expresses the extinction have been neglected. This is not to suggest half-width of the intrinsic color distribution which was that these effects are unimportant, but only that they derived from Figure 1 in de Vaucouleurs and de are currently not well enough understood to be incor- Vaucouleurs (1972). For successively fainter magni- porated in the models. Therefore, the predictions pre- tudes the observed color distributions become broader, sented here on the frequency distributions and colors of since the observed galaxies will have different redshifts galaxies are not necessarily expected to agree exactly and hence will have a range of ^-corrections. Figure 5 with the observations when z is greater than several shows how the mean and the frequency distribution of tenths, and conversely departures from predictions may the 1? — F color index vary with B magnitude for convey information on the neglected factors. E-SO and Sdm-Im type galaxies and for galaxies of all types combined. The bimodal distribution of the I am indebted to Dr. Gerard de Vaucouleurs for his latter is probably an artifact of the calculation as a many discussions and helpful suggestions, and to Dr. result of using only five discrete morphological types of Donald Wells whose dissertation results made this galaxies instead of a whole continuum of types. project possible. REFERENCES Azusienis, A., and Straizys, V. 1966, Bull. Vilnius Obs., No. 16, de Vaucouleurs, G., and de Vaucouleurs, A. 1972, Mem. 3. R.A.S., 77, 1. Brown, G. S. 1974, unpublished Ph.D. thesis, University of Holmberg, E. 1958, Medd. Lunds Astr. Obs., Ser. 2, No. 136. . 1974, Astr. and Ap., 35, 121. Code, A. D., Welch, G. A., and Page, T. L. 1972, The Scientific Johnson, H. J. 1966, Ap. J., 143, 189. Results from the Orbiting Astronomical Observatory, ed. Knapp, G. R. 1975, A.J., 80, 111. A. D. Code, National Aeronautics and Space Administra- Matthews, T. A., and Sandage, A. R. 1963, Ap. J., 138, 30. tion, Washington, p. 559. Mattig, W. 1958, Astr. Nach., 284, 109. de Vaucouleurs, G. 1958, A.J., 63, 261. . 1959, ibid., 2S5, 1. . 1961, Ap. J. Suppl, No. 48, 5, 232. Oke, J. B. 1971, Ap. J., 170, 193. . 1962, Problems of Extra-galactic Research, I AU Sym- Oke, J. B., and Sandage, A. 1968, Ap. J., 154, 21. posium No. 15, ed. G. C. McVittie (New York: Macmillan), Sandage, A. 1973, Ap. J., 183, 711. p. 3. Schild, R. E., and Oke, J. B. 1971, Ap. J., 169, 209. . 1963, Ap. J. Suppl, No. 74, 8, 31. Stebbins, J., and Whitford, A. E. 1948, Ap. J., 108, 413. . 1974, The Formation and Dynamics of Galaxies, I AU van den Bergh, S. 1967, A.J.,12, 70. Symposium No. 58, ed. J. R. Shakeshaft (Dordrecht: Wells, D. C. 1972, unpublished Ph.D. thesis, University of Reidel), p. 1. de Vaucouleurs, G., and Malik, G. 1969, M.N.R.A.S., 142, Whitford, A. E. 1971, Ap. J., 169, 215. 387.

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