J/AJ/81/807 Deprojection of the r^1/4^ law tables (Young, 1976) ================================================================================ Tables of functions for a spherical galaxy obeying the r^1/4^ law in projection Young P.J. =1976AJ.....81..807Y ================================================================================ ADC_Keywords: Models ; Galaxies, photometry; Photometry, surface Abstract: A spherical galaxy with reduced surface brightness, J=B({alpha})/B_e_, obeying the r^1/4^ law, logJ=-3.3307 ({alpha}^1/4^-1), where {alpha} is the reduced radius, {alpha}=r/r_e_ (r_e_ is the effective radius), is deprojected to find the corresponding space density, mass, mean density, force, potential, escape velocity, and potential energy at each point in the galaxy. Numerical tabulations to five significant figures are given for 124 points in the range 10^-6^<=R/r_e_<=260. In addition the projected surface brightness B({alpha}) and integrated luminosity within {alpha} are tabulated for the range 10^-6^<={alpha}=r/r_e_<=260. Conversion factors to cgs units and to M_{sun}_, pc, km/s, L_{sun}_ units are given. Asymptotic expansions for the space density {rho}(s) in the ranges s=<10^-4^ and s>=10^-1^ are derived, and it is demonstrated that the projection of the expansion for s<=10^-1^ is almost indistinguishable from the r^1/4^ law itself, apart from a small excess of luminosity in the central regions. Formulae and numerical tables of the luminosity distribution are given for use in galaxy photometry. Relations between the total galactic mass M_T_, the effective radius r_e_, the velocity dispersion {sigma}_v_, the central density {rho}_c_ and the mass M_N_ and radius R_N_ of the nucleus are derived. Here the "nucleus" is defined as the region within which stars having a velocity equal to the mean velocity dispersion in space, {sigma}_v_. File Summary: -------------------------------------------------------------------------------- FileName Lrecl Records Explanations -------------------------------------------------------------------------------- ReadMe 80 . This file table2.dat 117 124 Quantities in reduced units for spherical galaxy obeying R^1/4^-law in projection -------------------------------------------------------------------------------- Table 1: Conversion factors. -------------------------------------------------------------------------------- Reduced quantity M_{sun}, pc, km/s, L_{sun} cgs system system -------------------------------------------------------------------------------- total galaxy mass M_T_*=1 M_T_ in M_{sun}_ M_T_' in grams projected effective radius s=1 r_e_ in pc r_e_' in centimeters total luminosity L_T_*=1 L_T_ in L_{sun}_ (B system) L_T_' in erg/s (bolometric) L_T_'' in erg/s (B band) mass-to-luminosity ratio K*=1 K=M_T_/L_T_ (B system) K'=M_T_'/L_T_' (bolometric) K''=M_T_'/L_T_'' (B band) {beta} index {beta}*=2.62175 {beta}=1.1277x10^-2^ {beta}'=2.62175 G=1.749x10^-7^ -------------------------------------------------------------------------------- Byte-by-byte Description of file: table2.dat -------------------------------------------------------------------------------- Bytes Format Units Label Explanations -------------------------------------------------------------------------------- 1- 7 E7.1 --- s/alpha *[1.0E-06,2.6E+02]+ Space/Plane radius 9- 18 E10.4 --- B*(alpha) *Reduced surface luminosity (specific intensity) 20- 29 E10.4 --- f*(alpha) *Reduced luminosity enclosed within (circular) isophote of radius {alpha} 31- 40 E10.4 mag Dm(alpha) *Magnitude difference 42- 51 E10.4 --- rho*(s) *Reduced space density 53- 62 E10.4 --- M*(s) *Reduced mass contained within sphere of radius s 64- 73 E10.4 --- *Reduced mean space density within sphere of radius s 75- 84 E10.4 --- F*(s) *Reduced force 86- 95 E10.4 --- psi*(s) *Reduced potential (with psi*(0)=1 and psi*(infinity)=0) 97-106 E10.4 --- w*(s) *Reduced escape velocity 108-117 E10.4 --- OMEGA*(s) *Reduced potential energy contained within radius s -------------------------------------------------------------------------------- Note on s/alpha: {alpha} = a/a_e_ = reduced radius (semi-major axis) in projection s = R/r_e_ = reduced radius in space (s is in units of the effective radius r_e_) Note on B*(alpha): B(r) = B*(alpha)M_T_K^-1^/r_e_^2^ in L_{sun}_/pc^2^ B'(r')= B*(alpha)M_T_'K'^-1^/r_e_'^2^ in erg/s/cm^2^ Note on f*(alpha): f(alpha)= f*(alpha)M_T_/K in L_{sun}_ f'(r') = f*(alpha)M_T_'/K' in erg/s Note on Dm(alpha): Difference in magnitudes between luminosity enclosed within {alpha} and the total luminosity Note on rho*(s): rho(R) = rho*(s)M_T_/r_e_^3^ in M_{sun}_/pc^3^ rho'(R')= rho*(s)M_T_'/r_e_'^3^ in g/cm^3^ Note on M*(s): M(R) = M*(s)M_T_ in M_{sun}_ M'(R')= M*(s)M_T_' in g Note on : (R) = *(s)M_T_/r_e_^3^ in M_{sun}_/pc^3^ '(R')= *(s)M_T_'/r_e_'^3^ in g/cm^3^ means rho{bar} Note on F*(s): F(R) = F*(s){beta}M_T_/r_e_^2^ in (km/s)^2^/pc F'(R')= F*(s){beta}'M_T_'/r_e_'^2^ in dyn Note on psi*(s): psi(R) = psi*(R){beta}M_T_/r_e_ in (km/s)^2^ psi'(R')= psi*(R){beta}'M_T_'/r_e_' in (cm/s)^2^ Note on w*(s): w(R) = w*(s)[{beta}M_T_/r_e_]^1/2^ in km/s w'(R')= w*(s)[{beta}'M_T_'/r_e_']^1/2^ in cm/s Note on OMEGA*(s): OMEGA(R) = OMEGA*(s){beta}M_T_^2^/r_e_ in (km/s)^2^ OMEGA'(R')= OMEGA*(s){beta}'M_T_'^2^/r_e_' in g(cm/s)^2^ -------------------------------------------------------------------------------- History: Prepared via OCR at CDS. ================================================================================ (End) James Marcout, Patricia Bauer [CDS] 08-Jun-2000