Star B-V color index to apparent RGB color

melhosseiny picture melhosseiny · Feb 24, 2014 · Viewed 9.3k times · Source

I'm trying to convert a star's B-V color index to an apparent RGB color. Besides look up tables and color ramps, it seems like there's no well known algorithm for doing this.

What's a B-V color index?

It's a number astronomers assign to a star to indicate its apparent color. Hot stars (low B-V) are blue/purple and cool stars (high B-V) are red with those white/orange stars in between.

Initial algorithm

B-V to Kelvin

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var t = 4600 * ((1 / ((0.92 * bv) + 1.7)) +(1 / ((0.92 * bv) + 0.62)) );

Kelvin to xyY

If you model a star as a blackbody, then you can use a numerical approximation of the Planckian locus to compute the xy coordinates (CIE chromaticity)

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// t to xyY
var x, y = 0;

if (t>=1667 && t<=4000) {
  x = ((-0.2661239 * Math.pow(10,9)) / Math.pow(t,3)) + ((-0.2343580 * Math.pow(10,6)) / Math.pow(t,2)) + ((0.8776956 * Math.pow(10,3)) / t) + 0.179910;
} else if (t > 4000 && t <= 25000) {
  x = ((-3.0258469 * Math.pow(10,9)) / Math.pow(t,3)) + ((2.1070379 * Math.pow(10,6)) / Math.pow(t,2)) + ((0.2226347 * Math.pow(10,3)) / t) + 0.240390;
}

if (t >= 1667 && t <= 2222) {
  y = -1.1063814 * Math.pow(x,3) - 1.34811020 * Math.pow(x,2) + 2.18555832 * x - 0.20219683;
} else if (t > 2222 && t <= 4000) {
  y = -0.9549476 * Math.pow(x,3) - 1.37418593 * Math.pow(x,2) + 2.09137015 * x - 0.16748867;
} else if (t > 4000 && t <= 25000) {
  y = 3.0817580 * Math.pow(x,3) - 5.87338670 * Math.pow(x,2) + 3.75112997 * x - 0.37001483;
}

xyY to XYZ (Y = 1)

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// xyY to XYZ, Y = 1
var Y = (y == 0)? 0 : 1;
var X = (y == 0)? 0 : (x * Y) / y;
var Z = (y == 0)? 0 : ((1 - x - y) * Y) / y;

XYZ to RGB

enter image description here

var r = 0.41847 * X - 0.15866 * Y - 0.082835 * Z;
var g = -0.091169 * X + 0.25243 * Y + 0.015708 * Z;
var b = 0.00092090 * X - 0.0025498 * Y + 0.17860 * Z;

Question

I ran this algorithm with the B-V color indexes: 1.2, 1.0, 0.59, 0.0, -0.29. This is what I got as output.

enter image description here

Why did I get this strange output? Hot stars are bluish but cold stars are brownish and there doesn't seem to be white/orange intermediate stars.

Update

Following on a comment by Ozan, it seemed like I was using a wrong matrix to convert XYZ to RGB. Since sRGB is the default color space on the web (or is it?), I'm now using the correct matrix followed by a gamma correction function (a = 0.055).

enter image description here

enter image description here

I now get this nice color ramp,

enter image description here

but there's still no red/violet at the extremities.

Demo

There's also a fiddle now that you can play with.

Update 2

If use a gamma of 0.5 and extend the range of B-V color indexes to be from 4.7 to -0.5, I get red at one extreme but still no violet. Here's the updated fiddle.

enter image description here

Answer

Spektre picture Spektre · Mar 25, 2014

I use tabled interpolation instead. Some years back I found this table somewhere:

     type     r   g   b    rrggbb        B-V

     O5(V)   155 176 255  #9bb0ff       -0.32 blue
     O6(V)   162 184 255  #a2b8ff
     O7(V)   157 177 255  #9db1ff
     O8(V)   157 177 255  #9db1ff
     O9(V)   154 178 255  #9ab2ff
   O9.5(V)   164 186 255  #a4baff
     B0(V)   156 178 255  #9cb2ff
   B0.5(V)   167 188 255  #a7bcff
     B1(V)   160 182 255  #a0b6ff
     B2(V)   160 180 255  #a0b4ff
     B3(V)   165 185 255  #a5b9ff
     B4(V)   164 184 255  #a4b8ff
     B5(V)   170 191 255  #aabfff
     B6(V)   172 189 255  #acbdff
     B7(V)   173 191 255  #adbfff
     B8(V)   177 195 255  #b1c3ff
     B9(V)   181 198 255  #b5c6ff
     A0(V)   185 201 255  #b9c9ff       0.00 White
     A1(V)   181 199 255  #b5c7ff
     A2(V)   187 203 255  #bbcbff
     A3(V)   191 207 255  #bfcfff
     A5(V)   202 215 255  #cad7ff
     A6(V)   199 212 255  #c7d4ff
     A7(V)   200 213 255  #c8d5ff
     A8(V)   213 222 255  #d5deff
     A9(V)   219 224 255  #dbe0ff
     F0(V)   224 229 255  #e0e5ff       0.31 yellowish
     F2(V)   236 239 255  #ecefff
     F4(V)   224 226 255  #e0e2ff
     F5(V)   248 247 255  #f8f7ff
     F6(V)   244 241 255  #f4f1ff
     F7(V)   246 243 255  #f6f3ff       0.50
     F8(V)   255 247 252  #fff7fc
     F9(V)   255 247 252  #fff7fc
     G0(V)   255 248 252  #fff8fc       0.59  Yellow
     G1(V)   255 247 248  #fff7f8
     G2(V)   255 245 242  #fff5f2
     G4(V)   255 241 229  #fff1e5
     G5(V)   255 244 234  #fff4ea
     G6(V)   255 244 235  #fff4eb
     G7(V)   255 244 235  #fff4eb
     G8(V)   255 237 222  #ffedde
     G9(V)   255 239 221  #ffefdd
     K0(V)   255 238 221  #ffeedd       0.82 Orange
     K1(V)   255 224 188  #ffe0bc
     K2(V)   255 227 196  #ffe3c4
     K3(V)   255 222 195  #ffdec3
     K4(V)   255 216 181  #ffd8b5
     K5(V)   255 210 161  #ffd2a1
     K7(V)   255 199 142  #ffc78e
     K8(V)   255 209 174  #ffd1ae
     M0(V)   255 195 139  #ffc38b       1.41 red
     M1(V)   255 204 142  #ffcc8e
     M2(V)   255 196 131  #ffc483
     M3(V)   255 206 129  #ffce81
     M4(V)   255 201 127  #ffc97f
     M5(V)   255 204 111  #ffcc6f
     M6(V)   255 195 112  #ffc370
     M8(V)   255 198 109  #ffc66d       2.00
  1. just interpolate the missing B-V indexes (linearly or better) before use
  2. then use linear interpolation to get RGB=f(B-V);
  3. find the closest two lines in table and interpolate between them ...

[edit1] heh just coincidentally come across this (original info I mentioned before)

[edit2] here is my approximation without any XYZ stuff

BV to RGB

So the BV index is from < -0.4 , 2.0 >

here is mine (C++) code for conversion:

//---------------------------------------------------------------------------
void bv2rgb(double &r,double &g,double &b,double bv)    // RGB <0,1> <- BV <-0.4,+2.0> [-]
    {
    double t;  r=0.0; g=0.0; b=0.0; if (bv<-0.4) bv=-0.4; if (bv> 2.0) bv= 2.0;
         if ((bv>=-0.40)&&(bv<0.00)) { t=(bv+0.40)/(0.00+0.40); r=0.61+(0.11*t)+(0.1*t*t); }
    else if ((bv>= 0.00)&&(bv<0.40)) { t=(bv-0.00)/(0.40-0.00); r=0.83+(0.17*t)          ; }
    else if ((bv>= 0.40)&&(bv<2.10)) { t=(bv-0.40)/(2.10-0.40); r=1.00                   ; }
         if ((bv>=-0.40)&&(bv<0.00)) { t=(bv+0.40)/(0.00+0.40); g=0.70+(0.07*t)+(0.1*t*t); }
    else if ((bv>= 0.00)&&(bv<0.40)) { t=(bv-0.00)/(0.40-0.00); g=0.87+(0.11*t)          ; }
    else if ((bv>= 0.40)&&(bv<1.60)) { t=(bv-0.40)/(1.60-0.40); g=0.98-(0.16*t)          ; }
    else if ((bv>= 1.60)&&(bv<2.00)) { t=(bv-1.60)/(2.00-1.60); g=0.82         -(0.5*t*t); }
         if ((bv>=-0.40)&&(bv<0.40)) { t=(bv+0.40)/(0.40+0.40); b=1.00                   ; }
    else if ((bv>= 0.40)&&(bv<1.50)) { t=(bv-0.40)/(1.50-0.40); b=1.00-(0.47*t)+(0.1*t*t); }
    else if ((bv>= 1.50)&&(bv<1.94)) { t=(bv-1.50)/(1.94-1.50); b=0.63         -(0.6*t*t); }
    }
//---------------------------------------------------------------------------

[Notes]

This BV color is blackbody of defined temperature illumination so this represents star color viewed from space relative with the star. For visually correct colors you have to add atmospheric scattering effects of our atmosphere and Doppler effect for fast mowing stars!!! for example our Sun is 'White' but after light scatter the color varies from red (near horizon) to yellow (near nadir ... noon)

In case you want to visually correct the color these QAs might help: