SAMPLE DENSITY AND LUMINOSITY FUNCTION
CALCULATIONS
Based on Kapetyn's Method (Absorption Ignored)
We begin with the "fundamental equation of stellar statistics":
A(m) = 7.014 × 10-4 òr3 D(r) φ(M) d(log r),
where the range of integration is from r = 0 to r = ¥. When a parallax catalog is complete to some limiting magnitude over a particular region of the sky, the density function D(r), and the luminosity function φ(M), can be extracted from the catalog data. Let the distribution over apparent magnitude, m, and distance, r, for the catalog stars be fm(r), i.e., fm(r) is the number of catalog entries with m in the range m-½ to m+½ and log r in the range log r - 0.1 to log r + 0.1. Except for a scale factor equal to the number of square degrees surveyed in the catalog, each entry fm(r) should be roughly equal to the product
fm(r) = Vr D(r) φ(M = m - 5 log r + 5),
where
Vr = (4π2 /(log10e 3602)) × r3 Δ(log r) = 7.014 × 10-4 r3 (0.2)
is the volume of a 1°×1° segment of a spherical shell of thickness 0.2 in log r. Thus the catalog stars can be sorted into fm(r) bins, counted, and tabulated according to fm(r) = Vr D(r) φ(M) bin as indicated in the table below:
| Distance Indicator | Counts | |||||||
| log r | r* | π ["] | m = -1 | 0 | 1 | 2 | 3 | ∙ ∙ ∙ |
| 1.6 | 40 | .025 | V40 D(40) φ(-4) | V40 D(40) φ(-3) | V40 D(40) φ(-2) | V40 D(40) φ(-1) | V40 D(40) φ(0) | ∙ ∙ ∙ |
| 1.4 | 25 | .040 | V25 D(25) φ(-3) | V25 D(25) φ(-2) | V25 D(25) φ(-1) | V25 D(25) φ(0) | V25 D(25) φ(+1) | ∙ ∙ ∙ |
| 1.2 | 16 | .063 | V16 D(16) φ(-2) | V16 D(16) φ(-1) | V16 D(16) φ(0) | V16 D(16) φ(+1) | V16 D(16) φ(+2) | ∙ ∙ ∙ |
| 1.0 | 10 | .100 | V10 D(10) φ(-1) | V10 D(10) φ(0) | ∙ ∙ ∙ | ∙ ∙ ∙ | ∙ ∙ ∙ | ∙ ∙ ∙ |
| 0.8 | 6.3 | .160 | V6.3 D(6.3) φ(0) | ∙ ∙ ∙ | ∙ ∙ ∙ | ∙ ∙ ∙ | ∙ ∙ ∙ | ∙ ∙ ∙ |
| 0.6 | 4.0 | .250 | V4 D(4) φ(+1) | ∙ ∙ ∙ | ∙ ∙ ∙ | ∙ ∙ ∙ | ∙ ∙ ∙ | ∙ ∙ ∙ |
| 0.4 | 2.5 | .400 | ∙ ∙ ∙ | ∙ ∙ ∙ | ∙ ∙ ∙ | ∙ ∙ ∙ | ∙ ∙ ∙ | ∙ ∙ ∙ |
| Σ = | A(m=-1) | A(0) | A(1) | A(2) | A(3) | ∙ ∙ ∙ | ||
Note that the table diagonals are the loci of constant M = m - 5 log r + 5.
Now let us suppose that a hypothetical catalog has the following fm(r) distribution:
| Distance Indicator | Counts | |||||||
| log r | r* | π ["] | m = 1 | 2 | 3 | 4 | 5 | 6 |
| 1.6 | 40 | .025 | 2 | 8 | 30 | 130 | 520 | 1353 |
| 1.4 | 25 | .040 | 2 | 9 | 38 | 156 | 380 | 900 |
| 1.2 | 16 | .063 | 3 | 13 | 50 | 120 | 250 | 500 |
| 1.0 | 10 | .100 | 3 | 12 | 32 | 60 | 130 | 200 |
| 0.8 | 6.3 | .160 | 3 | 8 | 16 | 35 | 60 | 80 |
| 0.6 | 4.0 | .250 | 2 | 5 | 10 | 18 | 25 | 35 |
| Σ = | A(m= +1) | A(2) | A(3) | A(4) | A(5) | A(6) | ||
|
*The r values are approximate. The exact values are 101.6 =39.81, 101.4 =25.12, 101.2 =15.85, 101.0 =10.00, 100.8 =6.31, 100.6 =3.98 . |
||||||||
Since the diagonals are the loci of constant M it follows that:
[V40 D(40)] / [V25
D(25)] = {V40 D(40)[φ(-1)+φ(0)+
∙ ∙ +φ(+3)]}
/ {V25 D(25)[φ(-1)+φ(0)+
∙ ∙ +φ(+3)]}
=
(8+30+130+520+1353)/(2+9+38+156+380) = 3.48
± 0.16,
[V25 D(25)] / [V16
D(16)] = {V25 D(25)[φ(0)+φ(1)+
∙ ∙ +φ(4)]}
/ {V16 D(16)[φ(0)+φ(1)+
∙ ∙ +φ(4)]}
=
(9+38+156+380+900)/(3+13+50+120+250) = 3.40
± 0.18,
and similarly,
[V16 D(16)] / [V10 D(10)] = 3.93 ± 0.28, [V10 D(10)] / [V6.3 D(6.3)] = 3.55 ± 0.36, [V6.3 D(6.3)] / [V4 D(4)] = 3.31 ± 0.48.
The values of the relevant Vr s are determined from the relationship Vr = 7.014 × 10-4 r3 (0.2) (using the exact values of r) to be V40 = 8.85, V25 = 2.22, V16 = 0.599, V10 = 0.140, V6.3 = 0.0352, V4 = 0.00885. If we normalize the density function to D(4) = 1, we can calculate D(6.3) = 0.833, D(10) = 0.744, D(16) = 0.736, D(25) = 0.629, and D(40) = 0.551. for simplicity we have dropped our estimates of error ranges.
With the known density function and Vr s we can calculate a point in the luminosity function for each table entry. We do so in the following table, at the same time shifting the ith row in the newly generated table i-1 columns to the right so that each column in the new alignment is characterized by a common value of M rather than m. Entries in the new table are thus φ(M) = fm(r) / [Vr D(r)]. The number of stars counted for each entry is given in parentheses.
| r | φ(M) | ||||||||||
| M = -2 | -1 | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | |
| 40 | .410 (2) |
1.64 (8) |
6.15 (30) |
26.6 (130) |
107 (520) |
277 (1353) |
- | - | - | - | - |
| 25 | - | 1.43 (2) |
6.44 (9) |
27.2 (38) |
112 (156) |
272 (380) |
644 (900) |
- | - | - | - |
| 16 | - | - | 7.30 (3) |
31.6 (13) |
122 (50) |
292 (120) |
608 (250) |
1220 (500) |
- | - | - |
| 10 | - | - | - | 28.7 (3) |
115 (12) |
307 (32) |
575 (60) |
1250 (130) |
1920 (200) |
- | - |
| 6.2 | - | - | - | - | 102 (3) |
273 (8) |
545 (16) |
1190 (35) |
2040 (60) |
2730 (80) |
- |
| 4 | - | - | - | - | - | 226 (2) |
565 (5) |
1130 (10) |
2030 (18) |
2820 (25) |
3950 (35) |
| φ(M) = | .410 (2) |
1.60 (10) |
6.29 (42) |
27.1 (184) |
109 (741) |
277 (1895) |
631 (1231) |
1223 (675) |
1953 (278) |
2750 (105) |
3950 (35) |
The adopted luminosity function in the last row has been obtained by weighting each table entry by the number of stars it represents and combining all the entries in each column. {The result is actually a relative φ(M) which can be converted to an absolute φ(M)[stars mag-1 pc-3] by dividing by the number of square degrees of sky surveyed.}