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Counting stars and an empirical formula for estimating the number of stars in an area

In a planetarium program you have to control how many stars you display. So if you display 1000 stars in an area of one square degree, down to which magnitude you have to plot? This value lies close to magnitude 15.  Below an modern star count table and empirical formula valid from magnitude 0 down to magnitude 16. It is based on counting the number of stars in the combined Tycho-2 and UCAC4 star catalogue using a HNSKY tool.

Table 1,  number of stars in the combined Tycho-2 and UCAC4 star catalogue:

_____________
Star magnitude
Total number of Tycho-2 or UCAC4 stars down to this magnitude
 ln(degree2/star) Star density in stars/degree2
0 3 9.529 0.000072722
1 14 7.988 0.000339369
2 55 6.620 0.001333236
3 191 5.375 0.004629966
4 558 4.303 0.013526289
5 1744 3.164 0.042275713
6 5326 2.047 0.129105762
7 16503 0.916 0.400043633
8 48559 -0.163 1.177102271
9 139375 -1.217 3.378542167
10 382925 -2.228 9.282355223
11 1048084 -3.235 25.40624924
12 2955951 -4.272 71.65420697
13 7346735 -5.182 178.0897147
14 17214116 -6.034 417.2815553
15 39387795 -6.861 954.7861974
16 88896114 -7.675 2154.900589
The remainder of the 113 million UCAC4 stars are fainter then magnitude 16

So down to magnitude 10,  there are average 9.28 stars (=9 stars) in an area of one degree2. Simply calculated by dividing the total number of stars down to magnitude 10 equals 382924 by the area of the whole sky which is 41253 degree2.

In Excel graph below, the natural log of the area available for each star is plotted against the required limiting magnitude (blue line). It is pretty linear down to to magnitude 16. Using Excel a trend line and polynomial was determined. So the limiting star magnitude y:= 0.0128x2 - 0.9547x + 7.8866  , where "x" is is the natural logarithm of the number of stars divided by the total sky area of 41253 degree2. The star density is therefore average but in practice is varies in which direction you are looking. In direction of the milky way the star density is higher



The above table also indicates how many stars you can observe.


Update. This is the star count I get  using the Gaia eDR3 extended with Tycho2 for the bright stars:

Star BP magnitude
Total number of Tycho-2 or Gaia  stars down to this magnitude
 Star Johnson-V magnitude (calculated)Total number of Tycho-2 or Gaia  stars down to this magnitude
0 1 01
1 11 114
2 43 252
3 111 3185
4 460
 4550
5 1515 51720
6 4645 65173
7 14081 716064
8 41263 847405
9 117015 9136017
10 318750 10373520
11 829419 11973888
12 2067520 122426684
13 4935106 135792758
14 11290234 1413185761
15 24701075 1528678640
16 51907518 1659929965
1710538744317120447322
18204758304


Date: 2023-4-12
email:i Han