Help sought in debugging Schaefer TLM calculator

For some years I have been using the Larry Bogan's javascript
implementation -

http://www.go.ednet.ns.ca/~larry/astro/maglimit.html

- of Schaefer's limiting magnitude calculator. Schaefer' TLM
calculator was originally published in S&T in BASIC code in 1989:

Schaefer, B. Nov. 1989a. Your telescope's limiting magnitude. Sky &
Telescope 78(5):552
http://adsabs.harvard.edu/cgi-bin/np...6T....78..522S

Schaefer, B.E. Feb. 1990. Telescopic Limiting Magnitude. PASP
102:212-229
http://adsbit.harvard.edu/cgi-bin/np...ASP..102..212S

In using Schaefer's alogrithm (as implemented by Bogan), I notice that
if you put in red color stars - those with positive color indices
between 0.75 and 2.0 - the calculator returns a brighter TLM and not a
fainter TLM.

This is counter to my understanding of how the human eye sees faint
stars. The human eye should see red colored K and M stars at a fainter
TLM than white colored 0 index stars.

I have compared Schaefer's code with Bogan's Javascript port and I am
reasonably certain is a faithful translation.

In comparing Schaefer's 1990 paper with Bogan's Javascript port with
respect to color index, I notice the following difference:

Bogan's code: FC=Math.pow(10,0.4*(CI/2-1)); // COLOR OF STAR

Equation 13 in Schaefer 1990 at 215
-2.5 log (Fc) = 1-(B-V)/2 if log(B)3.17

which implies:

Fc = 10 ^ (0.4 * (1-(CI/2)
)

Is CI handled properly in the calculator?

My question is this. Am I properly interpreting the computed results of
the calculator as being improper, or am I missing something?

If the code and result are incorrect, can it be fixed?

Any help would be appreciated. - Canopus56

canopus56 wrote:

In using Schaefer's alogrithm (as implemented by Bogan), I notice that
if you put in red color stars - those with positive color indices
between 0.75 and 2.0 - the calculator returns a brighter TLM and not a
fainter TLM.

This is counter to my understanding of how the human eye sees faint
stars. The human eye should see red colored K and M stars at a fainter
TLM than white colored 0 index stars.

Are you sure about that? Schaefer makes the opposite assumption.

...[C]onsider the case of two stars with equal V magnitude but
different color. An observer using day vision would pronounce the
two stars to be of equal brightness, whereas if night vision were
being used the redder of the two stars would appear fainter. (1990
p. 212)

This is what I'd expect, since the scotopic sensitivity peak is bluer
than the photopic peak.

Keep in mind what this implies about limiting magnitude. The limiting
magnitude is the dimmest thing you can see. As you turn the magnitude
dial toward the dim end, the red stars disappear first. These red stars
have the same V magnitude as bluer stars you can still see, meaning that
your limiting magnitude is *brighter* for redder stars.

I have compared Schaefer's code with Bogan's Javascript port and I am
reasonably certain is a faithful translation.

In comparing Schaefer's 1990 paper with Bogan's Javascript port with
respect to color index, I notice the following difference:

Bogan's code: FC=Math.pow(10,0.4*(CI/2-1)); // COLOR OF STAR

Equation 13 in Schaefer 1990 at 215
-2.5 log (Fc) = 1-(B-V)/2 if log(B)3.17

which implies:

Fc = 10 ^ (0.4 * (1-(CI/2))

Is CI handled properly in the calculator?

You're missing a minus sign in that last equation. It should be

Fc = 10 ^ (-0.4 * (1 - ( CI / 2 )))

which is identical to the equation you've attributed to Bogan.

Fc is larger for redder stars and smaller for bluer ones.

B-V Fc
mu Cep 2.35 1.17
alp Ori 1.85 0.93
alp Tau 1.50 0.79
bet Gem 1.00 0.63
alp CMi 0.38 0.47
bet Ori -0.03 0.39
gam Ori -0.22 0.36

This is consistent with Schaefer's assertion. Using his nomenclature,
the V brightness I* is proportional to the perceived brightness I
multiplied by his color correction factor Fc,

I* ~ IFc

in some linear brightness units (e.g. lamberts), not magnitude. Turning
this around, the perceived brightness is the V brightness divided by Fc,
so bigger Fc (redder stars) implies dimmer perceived brightness for the
same V brightness.

Converting to magnitudes, the limiting magnitude for redder stars is a
smaller number.

- Ernie http://home.comcast.net/~erniew

Ernie Wright wrote:
canopus56 wrote:
snip all
Are you sure about that? Schaefer makes the opposite assumption.
...[C]onsider the case of two stars with equal V magnitude but
different color. An observer using day vision would pronounce the
two stars to be of equal brightness, whereas if night vision were
being used the redder of the two stars would appear fainter. (1990
p. 212)

Thanks Ernie. Shortly after posting I realized I had read the
correcting factors backwards and deleted the post (at least out of the
Google groups archive). The point of the color index correcting factor
in Schaefer's algorithm is to adjust what the eye sees to a V-band
standard. The Schaefer algorithm and Bogan's javascript calculator
should adjust the perception of red color index stars (which the eye
overreports in apparent brightness due to the Purkinje effect) to a
dimmer white color in the V band. Bogan's javascript calculator is
right. I just plain had a brain fart on this one and read it wrong. -
Canopus56