An investigation into the photopic threshold limit in a telescope
by Chris Lord
The photopic threshold limit in a telescope
I have recently been working on the thorny problem of trying to devise an algorithm that will predict the faintest star that may be observed with direct vision.
My project was sparked by a couple of web articles about Star Colours written by Andrew James:
Star Colours 1
Star Colours 2
The earliest useful work that impacts upon the mesopic threshold ( light levels below which vision transitions from photopic (i.e. foveal) to parafoveal ) is that of Harold H. Peterson [Sky & Telescope, 1954, Sep. p396 & Nov. 1980, p380]. Peterson examined 323 double stars with companion brightnesses from 4.5mv to 11.5mv and separations between 1"arc & 100"arc, using a 3-inch refractor and a low power eyepiece (x45). Peterson plotted a chart showing resolved and unresolved pairs, for companion magnitude against apparent separation.
His diagram revealed two zones, one in which pairs were bright enough for the magnitude of the companion not to effect resolution, and a second zone in which the magnitude of the companion had an influence on the resolution of the pair. The zone boundary marks the threshold between photopic (foveal) and scotopic (peripheral) vision.
James has attempted to derive a colour vision limit based on personal observation and has also provided a best fit solution in the form m(v)=5.42+3.32logD(cm) with which I do not altogether concurr. The slope should be of the form 2.5logD for a V magnitude relationship.
Coincidental to James´ work I came across a very interesting web page discussing the resolution of unequal binaries which considers the effect of decreasing visual acuity with brightness of the companion.
Both these web pages set me to thinking about how I could determine the magnitude limit to which my unequal doubles algorithm can be applied. My work and also that of Lewis and Treanor before me, is based on doubles observed with direct (photopic) vision. I picked pairs with companions 4 mags above limit. Peterson having already shown that the scotopic threshold occurs 3 mags above limit. So the problem essentially boils down to how faint does a star have to be before you need averted vision to see it in a telescope?
This is a question with no straightforward answer because as the observed brightness of a point source decreases the eye adjusts its dark reference. Star colours are only about 10% saturated and colour contrasts are easier to percieve than colour per sé.
It is generally accepted a dark adapted observer can see stars down to roughly 6mv. The pupilliary opening will be approximately 7mm, although it varies with age, from about 9mm down to less than 5mm. A star, apparent brightness 0mv has a luminance of 2.65E-6 lumens/sq.m or 3.33E-5 cd/sq.m. Colour can be detected in stars without a telescope down to roughly 3.25mv, or 1.67E-6 cd/sq.m.
My initial approach was to ignore complicating factors such as visual acuity, sky background, and light losses due to the seeing, zenith distance and telescope, and to see what values would be produced by simply considering the effect of light grasp.
A telescope of aperture D gathers (D/Dp)^2 more light than the eye, assuming no light losses. Taking the pupilliary aperture Dp=5mm when D=25mm (D/Dp)^2=25 times more light is gathered, so a star of 6.7mv will produce a luminance of 1.67E-6 cd/sq.m. A 50mm telescope will gather 100 times more light, so a star of 8.3mv will produce the same threshold luminance. A 75mm telescope like Peterson´s aught to have 225 times the light gathering power of a 5mm pupil, taking the threshold luminance down to 9.1mv. Peterson´s Zone Diagram shows the break point lying at 8.7mv corresponding to a pupil opening of 6mm.
I then plotted the threshold magnitude above which colour may be perceived, against aperture ranging from 25mm to 500mm in 25mm intervals, and fitted a log-linear function curve to the calculated results using regression analysis.
The regression fit is: m(v)=5lgD(mm)-0.25 where m(v) is the photopic threshold. However there is a wide variance in the photopic threshold, ranging from 1.0E-02 cd/sq.m to 1.0E-06 cd/sq.m. I therefore calculated photopic thresholds corresponding to luminance thresholds between these limits at 1.0E-01 intervals, and added my fitted power curve and also critical visual angle data from Blackwell, given as 4.67E-04 cd/sq.m.
Photopic Threshold Table of Results
Photopic Threshold Plot
My next task was to take into account the various visual and astronomical factors that modify this ideal model. For this I took as a suitable reference a paper published by Bradley E. Schaefer and commented upon by
Nils Olaf Carlin
Schaefer´s paper lists the various factors that effect the scotopic limiting telescopic magnitude, i.e. the faintest star visible using averted vision. But he also includes photopic threshold factors, not used in his TLM algorithm. He omitted one factor, the system Strehl Intensity.
I worked through his algorithm using Xcel, and having satisfied myself that my spreadsheet gave results that closely matched his, and a pair of Java engines based upon it, modified his algorithm with the photopic threshold factors.
Limiting Naked Eye Magnitude
Limiting Telescopic Magnitude
Schaefer´s telescopic limiting magnitude algorithm, as modified slightly, is derived from:
Telescope aperture: D (mm)
Telescopic Limiting Magnitude: m(v)=-2.5lgD(mm) -16.57+0.16(e-6)-(8.68-M(l))
Observer´s experience: e = 0 to 10
Limiting Zenithal Magnitude (LZM): M(l)=8.68-2.5lgFs-1.2kv-5lg(1+0.158Bs^05)
V sky brightness: Bs=136E-09 Lamberts (varies with LZM)
Telescopic Limiting Magnitude: m(v)=-16.57-2.5lgI*
Perceived star brightness: I*=I.Fb.Fe.Ft.Fp.Fa.Fv.Fsc.Fc.Fs.FST
Star brightness: I=C(1+(KB)^0.5)^2 lgC=-9.80; lgK=-1.90 if lgB<3.17
V sky brightness: Bs=B.Fb.Ft.Fp.Fa.FSC.Fm.Fc
Sky brightness @ (z) Bs(z=0)(1+(z)^2/2) nanoLamberts
Sky brightness: Bs'=26.331-2.5lgBs mag /sq arcsec
Definition of factors:
Fb = binocular factor = 2^0.5 =1.414 (used for monocular observation)
Fe = atmospheric extinction factor
Extinction coefficient: kv mag/airmass
Extinction factor: Fe=lg^-1(0.4(q.kv.secZ)) where q=1.2 if lgB<3.17
Ft = transmission factor =(t^n(1-Ds/D)^2))^-1 where n = number air-glass surfaces
Fp = pupil factor =(D/De.M)^2 where De=7exp(-0.5(A/100)^2)
Fa = aperture factor = =(De/D)^2
Fr = resolution factor: Fr=2*seeing disc(arcsecs)*M/900 when 2*seeing disc*M>900"arc
Fr=1.0 when 2*seeing disc*M<900"arc
FSC = Stiles Crawford factor: FSC=(1-(D/12.4M)^4)/(1-(De/12.4)^4) if De>D/M & lgB<3.17
Fc = Colour Index factor:Fc=lg^-1(-0.4(1-(B-V)/2)) if lgB<3.17
Fs = Visual Acuity factor:Fs=20:n/20:20=n/20 = Snellen Ratio
Fm = Magnification factor = M^2
FST = Strehl Intensity factor = I(ST) when D< Fried Length
The modification for a photopic threshold entails:
Star brightness: I=C(1+(KB)^0.5)^2 lgC=-9.80; lgK=-1.90 if lgB>3.17
Extinction factor: Fe=lg^-1(0.4(q.kv.secZ)) where q=1.0 if lgB>3.17
FSC = Stiles Crawford factor: FSC=(1-exp(-0.026(D/M)^2))/(1-exp(-0.026De)^2) if De>D/M & lgB>3.17
Fc = Colour Index factor: -2.5lgFc=0 if lgB>3.17
The sky brightness threshold corresponds to 18.4 mag / sq.arcsec and a
Photopic Telescopic Limiting Magnitude: m(v)=-19.57-2.5lgI* when factored to Peterson´s threshold of 8.7mv in a 75mm telescope. A photopic threshold 3 mags above the scotopic limit.
Telescopic Limiting Magnitude Spreadsheet
Telescopic Photopic Limiting Magnitude Spreadsheet
Photopic Threshold Chart
The model has input parameters Bs'=21 mag/sq.arcsec; kv=0.3 mags/air mass;
seeing disc 1".0 arc & M=x70.
The modification to Schaefer´s TLM algorithm gives photopic thresholds fairly well in line with naked eye, binocular and small telescope observations, but seems too optimistic for apertures greater than 100mm. Neither is it consistent with the log-linear law m(v)=5lgD(mm)-0.25 derived from relative light grasp, neglecting losses.
The reason for this could be the rather ill defined transition from photopic to mesopic vision. Another possible cause of the steady increase in threshold luminance with increasing aperture is Seeing. A seeing disc measured at say 3 arcsecs in a 100mm aperture may be bigger in a larger aperture. I have factored in Strehl Intensity as a measure of optical quality, but not something called the Fried length, which effects the seeing cell size. When telescope aperture equals or is less than the Fried length, stars do not puff up but flit about the field of view. The Fried length in the UK is typically 4-inches to 8-inches. Bigger apertures than the Fried length see stars swell into seeing discs. You don't see the Airy disc or the PSF (diffraction pattern). Light is therefore spread out and not confined to the PSF, making the equivalent star fainter than it otherwise would appear.
The only way to pin the photopic threshold function down is by analysis of actual observations. Those interested in participating should select stars whose magntiudes are roughly 3mags to 4mags above the threshold of averted vision, and comment on whether or not it can be held continuously in direct vision, and whether or not any colour could be perceived.
e-mail your observations to: chris lord