Radiometry and photometry in astronomy

By Paul Schlyter, Stockholm, Sweden
email: pausch@stjarnhimlen.se or WWW: https://stjarnhimlen.se/

Break out of a frame

1997-09-10 First version
1998-03-29 Added a paragraph about photography + some minor corrections
1998-05-06 Added Jeff Medkeff's paragraph about visual observing and the physiology of the eye. Also added several new references.
1998-06-10 Some corrections to the relation between candela and lumen. Also added a paragraph about Lambert's law.
2000-03-11 Added a table of horizontal illumination from sunlight during clear weather at different solar altitudes above the horizon.
2000-11-12 Added a link to another radiometry/photometry FAQ + removed some dead links.
2000-11-15 Added a list of the brightness of the radio sky. Also added something about conversion from radiometric from photometric units.
2001-01-18 Corrected a confusion of luminance and illuminance at a few places. Also added a table of the correspondence between radiometric and photometric units.
2001-02-02 Added Tom Polakis paragraph about big telescopes and light pollution. Also rearranged the paragrahs at the end.
2003-04-24 Corrected a typo: 1 stilb isn't 1 lumen/cm/ster but 1 lumen/cm2/ster
2004-02-18 Added a reference to the Explanatory Supplement
2004-06-16 Updated or removed outdated links
2006-12-08 Adjusted the luminance of a candle down from 6E+5 to 2E+4 cd/m2 (thanks to Jan Hollan for measuring this)
2009-03-18 Corrected a small error regarding units for photon flux
2010-04-13 Corrected an error regarding converting between watts and lumens
2017-03-05 Updated numerous obsolete links and removed those for which no update could be found. Corrected a few typos as well.
2023-01-02 Updated a few obsolete links.

1. What is photometry?

Photometry is the measurement of the intensity of electromagnetic radiation in photometric units, like lumes/lux/etc, or magnitudes. The measurement is done with an instrument with a limited, and carefully calibrated, spectral response. Lux/lumens/etc uses the standard response curve for the eye, while astronomical photometry uses standard filters, with standard spectral response curves, e.g. UBVRI photometry.

2. What is radiometry?

Radiometry is the measurement of the intensity of electromagnetic radiation in absolute units, such as watts/m2, or photons/sterad/sec.

3. What is solid angle?

A plane angle is well-known: one full revolution is 360 degrees or 2*pi radians. The solid angle expands this concept over the surface of a sphere.

4. What is a steradian? What is a square degree?

These are units of solid angle, just like radians and degrees are units of (plane) angle.

The full sphere has a solid angle of 4*pi steradians. Half the sphere (e.g. the sky above the horizon) is 2*pi steradians, etc.

Another unit of solid angle is the square degree. It's an imaginary square on e.g. the celestial sphere, having a side of 1 degree. How many square degrees is the entire sphere? Well, the circumference of the sphere is of course 360 degrees. Divide by 2*pi to get its radius: r = 180/pi degrees. Yep! The celestial sphere has an undetermined radius in e.g. meters, but a well-determined radius in degrees! Finally compute the area of the sphere as 4*pi*r^2 and set r=180/pi degrees, to get: 4*180^2/pi = 41252.961... square degrees.

Since the entire sphere is 4*180^2/pi square degrees = 4*pi steradians, it follows that one steradian is (180/pi)^2 square degrees. This appears natural, since one radian is 180/pi degrees.

5. What units are used in photometry and radiometry?

There is a confusing set of units for radiation. Some measure the flux of energy, some measure "brightness", and some measure the flux of photons. Some measure the total flux incident on some surface, from all directions, while others measure flux only from a specified direction.

But let's try to pick this apart. First we must learn to distinguish between Intensity, i.e. flux from a given direction, and Flux, i.e. the total flux from all directions:

I: intensity (specific intensity): the flux of radiation at a given point in a given direction across unit surface normal to that direction per unit time and per unit solid angle.

F: flux of radiation through unit surface: surface flux or flux density. 
              4*pi
             /
            |
      F  =  |  I cos(theta) dw 
            |
           /
         0
where theta = angle to normal of area, and w = solid angle


These are the basics. Now we can elaborate somewhat:

Emittance: flux of radiation emitted from a unit surface

Irradiance: incoming flux of radiation falling on a unit surface, from the entire hemisphere.

              2*pi
             /
            |
      F  =  |  I cos(theta) dw 
            |
           /
          0
For isotropic raditaion, i.e. if I is independent of the direction:
                   F = pi*I
(note: the factor is pi instead of 2*pi, because the integral of cos(theta) over the entire half-sphere becomes pi)

Radiance is another name for I (intensity, or specific intensity).

6. What units are used in photometry?

Now, let's review various photometric and radiometric units. We'll start with the common photometric units, which assumes a detector having the same spectral response as the human eye in daylight vision.

6.1. The Lumen

The basic SI photometric unit is the Lumen, which is the flux of one Candela into one steradian. One watt of monochromatic radiation at 5550 Angstroms is defined as 680 lumens. This refers to the daylight vision of the eye. (There is also a "scotopic lumen" (which refers to the night vision of the eye), where 1 Watt of 5100 Angstrom radiation is defined as 1720 lumens.)

6.2. The Candela

One Candela was formerly defined as 1/60 of the luminous intensity of 1 projected cm2 black body at the temperature of melting platinum (2044 K). However, the candela has now been re-defined in terms of lumens: 1 Candela = 1 Lumen per steradian. Note that the Candela is defined as intrinsic brightness as seen from one particular direction. Most light sources will have an intrinsic brightness, expressed in Candelas, which varies with direction.

6.3. Lambert's law

A Lambertian emitter is a light source which follows Lambert's law, which says that the surface brightness is independent of direction. The surface brightness can be expressed in Candelas per projected square meter (i.e. per square meter perpendicular to the direction of view). An illuminated perfect diffusor is one example of a Lambertian light source. The brightness of a flat Lambertian light source, in Candelas, is: 
        IB  =  PIB  *  cos(theta)
where IB = Intrinsic Brightness from any direction less than 90 degrees from the normal to the surface, PIB = Perpendicular Intrinsic Brightness i.e. the IB as seen from a direction normal to the surface), and theta is the angle of the direction of view from the normal to the surface. This is the "cosine factor" which applies to all Lambertian emitters, and which also serves as a good approximation to many real emitters.

Since 1 lumen (lm) is the flux of 1 candela (cd) into 1 steradiam, the flux from a flat Lambertian emitter having a PIB of 1 cd into the entire half-sphere, becomes pi lumen (not 2*pi lumen, because of the cosine factor).

A sphere which is a perfect diffusor, and which is illuminated only by one point light source at infinite distance, will (other things being equal) have a brightness exactly pi times smaller when "half" compared to when "full". Cloud-covered Venus gets farily close to this ideal. Our Moon deviates a lot from it: the Half Moon is only about 1/10 as bright at the Full Moon.

6.4. Other photometric units

We also have some other photometric units:

SI units:

1 lm (lumen) = flux from 1 cd into 1 ster
1 cd = 1 lumen/ster
1 nit = 1 cd/m2 (= 1E-4 stilb)
1 apostilb = 1/pi cm/m2 = 1 lm/m2 for perfect diffusor = 1E-4 lambert
1 lux = 1 lm/m2 (= 1E-4 phot = 1 "metre-candle")
CGS units:

1 stilb = 1 cd/cm2 = pi lambert = 1 lumen/cm2/ster
1 lambert = 1/pi cd/cm2 = 1 lumen/cm2 for perfect duffusor
1 milli-lambert = 10/pi cd/m2
1 skot = 1 milli-blondel = 1E-3/pi nit = 1E-7 lambert
1 phot = 1 lumen/cm2
English units:

1 foot-candle = 10.76 lux = 1 lumen/ft2
1 cd/ft2 = 10.76 nit
1 foot-lambert = 1/pi cd/ft2 = 3.426 nit
The factor 1/pi which enters sometimes is due to the difference between isotropic and lambertian radiation. Isotropic implies a spherical source which radiates the same in all directions. Lambertian refers to radiation from a flat surface (an active radiator or a passive reflector) which appears to have the same "surface brightness" (i.e. radiance) from any direction; the distribution of such radiation follows Lambert's law. The ratio of W/m2 to W/m2 sr of a lambertian surface is a factor of pi, not 2*pi, due to the cosine factor in the integration over the hemisphere.

7. What units are used in astronomical photometry?

The well-known magnitude scale of course, which has been calibrated using standard stars which (hopefully) do not vary in brightness.

But how does the astronomical magnitude scale relate to other photometric units? Here we assume V magnitudes, unless otherwise noted, which are at least approximately convertible to lumes, candelas, and lux'es.


1 mv=0 star outside Earth's atmosphere  = 2.54E-6 lux = 2.54E-10 phot

1 mv=0 star per sq degree outside Earth's atmosphere    = 0.84E-2 nit
                                                        = 0.84E-6 stilb

1 mv=0 star per sq degree inside clear unit airmass     = 0.69E-2 nit
                                                        = 0.69E-6 stilb

(1 clear unit airmass transmits 82% in the visual, i.e. it dims 0.2 magnitudes)

One star, Mv=0 outside Earth's atmosphere  =  2.45E+29 cd

Apparent magnitude is thus an irradiance or illuminance, i.e. incident flux per unit area, from all directions. Of course a star is a point light source, and the incident light is only from one direction.

Apparent magnitude per square degree is a radiance, luminance, intensity, or "specific intensity". This is sometimes also called "surface brightness".

Another unit for intensity is: 1 S10vis = the intensity (surface brightness) corresponding to one star of 10th (visual) magnitude per square degree of the sky.

1 S10vis = 0.69E-6 nit = 0.69E-10 stilb (inside clear unit airmass)
Still another unit for intensity is magnitudes per square arcsec, which is the magnitude at which each square arcsec of the extended light source shines:

 Magnitudes per     S10vis               Nit = Candelas/m2
  square arcsec                 inside unit airmass   outside atmosphere

         0          1.30E+11         9.0E+4              10.9E+4
        +5          1.30E+9          9.0E+2              10.9E+2
       +10          1.30E+7          9.0                 10.9
       +15          1.30E+5          9.0E-2              10.9E-2
       +20          1.30E+3          9.0E-4              10.9E-4
       +25          1.30E+1          9.0E-6              10.9E-6
Absolute magnitude is a total flux, expressed in e.g. candela, or lumens.

Only visual magnitudes can be converted to photometric units. U, B, R or I magnitudes are not easily convertible to luxes, lumens and friends, because of the different wavelengths intervals used. The conversion factors would be strongly dependent on e.g. the temperature of the blackbody radiation or, more generally, the spectral distribution of the radiation. The conversion factors between V magnitudes and photometric units are only slightly dependent on the spectral distribution of the radiation.

Bolometric magnitudes can of course be converted to energy flux: One star of Mbol=0 radiates 2.97E+28 Watts.

8. What units are used in radiometry?

Here we're not interested in the photometric response of some detector with a well-known passband (e.g. the human eye, or some astronomical photometer). Instead we want to know the strength of the radiation in absolute units: watts etc. Thus we have:

Radiance, intensity or specific intensity:

   W m-2 ster-1  [A-1]              SI unit
   erg cm-2 s-1 ster-1 [A-1]        CGS unit
   photons cm-2 s-1 ster-1 [A-1]    Photon flux, CGS units
Irradiance/emittance, or flux:

   W m-2 [A-1]               SI unit
   erg cm-2 s-1 [A-1]        CGS unit
   photons cm-2 s-1 [A-1]    Photon flux, CGS units
Note the [A-1] within brackets. Fluxes and intensities can be total (summed over all wavelengths) or monochromatic ("per Angstrom" or "per nanometer").

We also have a somewhat oddball unit, the Rayleigh, which often is used in atmospheric radiometry where monochromatic radiation is involved, e.g. in aurora research:

1 Rayleigh =  1E+6/(4*pi) photons cm-2 s-1 ster-1
In Radio Astronomy, the unit Jansky is often used as a measure of irradiance at a specific wavelength, and is the radio astronomer's equivalence to stellar magnitudes. The Jansky is defined as: 
1 Jansky  =  1E-26 W m-2 Hz-1

9. How do I convert between radiometric and photometric units?

This is often not easily done. The conversion depends strongly on the spectral distribution of the light. At 5300 Angstroms (i.e. for monochromatic radiation at 5300 A, or narrow-band radiation around 5300 A) these conversion factors apply:

    1 erg/cm2/s/ster/A = 7.62E8 S10vis = 3.35E6 Rayleigh/A

    1 S10vis = 4.40E-3 Rayleigh/A  =  1.31E-9 erg/cm2/s/ster/A

    1 Rayleigh/A = 227 S10vis = 2.98E-7 erg/cm2/s/ster/A
There is a correspondence between radiometric and photometric units:
Photometric units              Radiometric units        Astronomical units
=================              =================        ==================

Luminous flux                  Power                    Absolute magnitude
(lumen)                        (watts)                        M

Luminous intensity             ----------               ---------
(candela = lumen/ster)         (watts/ster)             ---------

Illuminance                    Irradiance/emittance     Apparent magnitude
(lux = lumen/m2)               (watts/m2)                     m

Luminance, Intensity           Radiance, Intensity      "Surface brightness"
(nit = lumen/ster/m2 = cd/m2)  (watts/ster/m2)          (m/arcsec2, S10vis)

Photometric units are obtained by integrating the corresponding radiometric unit multiplied with the wavelength sensitivity of the eye, over all visible wavelengths, and then multiplying by a suitable factor. Detailed tables of the wavelength response of the human eye be obtained from the Color Vision Lab at UCSD at http://cvision.ucsd.edu/index.htm. A good approixmation to these tables can be obtained from these functions (where lambda is given in micrometers):

Photopic vision ("Day vision"): cones active. To be used above 3 cd/m2 
   V(lambda) = 1.019 * exp( -285.4*(lambda-0.559)^2  )
Scotopic vision ("Night vision"): only rods active. To be used below 3 cd/m2 
   V(lambda) = 0.992 * exp( -321.9*(lambda-0.503)^2  )
The conversion is performed using the integral: 
                 infinity
                    /
                   |
        Xp  =  K * |  Xr(lambda) * V(lambda) * dlambda
                   |
                  /
                zero
where Xp is the photometric quantity, Xr(lamnda) the corresponding radiometric quantity and V(lambda) the sensitivity function of the eye (photopic or scotopic, depending in the light level). K should be chosen so that one watt corresponds to 683 lumens at a frequency of 540 THz (= a wavelength of 555 nanometers) for photopic vision, and 1700 lumens at 507 nanometers for scotopic vision. Since V(lambda) equals unity at these peak wavelengths, K simply becomes 683 for photopic vision and 1700 for scotopic vision.

In principle, the integration should be performed from zero to infinity. In practice, it's sufficient to integrate from 360 to 830 nanometers.

Naturally, photometric quantities are meaningful only to sources of visible light. It's meaningless to try to figure out how many lumens e.g. a radio source emits, because it doesn't emit in the visual wavelength range.

10. How bright are natural light sources?

The table below shows common illumination levels of natural light sources:

                                 Stellar magnitude    Illuminance
                                                         Lux

Sun overhead                            -26.7          130000
Full daylight (not direct sun)      -24 to -25          10000-25000
Overcast day                            -21              1000
Very dark overcast day                  -19               100
Twilight                                -16                10
Deep twilight                           -14                 1
1 Candela at 1 meter distance           -13.9               1.00
Full Moon overhead                      -12.5               0.267
Total starlight + airglow                -6                 2E-3
Total starlight only                     -5                 2E-4
Venus at brightest                       -4.3               1.4E-4
Total starlight at overcast night        -4                 1E-4
Sirius                                   -1.4               1E-5
0th-mag star                              0                 2.7E-6
1st-mag star                             +1                 1.0E-6
6th-mag star                             +6                 1.0E-8


Solar illumination on horizontal surface at
various solar altitudes above the horizon

    Solar altit     Illumination
     degrees    log10 Lux     Lux

      90.0         5.11    129000
      80.0         5.09    122000
      70.0         5.06    114000
      60.0         5.01    103000
      50.0         4.94     87400
      45.0         4.89     77800
      40.0         4.83     67500
      35.0         4.75     56900
      30.0         4.67     46300
      25.0         4.56     36300
      20.0         4.44     27400
      15.0         4.28     19200
      14.0         4.25     17600
      13.0         4.20     15900
      12.0         4.16     14300
      11.0         4.10     12700
      10.0         4.05     11100
       9.5         4.02     10400
       9.0         3.98      9610
       8.5         3.95      8880
       8.0         3.91      8170
       7.5         3.87      7490
       7.0         3.84      6840
       6.5         3.79      6220
       6.0         3.75      5620
       5.5         3.70      5060
       5.0         3.66      4540
       4.5         3.60      4010
       4.0         3.55      3550
       3.5         3.49      3110
       3.0         3.43      2690
       2.5         3.36      2290
       2.0         3.28      1920
       1.5         3.20      1580
       1.0         3.10      1270
       0.5         3.00       994
       0.0         2.88       759
      -0.5         2.75       562
      -1.0         2.61       405
      -1.5         2.45       281
      -2.0         2.28       189
      -2.5         2.09       124
      -3.0         1.90        79.1
      -3.5         1.69        49.2
      -4.0         1.48        29.9
      -4.5         1.25        17.8
      -5.0         1.02        10.4
      -5.5         0.78         5.99
      -6.0         0.53         3.41
      -6.5         0.29         1.93
      -7.0         0.04         1.09
      -7.5        -0.21         0.613
      -8.0        -0.46         0.348
      -8.5        -0.70         0.200
      -9.0        -0.93         0.116
      -9.5        -1.16         0.0692
     -10.0        -1.38         0.0421
     -10.5        -1.58         0.0264
     -11.0        -1.77         0.0171
     -11.5        -1.94         0.0115
     -12.0        -2.09         0.00806
     -12.5        -2.22         0.00597
     -13.0        -2.34         0.00456
     -13.5        -2.44         0.00360
     -14.0        -2.54         0.00292
     -14.5        -2.62         0.00241
     -15.0        -2.69         0.00202
     -15.5        -2.77         0.00171
     -16.0        -2.84         0.00144
     -16.5        -2.92         0.00121
     -17.0        -3.00         0.00100
     -17.5        -3.09         0.000815
     -18.0        -3.19         0.000645
The table below gives approximate intensities (surface brightnesses) of some natural light sources:

                                    Luminance       Magnitudes per square
                                    Nit = cd/m2       arcsec   arcmin

Sun                                   3E+9            -10.7    -19.6
Venus (max elong)                     15000            +1.9    -7
Clear daytime sky (at horizon)        10000            +3      -6
Full Moon                              6000            +3.6    -5.3
Mars at perihelion                     4000            +3.9    -5.0
Overcast daytime sky (at horizon)      1000            +5      -4
Jupiter                                 800            +5.7    -3.2
Saturn                                  700            +5.9    -3.0
Heavy daytime overcast (at horiz)       100            +8      -1
Uranus                                   60            +8.6    -0.3
Neptune                                  30            +9.3    +0.4
Sunset at horizon, overcast              10           +10      +1
Clear sky 15 min after sunset (horiz)     1           +13      +4
Clear sky 30 min after sunset (horiz)     0.1         +15      +6
Fairly bright moonlight (at horizon)      0.01        +18      +9
Moonless, clear night sky (at horiz)      1E-3        +20     +11
Moonless, overcast night sky (at horiz)   1E-4        +23     +14
Dark country sky between stars (zenith)   3E-5        +24     +15
The table below gives approximate intensities (surface brightnesses) of some artificial light sources:

                                    Luminance       Magnitudes per square
                                    Nit = cd/m2       arcsec   arcmin

Arc crater (plain carbon)             1.6E+9          -10      -19
Tungsten lamp filament                8E+6             -4      -13
High-pressure mercury vapor lamp      1.5E+6           -2      -11
Sodium vapor lamp                     7E+5             -1.6    -10.5
Acetylene burner                      1.1E+5           +0.4     -8.5
Candle                                2E+4             +2.3     -6.6

11. How bright is the night sky?

The table below give approximate intensities of components of the night sky:

                                       Luminance     Magnitudes per square
                                        S10vis          arcsec   arcmin

Airglow (not OH in IR)                     50           +23.5    +14.6
Aurora
    IBC I                                  70           +23      +14
    IBC II                                700           +21      +12
    IBC III                              7000           +18       +9
    IBC IV                              70000           +16       +7

The zodiacal light:
    Ecliptic pole                          78           +23.1    +14.2
    Ecliptic
      180 deg from Sun (Gegenschein)      205           +22.0    +13.1
      140 deg from Sun                    164           +22.2    +13.3
       90 deg from Sun                    250           +21.8    +12.9
       60 deg from Sun                    500           +21.0    +12.1
       30 deg from Sun                   2330           +19.4    +10.5

The Milky Way - integrated starlight:
    Galactic pole                          30           +24      +15
    Galactic equator               160 to 640      +21 to +22   +12 to +13

Diffuse galactic light (estimate):          9           +25      +16

Extragalactic light (estimate):           0.9           +28      +19

12. How bright is the radio sky?

The table below give approximate intensities of some of the strongest natural radio sources in the sky (1 Jansky = 1E-26 W m-2 Hz-1):

                       Flux (in Janskys) at:
                 10 MHz  100 MHz    1 GHz   10 GHz

Cassiopeia A     100000    19500     3300     1000      Supernova remnant
Cygnus A          70000    13800     2340      300      Radio galaxy
Sagittarius A               4000     2000               Center of our galaxy
Centaurus A                 3000     2000               Peculiar galaxy
Virgo A           10000     1800      250      100      M87, galaxy with "jet"
Taurus A                    1700      955               M1 - Crab Nebula - SN remnant

Sun, quiet          100   10 000  100 000     1E+6
Sun, disturbed     1E+7     1E+8     1E+8     1E+8

Moon                0.1        3       50    10000

Jupiter            1E+7        0        0       50

Sky background     2E+7     3E+6     1E+6     3E+5
Source: "Radio Astronomy" by Kraus, McGraw-Hill 1966

13. Photography and photometry

In photography we let a certain amount of illumination fall upon a film or a plate, and after development we get a blackening of the film.

13.1 Some fundamental definitions

During exposure:
H     = Illumination in lux-seconds (1 lux-sec = 1 lux during 1 sec)
After development:

Phi0  = incident light flux
Phi   = transmitted light flux

T     =  Phi/Phi0  = transmission (never larger than 1.0)
1/T   =  Phi0/Phi  = opacity (never smaller than 1.0)

S     =  log10(1/T)  = blackening (never smaller than 0.0)
Ss    =  minimum blackening, i.e. blackening of unexposed and developed film

13.2 Definition of film sensitivity

S01   =  Ss + 0.1
S09   =  Ss + 0.9

Hm    =  illumination yielding a blackening of S01, lux-seconds
Hn    =  HM * 10^1.3
The film should be developed such that an illumination of Hn yields a blackening of S09. Hm corresponds to "minimum blackening" S01, i.e. the minimum required for the blackening to be visible at all.

DIN number  =  10 * log10(1/Hm)
ASA aritm   =  0.8 / Hm
ASA log     =  log2( 0.24 / Hm )
Hm = illumination (lux-seconds) needed for "mimimum blackening".

(There's also a Russian unit - GOST - which is based on a point Ss+0.2)

Today film sensitivities are given in ISO, which is the same as ASA aritm.

ISO = International Standards Organisation
ASA = American Standards Association (nowadays ANSI)
DIN = Deutsche Industrie Normen (German Industry Norms)

Approximate conversion of film sensitivities between different units:

DIN       =  10 * log10(ISO) + 1
ASA aritm =  ISO
ASA log   =  log2(0.3 * ISO)
GOST      =  0.8 * ISO

13.3 Worked example: computing exposure time of celestial object

I want to photograph the Orion Nebula, using 200 ISO film, a 6-inch (15 cm) telescope of focal length 1 meter. I'm going to use prime focus. What exposure time should I use?

First a caution: the exposure time computed here is only very approximate, because the magnitude and apparent size of the nebula are both only approximate, and also because at long exposure times film often lose sensitivity due to reciprocity failure. Thus one should consider our computed values here as only an approximate guide, and when photographing the nebula one should bracket with exposures both below and above this value.

Nevertheless it's interesting to carry out the computation, because it will illuminate conversions from magnitudes to lumens, and then use ISO numbers of the film to compute exposure times. So here we go:

First let's assume the Orion Nebula shines at 4th magnitude, having an apparent diameter of 10 arc minutes.

One mv=0 star illuminates at 2.54E-6 lux outside the Earth's atmosphere. The Orion nebula shines at 4th magnitude, i.e. it illuminates at
    2.54E-6 * 10^( -0.4 * 4 )  =  6.38E-8 lux
If the Orion nebula is 30 deg above the horizon, the light will have to pass 2 air masses, where each air mass transmits 82% of the light. Thus at the ground the Orion nebula illmuminates at
    6.38E-8 * 0.82^2  =  4.29E-8  lux
Our reflector has an entrance 15 cm across. If we assume a central obstruction of 20% the entrance, we get an entrance area of 170 cm^2

At a focal length of 1 meter the Orion nebula, 10 arcmin across, will be imaged at a size of 0.291 cm, i.e. an area of 0.085 cm^2

Thus the 170 cm^2 entering the telescope will be imaged on 0.085 cm^2, increasing the illumination by a factor of 170/0.085 = 2000

If we assume 90% of the light entering the telescope will be transmitted, we find that the illumination at the film will be:
     4.29E-8 lux * 2000 * 0.9  =  7.7E-5 lux
A 200 ISO film will require an illumination for minimum blackening:
        Hm  =  0.8 / 200  =  0.004 lux-seconds
For "normal" blackening more will be needed -- let's assume 10 times more illumination will be needed - this is 0.04 lux-seconds.

Thus we have an illumination at the film of 7.7E-5 lux, and we need 0.04 lux-seconds. This requires approx. 520 seconds or about 9 minutes exposure time.

14. Visual Observing and Physiology of the Eye

By Jeff Medkeff (1968-2008)

Q: How does the eye work?
A: Simply put, light passes through the eye's lens, which focuses it onto a curved surface called the retina which is covered with light-sensitive cells called rods and cones.

Q: Is it true that the rods and cones are on the "wrong" side of the retina?
A: Almost. The rods and cones are all underneath the eye's bipolar cells, ganglion cells, and a mat of nerve fibers, which are at the front surface of the retina. All this stuff, along with the vitreous humor, lens, and cornea, absorbs about 50 per cent of the light entering at the eye's lens. Fortunately, there is plenty left to stimulate the sensitive cells.

Q: What is the difference between photopic and scotopic vision?
A: Photopic vision is used in bright-light environments. Cells on the retina called "cones" are used in this vision, and these cells are the only ones which can detect color. The concentration of cones on the retina co-incides with direct vision, so looking directly at a bright object will give the best view.

Scotopic vision is dark adapted vision. Rods on the retina are the eye's low-light detectors, but they are color-blind. The concentration of rods is greatest at 20 degrees off the axis of direct vision. The astronomical technique of "averted vision" takes advantage of this physiology.

Q: How long does it take to get dark adapted?
A: Probably longer than you think. The cones achieve dark adaptation fairly quickly - after ten minutes, the cones will cease to become any more sensitive to light. The rods take much longer, with large gains in sensitivity still being made up to 45 minutes after the beginning of dark adaptation, and small gains continuing after even four or five hours of adaptation.

Q: Does dark adaptation change with age?
A: Yes. Unfortunately, in most people the eye loses its ability to become dark adapted as we age. The log of threshold intensity while dark adapted is .5 for an average teenager, while it is up around 3 for an eighty year old. This means that the teenager can see more than two magnitudes dimmer than the eighty year old, assuming dark adaptation is the only factor influencing the limits of vision (it is not).

Q: What about pupils? Don't they get smaller with age?
A: The pupils of the eye do tend to lose their ability to open widely as we age. This effect, however, is minuscule in relation to the chemical effects of dark adaptation on the retina. If your pupils don't open as large as you wish, simply select a smaller exit pupil on your optical instruments.

Q: You mentioned chemical effects of dark adaptation. What do you mean?
A: As the eye dark adapts, a chemical called rhodopsin (or "visual purple") plays a more and more important part in vision. This chemical is most sensitive to light at 5100 angstroms. The greater use of rhodopsin by the eye when dark adapted is more important than pupillary size - far more important.

Q: What are some things that can affect how sensitive my vision is?
A: Blood sugar, smoking, alcohol, altitude above sea level, diet, pre-exposure to sunlight, dehydration, and general fitness and health all contribute.

High blood sugar levels can counteract to a small degree a loss of retinal sensitivity, but only for a short time and with longer-lasting degradation in the end. In other words, it isn't worth eating candy bars at a star party in order to see better. But if you have low blood sugar, eating something to normalize it will certainly help your vision.

Smoking and alcohol degrade vision to varying degrees. Some smokers have such poor nighttime vision that it is dangerous for them to drive, while others notice little difference when away from the telescope. Alcohol will reduce scotopic sensitivity and photopic acuity, too. In an informal experiment called the "Killians Red Ale Study", several people in Arizona observed stars with the naked eye and noticed a degradation of two magnitudes in naked-eye limits after only two (12-oz) beers in two hours (as measured against non-drinking "controls"). The beer used was an American beer, with a low to average amount of alcohol - some popular European and Japanese beers are somewhat stronger in this regard.

Altitude above sea level affects the availability of oxygen to the body. When the body doesn't have enough oxygen, the eyes lose sensitivity quickly. Too much oxygen, however, can also be permanently damaging to the retina, so care should be taken when using supplemental oxygen at high altitudes.

Pre-exposure to ambient sunlight can interfere with the retina's ability to dark adapt for several days. This is because bright sunlight removes most of the rhodopsin from the rods. This effect can be attenuated by wearing yellow-tinted sunglasses. (See also "Q: You mentioned chemical effects of dark adaptation. What do you mean?")

Diet and general fitness also contribute to retinal sensitivity. Those on junk food diets or who get little exercise are generally not as sensitive with their eyes as others who do.

The eyes seem to detect dehydration well before one gets thirsty. When in doubt, have a glass of water and see if that helps your vision.

Q: What about bilberry?
A: It has never worked for me. It seems to be essentially a "wive's tale".

Q: I did some observing at a dark country location the other night. It was really dark there! I probably had really good dark adaptation, don't you think?
A: Nope. The darkest country sky is still pretty bright, at about 22 magnitudes per square arc second, due to solar system dust illuminated by the sun and natural skyglow. Over the whole sky, this alone can interfere with dark adaptation. Add in the effects of a starry sky, especially when the milky way is above the horizon, and this will interfere with dark adaptation to the extent of inhibiting your eyes by one magnitude or more. If there is any light pollution at all - either in the form of direct but distant lighting, or a dome of skyglow from a distant city - your dark adaptation will suffer even more.

The best dark adaptation is achieved in domed observatories, or when exceptional measures are taken to plunge the eyes into prolonged darkness.

Q: But the sky was really black there!
A: No it wasn't. Next time, hold up your hand against the sky. You will see a clear outline, and the sky will have a dull gray appearance.

Q: What is the faintest star I can see with my naked eye - when dark adapted as well as I can be?
A: It depends. There are probably no absolute limits, and variations between observers are great. That means that what one person sees cannot always be used to decide what another person can see.

- From light polluted skies, frequently the limit will be at 2.5 to 4th magnitude, depending on the severity of the problem.

- From average suburban skies, fifth or sixth magnitude seems to be a good rule of thumb, if not taken too hard and fast.

Most observers from a dark country sky on an excellent night can see somewhere a little deeper than 7th magnitude. Local conditions at the observing site will affect this number greatly, of course. In particular, altitude will make the magnitude limit deeper, until the observers encounter diminishing returns due to oxygen deprivation.

There have been a few studies where test subjects have seen stars of 8 and 8.2 magnitudes. The conditions of the testing were "blind", so to speak, and observing conditions were excellent. If anything can be taken as an absolute limit, this is probably it.

Q: Ok, what are the magnitude limits for various sizes of telescopes?
A: There are no hard and fast "limits". But most of the tables quoted in various books should be discarded in favor of a newer theory by Bradley Schaefer published in 1990PASP..102..212S. But by far the best thing to do is determine this by viewing photometric sequences telescopically.

Q: Where can I get deep photometric sequences to test my magnitude limits?
A: Also, there are some sequences in Sky & Telescope, January 1984, p. 28ff. These are considered to be very rough, and should not be used as very accurate indicators.

Above all, avoid the magnitudes listed in the Hubble Guide Star Catalog or the USNO A1.0 (or its subset, SA1.0). These magnitudes are absolutely useless for this purpose.

Q: Why can two stars with the same V magnitude appear to be slightly different in brightness?
A: The V magnitude band is calibrated to the spectral response of the average human eye' photopic vision. Photopic, or daylight, vision utilizes the cones, whose sensitivity peaks at a wavelength of 5600 angstroms, and drops to half sensitivity around 6100 and 4900 angstroms. Dark adapted vision peaks at 5100 angstroms and is half-sensitive at about 4200 and 5500 angstroms. As a result, stars with slightly different colors (different spectral types, or different photometric colors as expressed by a B-V magnitude) can appear to differ in apparent brightness in the telescope.

Schaefer in "Telescope limiting magnitudes" claims the scotopic magnitude can be estimated by adding a factor (B-V)/2 - 1 to the V magnitude [(B-V)=color index] - that gives a scotopic magnitude of (B/2 + V/2 - 1). A red star with (B-V)=2 will have the same visual as scotopic magnitudes (Betelgeuse has (B-V)=1.86), an A0 star where (B-V)=0 will seem -1 mag brighter scotopic. Brian Skiff uses the formula m(vis) = V + 0.2(B-V) in his "Observational Data for Galactic Globular Clusters", see http://www.ngcic.com/gctext.htm

Also, such effects as the Purkinje Effect can change the apparent brightness of certain colored stars to a degree much greater than the star's B-V alone could account. (See "What is the Purkinje Effect" below.)

Q: What is the Purkinje Effect?
A: The Purkinje effect is an interesting and insidious effect relating to the perceived brightness of red stars. The effect can be understood by imagining the estimation of the brightness of a red star and a white star. Suppose the observer estimates the red star and the white star to be equally bright. If the brightness of each star is then somehow doubled, the observer will estimate the red star to be up to .5 magnitudes brighter than the white one. Thus, the phenomenon speaks directly to the estimation of the brightness of red objects, or of an object of any color if estimated in relation to red comparison objects.

The cause is apparently the runaway excitation of the cones by red objects. It is possible that other neurological effects also contribute.

To a certain extent this phenomenon can be attenuated by the observer through the use of averted vision when making such estimates.

Q: What is the Troxler Phenomenon?
A: It is a neurological trait whereby the brain tends to ignore an image that remains stationary on the retina for any length of time. There is an unconscious reflex that moves the eye slightly about ten times per second to avoid it. During astronomical observations, the low light levels and the concentration of the observer tend to inhibit this reflex. The observer will notice this as a fading of the object they are observing. Astrophotographers concentrating on guide stars often notice this phenomenon, but it is not limited to their experience only.

The Troxler phenomenon can be avoided by consciously moving the eye about periodically while observing or by sweeping the telescope from side to side slightly while observing.

Q: What is the "integration time" of the rods?
A: The rods can "collect" more and more light over a period of perhaps up to 15 to 20 seconds. This means that if the eye is held still for that long, an image can 'build up' in a rough analogy with the way that photographic film works.

However, long before the physiological limit of the rods is reached, the brain will rebel and the textbook Troxler phenomenon effects will dominate what is seen. The point at which this happens varies greatly from observer to observer, and there are strong indications that the suppression of the Troxler phenomenon can be learned. In any case, an observer will have to determine for themselves how long they can last before the Troxler phenomenon becomes bothersome. For the most remarkable observers, ten or twelve seconds seems to be the limit; typical cases are probably only a few seconds.

Q: Why does moving the telescope a little bit sometimes show a faint galaxy where I could not see it before?
A: There is a tendency for the eye to notice that which moves. The trick of gently sweeping the telescope back and forth that experienced observers have learned has its roots in the eye's physiology.

Q: I should use wide field, low power eyepieces to see faint deep sky objects, right?
A: Wrong. The eye is most sensitive to faint astronomical objects when they are magnified to an apparent size of one to three degrees. In many cases, this means that medium or high power oculars need to be used to have the best chance of seeing something, especially for the smaller galaxies and most planetary nebulae.

Q: I see that some observers are now using dim green or blue flashlights to preserve dark adaptation. Is this a good idea?
A: No. There are various reasons for this. Red light not only affects the eye's dark adaptation less at a given intensity, but it also affects the eye's ability to re-adapt the least. And a much brighter red can be used before degradation becomes extreme. There are numerous studies that show that red light is best for preserving dark adaptation, arguments based on green airplane cabin lighting notwithstanding. See: Q: How bright should my red flashlight be, then?
A: If you can see the color when you shine it on a chart page, it is too bright.

15. Does a telescope make celestial objects appear brighter?

Yes - and no.

The flux density - expressed in e.g. lumens per square centimeter, will of course increase, because the telescope squeezes the light from its large input pupil through its much smaller output pupil. This is the reason why the telescope will show stars which are too faint to be seen naked-eye.

But - the intensity, i.e. the flux density per steradian, AKA the "surface brightness", will not get brighter in any telescope. This is because the objects are also magnified, and thus the greater flux density is spread out over a larger solid angle. This is the reason why extended objects (e.g. the zodiacal light) won't be any easier to see in a telescope.

16. Can viewing the full moon through a large telescope damage the eye?

Short answer: No Way !!!!

Long answer (let's pretend we have a conversation with someone who believes viewing the full moon through the 5-meter Hale telescope will damage the eye):

Q: The Hale telescope will gather 500,000 times more light than the 7 mm pupil of the naked eye. The Sun is 500,000 times brighter than the Full Moon. Doesn't that mean that a view of the Full Moon through the Hale telescope would look as bright as the Sun and therefore could damage your eye, as viewing the Sun naked-eye can do?

A: No. If you view the Moon through the Hale telescope, using a magnification of 100x, then the exit pupil will be 50 mm, which means that most of the lunar light will be blocked by the 7 mm pupil of your eye. You'll really be using only some 0.7 meters of the 5-meter aperture of the Hale telescope at 100x magnification.

Q: OK, then let's increase the magnification to 700x, in order to bring down the exit pupil to 7 mm such that all the light can enter the eye's pupil. Won't that damage the eye?

A: No. The apparent size of the naked-eye Moon is 1/2 degree. If you magnify this 700x, the apparent diameter of the magnified Moon should be some 350 degrees! No eyepiece will have such a large apparent field of view, and even the eye itself is able to see a field of view only some 180 degrees large. The largest field of view in eyepieces is some 120 degrees, which means that, at the very most, only about 1/10 of the light of the Full Moon would ever reach your eye, if viewed at 700x magnification through the Hale telescope.

Q: OK, but 1/10 of the brightness of the Sun is still pretty bright. Doesn't that impose a risk of damaging the eye?

A: No, since the entire field of view will be filled with this light, while a naked-eye view of the Sun will see all the solar light concentrated in a small circle only some 1/2 degrees across. In fact, viewing the Full Moon though the Hale telescope or through any large telescope, at any magnification, would be no brighter than a naked-eye view of a bright overcast sky in daytime - and that won't damage your eyes, will it?

17. Does a big telescope suffer more from light pollution than a small telescope?

By Tom Polakis

Short answer: No!!!!

Long answer: The simple-minded claim is that under conditions of bad light pollution, beyond some certain aperture, you gain very little or nothing over a smaller telescope. The claim also is that this crossover aperture becomes smaller under progressively worse conditions of light pollution.

The physics is pretty simple: "aperture always wins". It is a simple matter of signal-to-noise: in any scene a telescope does not magically brighten the sky background while not brightening the celestial target at the same time. Yet the claim being made by most folks is that the background is indeed enhanced.

Take this notion to an extreme of really bad light pollution: daytime! If the expections of this urban legend hold, then "theoretically" the naked eye should show things in the sky better than any telescope. But you can see no stars with the naked eye in daylight (let's talk noontime here, not 10 minutes before sunset), whereas you can see stars in even small a telescope without much difficulty. (Try it! First a bright star like Vega. Figure out how to get to it with your telescope, and go look at it some clear day at 11 in the morning. Is Vega easier to see in a 6x30mm finder or a 16-inch? Now go for Altair, then Deneb...and how much fainter?)

If you wish to learn a lot more about this subject -- which will help you become a better observer -- go to Mel Bartels' Web site http://www.efn.org/~mbartels/aa/visual.html and read the stuff by him and by Nils Olof Carlin about visual detection thresholds. Apply the equations there to dark, light-polluted, and daytime sky brightness values to see what happens, and to prove to yourself that this light-pollution/aperture claim is pure baloney.

A. References: Books

B. References: Web sites