Photomultiplier Tube Saturation
It is possible to saturate a photomultiplier tube with light too bright. When this happens, the lineraity decreases to the point where data will be worthless. If the light is bright enough, permanent damage to the tube may result. For a 1P21 with high voltage set to -950 VDC, the maximum count should be under 5,000,000 counts per second or 50,000,000 counts per 10 seconds. For an 8" telescope this relates to around a 1st or 2nd magnitude star.

Dark Counts
No calibration is required concerning Dark Counts or system noise. Dark counts are the counts seen when no light is falling on the powered photomultiplier tube. For better tubes and at lower temperatures the dark counts approach zero. For all but very faint stars in small telescopes, the dark counts are usually insignificant and are automatically subtracted when the sky counts are subtracted from the star + sky counts. As noted earlier typical dark counts for a 1P21 at -950 VDC and 50 degrees F are 50 counts per second . At 85 degrees F the counts increase to over 200 counts per second. Dark counts in the low hundreds are acceptable for all but very faint star work. Dark counts approaching the thousands may be a problem and a better tube may be in order or wait for a cooler time. Remember, some tubes may exhibit high counts when first turned on, but after several minutes the counts should go down significantly. Actual dark count values need not be determined as when the sky counts are subtracted fromt the star + sky counts, the dark counts are also subtracted.

Dead Time (Photon Counting)
For photon counting systems, the dead time for the system must be determined. Since analog systems (with a DC amplifier, voltage-to-frequency converter) use an integrated current, there are no dark counts to be concerned about. Tube saturation on bright objects is still a concern, however.

When observing bright sources, a photon-counting system tends to under-count the number of photons, i.e., the number of pulses counted becomes increasingly non-linear as the number of photons per second increases. This is because the output pulse of the tube has a given pulse width (about 150 nanoseconds). When the pulse rate increases to the point where there are more pulses than one per 150 nanoseconds then the additional pulses do not get counted. Since pulse counters use a rising or falling edge as an event to count, a second edge occurring during a pulse will not be seen. To correct for this a dead time factor is used.

Assume the following:

N= True number of pulses per integration time.
n= Number of observed pulses per integration time.
t= Integration time.
= Dead Time.

The rate at which photons are counted is:

n= N(1 - N x / t)

Because the factor desired is the "true" number of pulses per second (N) an approximation of the above equation yields the following:

N= n / (1 - N x / t) ~ n / (1 - n x / t)
or
N= n(1 + n x / t)

For integration times of 1 second the equation reduces to:

N= n(1 + n x )

Perhaps the easiest way to determine a system's dead time is to use the complete system to measure a pair of stars. One star should be 3 to 4 magnitudes brighter than the other. It should also be bright enough to cause significant dead time (V= 3 for an 8" aperture), but not so bright as to cause saturation of the PMT. Saturation will produce a non-linear relationship that will invalidate the dead time calibration. A diaphragm with many small holes (0.5" to 1.0" diameter holes) that fits over the end of the telescope tube can be made out of card board or thin plywood. The diaphragm should reduce the effective aperture by about 80%, e.g., for an 8" aperture (area = 50 sq. in. minus the central obstruction or about 45 sq. in.) the sum area of all the small holes should be about 9 sq. in. (.20 X 45). For 1" holes there should be 12 - 15 holes. The holes should be place in an irregular pattern to avoid diffraction effects. See Figure 34.

Figure 34 Dead Time Calibration Mask

Pick stars that are close to the zenith and close together. Make all the measurements as quickly as possible to reduce any extinction change effects. First measure the bright star and then the faint star without the diaphragm. Next measure the faint star and then the bright star with the diaphragm. Finally measure the bright star and then the faint star without the diaphragm. Average the measurements of the faint star without the diaphragm (call this number of count F). Next average the counts for the bright star without the diaphragm (call these counts B). Determine the ratio (R) of the averaged faint star's count without the diaphragm to that of the faint star's count with the diaphragm (call this FD).

R= F / FD

The true counts (N) for the bright star without the diaphragm should now be the bright star's count with the diaphragm (BD) times the ratio (R).

N= BD * R

Make several determinations of R using different sets of stars and then average the values of R.

Example
Average faint star count without the diaphragm F= 27500
Average faint star count with the diaphragm FD= 1222

R= 27500 / 1222 = 22.50

Averaged bright star count with the diaphragm

BD= 40000

Averaged bright star count without the diaphragm

B= 787,500

True count for the bright star without the diaphragm

N= BD * R = 40000 * 22.50 =900,000

The approximate dead time can now be determined by:

= (N - B)/ B * B

or

= (900,000 - 787,500)/787500 * 787500
= 181 nanoseconds (nS)

Now to correct for system dead time use the following equation:

N= n * (1 + * n)

Where:
N= True counts in counts per second
n= Observed counts in counts per second
= Dead time in seconds

Example:
n= 750,500 counts per second
= 0.000000181 seconds (181 nanoseconds)
N= 750,500 * (1 + 0.000000181 * 750,500)
or
N= 852,609 counts per second

Determining Color Transformation Coefficients
To standardize your data you must determine your system's color transformation coefficients and use those coefficient when reducing the data.

Having determined the extinction coefficients that account for atmospheric effects, the next step is to transform the data to the standard UBV system. This is done by observing a set of standard stars with a wide range of known colors, applying the extinction corrections and then determining the coefficients using the following equations.

Where

Figure 35

Figure 36

Figure 37

Shortcuts.
The coefficients subject to the most variation are those for the first-order extinction. If several tests have shown that the remaining coefficients are stable, a quick shortcut can be used to find approximate values for k. This procedure is useful only for a quick check of the sky and should not be substituted for a more robust determination when undertaking serious all-sky observations.

From the above equations:

Although these look complicated, they can be used in the same manner as previous equations. The quantity on the left side of the equals sign can be plotted against the multiplier of the k' term on the right side. The result each plot should be a straight line with the slope k' and an intercept . It is important to note the change in algebraic sign of the intercepts!

So, where is the shortcut? It lies in the specific application of the first two equations to stars with instrumental colors that are approximately zero. Then, because the k'' terms are very small, the set of equations reduces to:

Plots of these quantities are much easier to construct.

The ultimate shortcut is to observe just two stars having instrumental colors about zero at very different air masses. Letting the subscripts 1 and 2 denote these observations, then the equations can be further reduced to isolate just the first-order extinction coefficients:

A quick estimate of nightly first-order extinction coefficients can be obtained from just two observations. Observers are encouraged to follow this two-star procedure on nights when they plan to use only differential photometry. The results of consistently monitoring extinction can be as valuable in subsequent data analysis as the observations of check stars!

Zero Points
The Zero points are a function of the telescope's aperture, mirrors, lens, electronics and gain settings (for analog systems).

Note: The zero points drop out when doing differential photometry. A rough approximation for the zero points is

= (Standard Magnitude of the star) + 2.5 * Log₁₀ (visual reading).

For a V = 4.95 magnitude star with a reading of 32,000 counts per second (dead time corrected)

= 4.95 + 2.5 (4.5051) = 16.2129

Thus if that V= 4.95 star was observed (ignoring color transformation and extinction coefficients)

v_o = observed brightness
v_o = -2.5 * Log₁₀ (averaged V filter star reading)
v_o = -2.5 * Log₁₀ (32,000)
v_o = - 11.2629
Thus
V = v_o +
V = -11.2629 + 16.2129
V = 4.95

For the same star with an analog reading of v_o = 79.25

= 9.697

For a more precise determination, V - v_o vs. (B - V) of several stars can be plotted. The (B - V) = 0.0 intercept of a straight line through the plotted points will produce . See Figure 32.



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