Magnitude Calculations

This topic describes how the Aperture Photometry package calculates the magnitudes and their random errors (or "uncertainties"). These quantities, as related to the table of photometry results, are also described in the topics Photometric Measurement Definitions and Photometric Error Definitions. This topic describes the math behind the calculations and how apertures are measured on an image.

Calculating the Magnitude

Mira computes the magnitude as

m = K – 2.5 log ( Net Flux )

where

K is the photometric "zero point".
Net Count = Total Object Aperture Count - Average Background Count * Object Area.
Net Flux = Net Count * Gain / Exptime.
Object Area is the area of the circular or elliptical object aperture.

Here are some addition remarks about the quantities listed above:

The Net Count is simply the total count inside the object aperture minus the background value inside the object aperture. Since the background value inside the object aperture cannot be determined, it must be estimated from pixels nearby.
The Total Object Aperture Count is the sum of all pixel values inside the object aperture, i.e., the sum of the object signal inside the object aperture plus the background signal inside the object aperture. Mira determines the background from pixels inside a concentric annulus formed by two rings located beyond the point where the star profile merges into the background noise. This is listed in the photometry output (measurement table or text editor) in the column labeled Ap Sum. This value may be used to manually calculate the magnitude using an alternative nearby background value, such as when the background annulus contains a very bright star.
The Average Background Count is listed in the photometry output in the column labeled Backgr. It is in units of counts.
The Object Area is calculated as described near the end of this topic.

The camera gain ("Gain") and exposure time ("Exptime") are obtained from the FITS keywords GAIN and EXPTIME. The Total Count,Net Count, and Background Count are measured in raw pixel value units (often called DN or ADUs"). The Net Flux therefore represents the total number of electrons per second attributable to the object, above the sky.

To compute the Flux, the image signal is summed over all pixels inside all whole and partial pixels inside the innermost "object" aperture (see below). From this sum is subtracted the estimated sky count computed from pixels in the annulus between the 2 outer apertures. If the GAIN and EXPTIME keywords are not present in the image header, or they are wrong, you can add them or edit them using the Image Information page of the Aperture Photometry Properties dialog. Also see the topic Fixing Header Problems in Photometric Data. You can also change the names used for fetching the GAIN, EXPTIME, and RDNOISE values using the Photometry Keywords dialog.

The value K is the photometric zero point. The zero point is assumed to be 0 or the value of the ZERO-PT keyword in the image header. If the value is 0, then the magnitude may be called a "raw magnitude". If a zero point value is computed for a particular instrumental setup and used in the calculation, the result is called "instrumental magnitude". If the zero point value, K, is unknown, it may be calculated using standard stars of known magnitude.

Magnitude Errors

The uncertainty of the magnitude measurement is calculated in two ways: an empirical error, listed in the Report Window as Error, and a theoretical value listed asError(T) . The reported values may or may not include the uncertainty in the photometric zero point, which is calculated when more than 1 standard star is used. This option is controlled by theInclude Zero Point Error check box on the Other Properties page. If checked, the reported error is the Root Sum Square ("RSS") value of the internal error of the measurement and the error of the mean of the zero point. The latter value is reported in the Photometry Measurements.

The empirical error involves the noise measured in the sky annulus as well as the values of GAIN, RDNOISE, and EXPTIME from the image header. The theoretical error uses the keywords but not the measured sky noise. The theoretical error estimate is the minimum possible uncertainty that could be measured at a given magnitude. You can change the values of GAIN, EXPTIME, and RDNOISE on the Image Information page. Doing so will change the error values by a small amount.

Description of the Apertures

The aperture photometry procedure uses 3 apertures to measure an object:

The central aperture measures the total signal for the object. It does not have to enclose 100% of the signal from the object, provided that all objects in the image are measured using the object same aperture shape and size. The total count inside the object aperture is listed as the Ap Sum quantity in the photometry output (measurements table or text editor).
The outer 2 apertures form a concentric ring, or annulus where the local sky (background) is measured. This is subtracted from the total signal inside the object aperture to produce theNet Count listed in the photometry output (measurements table or text editor).
In general, the object background does not have to be measured inside an annulus. The general rule is that the optimum background value is the value at the coordinates of the object centroid. In cases where the annulus would contain a bright star, you may wish to manually calculate the background near the object and use that to obtain the net count from the Ap Sum value. In this case, the following equation gives the net count from an alternate background value:

Diagram of a "measuring aperture" formed by an object aperture and a sky sampling region between two outer apertures.

The inner sky aperture should begin beyond the light in the star profile. However, the sky measuring ring should not be placed so far that it includes many other stars or that it samples sky that is not the same as that underneath the object.

The inner, object measuring aperture can be adjusted to any radius but, generally, it should not extend all the way to the point where the star appears to merge into the sky noise as this will lower the signal- to-noise ratio ("SNR") of the calculated magnitude. Use the following guidelines to determine a reasonable size for the object aperture:

The fraction of the object's total signal measured by the object aperture must not change from one object to another. This is accomplished by using apertures of the same radius, ellipticity, and orientation for every object on all images being measured together.

Note that the object aperture may be allowed to cut onto the wings of the star profile. You can be totally safe by making the object aperture large enough to contain "all the light", but this adds more sky noise, which lowers the precision of the measurement and makes the magnitude errors larger. On the other hand, if the star profile varies over the region being measured, you might use an object aperture large enough to hold all the light for bright objects, or as the only option for measuring the field of view.

The aperture photometry procedure uses 3 apertures to measure the object. The central aperture measures the total signal for the object while the local background is measure in the annulus between the outer two apertures. The apertures may be circular or elliptical, with any ellipticity and angle of rotation, but all three apertures have the same shape and are concentric. Elliptical apertures are used to measure objects when the Point Spread Function ("PSF") is elongated by poor tracking or other reasons. Matching the ellipse to the PSF shape gives a better measurement by maximizing the signal through the aperture of a given area, which in turn maximizes the signal to noise ratio.

Small Apertures

The general placement of a measuring aperture on the pixel grid is shown below. In general, the object aperture, whether elliptical or circular, or at whatever angle, will have partially sampled pixels along its rim. For small apertures, a good measurement depends upon correctly accounting for the partial pixel coverage. Mira properly accounts for this case using a mathematically complex, but exact partial pixel algorithm—the only one of its kind.

Enlarged view of the object aperture. The star's light profile is assumed to extend outside the picture. so the aperture does not contain all the light from the star. Many pixels fall along the rim and are only partially measured by the aperture.

Yellow: Fully sampled pixels.

Blue: Partially sampled pixels.

The Ellipticity parameter is defined in terms of the axis ratio of the ellipse: for semimajor axis a and semiminor axis b. The Ellipticity, e, is given by

Ellipticity e = 1 - b/a.

A circle has an ellipticity of 0. The major axis orientation and the Radius are adjustable using the Aperture Tool window. The orientation angle of the semimajor axis is only used to determine how pixels are added to the count sums. The ellipse properties that are independent of angle are related to its size and ellipticity as follows:

a = Semimajor axis.
b = Semiminor axis.
Radius r = square root( a * b ).
Ellipticity e = 1 - b/a
Area A = pi * r² (1 - e).

All 3 apertures use the same radius, ellipticity, and orientation for every measurement. If measuring an image set, Mira uses the same properties for all images.

Calculating the Magnitude using an Alternative Background

When the background cannot be reliably determined from pixels inside the background annulus, the Ap Sum value from the photometry output may be used to calculate the magnitude. This procedure might be implemented when the normal background annulus contains another object or a region of bad pixels that cannot be rejected by the built-in background method.

Before giving a procedure, let's look at how Mira computes the Net Flux as described at the beginning of this topic. The following equation uses the photometry output quantitiesAp Sum and Backgr . In addition, the object area is calculated using the equation for Area A in the previous section. Remember that Mira computes the value of Backgr using pixels inside the background annulus:

Net Count = Ap Sum - (Backgr * Object Area)

This calculation is made internally using the Object Area based on the aperture properties specified in the Aperture Tool window. The Ap Sum and Backgr are computed by Mira and listed in the photometry output. The GAIN andEXPTIME keyword values are applied to the Net Count to calculate theNet Flux and thence the magnitude ("Mag") listed in the photometry output.

Calculating the magnitude using an alternative background requires calculating two quantities:

The background value estimated by some means and evaluated at the centroid of the object;
The aperture area as calculated using the Area equation above.

The alternative background can be estimated in numerous ways. The background value may be estimated at the Image Cursor location using the method described in the Statistics Measurements topic. The estimator is chosen or configured using the Statistics Properties dialog. Since the optimal background is evaluated at the position of the object, obtaining the background value may involve averaging estimators obtained symmetrically on opposite sides of the object. Let the value Alt Bg be the alternative background value calculated in some way. Then the following equation gives the Net Count:

Net Count = Ap Sum - (Alt Bg * pi * r² (1 - e)).

The Net Flux and Magnitude may then be calculated from the Net Count. This can be calculated by hand, a spreadsheet, or a Mira script.

Using a Script to Implement an Alternative Background

A Mira script may be implemented to automate the process of utilizing an alternative background. The script would be called from the photometry measurements grid. Here is an outline for creating such a script: