## GNU Astronomy Utilities

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#### 8.1.3 Flux Brightness and magnitude

Astronomical data pixels are usually in units of counts116 or electrons or either one divided by seconds. To convert from the counts to electrons, you will need to know the instrument gain. In any case, they can be directly converted to energy or energy/time using the basic hardware (telescope, camera and filter) information. We will continue the discussion assuming the pixels are in units of energy/time.

The brightness of an object is defined as its total detected energy per time. This is simply the sum of the pixels that are associated with that detection by our detection tool for example NoiseChisel117. The flux of an object is in units of energy/time/area and for a detected object, it is defined as its brightness divided by the area used to collect the light from the source or the telescope aperture (for example in $$cm^2$$)118. Knowing the flux ($$f$$) and distance to the object ($$r$$), we can calculate its luminosity: $$L=4{\pi}r^2f$$. Therefore, flux and luminosity are intrinsic properties of the object, while brightness depends on our detecting tools (hardware and software). Here we will not be discussing luminosity, but brightness. However, since luminosity is the astrophysically interesting quantity, we also defined it here to avoid possible confusion between these two terms because they both have the same units.

Images of astronomical objects span over a very large range of brightness. With the Sun (as the brightest object) being roughly $$2.5^{60}=10^{24}$$ times brighter than the faintest galaxies we can currently detect. Therefore discussing brightness will be very hard, and astronomers have chosen to use a logarithmic scale to talk about the brightness of astronomical objects. But the logarithm can only be usable with a unit-less and always positive value. Fortunately brightness is always positive and to remove the units we divide the brightness of the object ($$B$$) by a reference brightness ($$B_r$$). We then define the resulting logarithmic scale as $$magnitude$$ through the following relation119

$$m-m_r=-2.5\log_{10} \left( B \over B_r \right)$$

$$m$$ is defined as the magnitude of the object and $$m_r$$ is the pre-defined magnitude of the reference brightness. One particularly easy condition is when $$B_r=1$$. This will allow us to summarize all the hardware specific parameters discussed above into one number as the reference magnitude which is commonly known as the Zero-point120 magnitude.

### (116)

Counts are also known as analog to digital units (ADU).

### (117)

If further processing is done, for example the Kron or Petrosian radii are calculated, then the detected area is not sufficient and the total area that was within the respective radius must be used.

### (118)

For a full object that spans over several pixels, the telescope area should be used to find the flux. However, sometimes, only the brightness per pixel is desired. In such cases this book also loosely uses the term flux. This is only approximately accurate however, since while all the pixels have a fixed area, the pixel size can vary with camera on the telescope.

### (119)

The $$-2.5$$ factor in the definition of magnitudes is a legacy of the our ancient colleagues and in particular Hipparchus of Nicaea (190-120 BC).

### (120)

When $$B=Br=1$$, the right side of the magnitude definition will be zero. Hence the name, “zero-point”.

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