Understand the relationship
The reasoning behind the result
A surface average divides flux by area
Fper arcsec²=Ftotal/A; μ=m+2.5 log₁₀A
Here A is the numerical area in square arcseconds. Dividing total flux by A and applying the magnitude logarithm adds 2.5 log10(A) to integrated magnitude. A larger numerical surface-brightness magnitude means a fainter average per unit angular area.
The area must match the region whose integrated light was measured. Combining a whole-source magnitude with an unrelated half-light aperture would not describe the mean inside that aperture.
Radius and full axes use different geometry
Acircle=πr²; Aellipse=πab/4; Arectangle=wh
The circular input is a radius r. Elliptical inputs a and b are full axis lengths, so each must be halved before calculating the ellipse area. Rectangular inputs are complete side lengths.
An arcminute contains sixty arcseconds, making one square arcminute 3,600 square arcseconds. These are local image-plane angular areas, not exact large-field spherical solid angles. Direct area is preferable when masking, distortion or fractional pixel overlap changes the effective footprint.
Pixel magnitude describes a mean over a chosen pixel area
Apixel=sx sy; Nequivalent=A/Apixel; mpixel=μ−2.5 log₁₀Apixel
The two entered pixel scales allow rectangular pixels or different angular scales along the axes. Multiplying them gives the angular area per pixel. The footprint divided by that area is an equivalent pixel count and may be fractional.
Average pixel magnitude describes the same mean flux density multiplied by one pixel area. It does not say every pixel has that brightness or model a point-spread function, central concentration, background noise or saturation.
Equal integrated light can have different surface brightness
Spreading the same total light over one hundred times the area makes the mean surface brightness five magnitudes fainter. Conversely, at fixed surface brightness, one hundred times the area contains one hundred times the integrated flux and has an integrated magnitude five lower.
The graph holds the calculated integrated magnitude fixed to show how the chosen averaging footprint changes the mean. It is an explanatory redistribution scenario, not a measured radial profile or an assertion that extra source light is absent outside the aperture.
Follow the numbers
Magnitude 10 over 100 square arcseconds
- The same-region inputs are m=10 and A=100 arcsec².
- The area term is 2.5 log10(100)=5 mag, giving μ=15 mag/arcsec².
- A 0.5 by 0.5 arcsec pixel covers 0.25 arcsec². Its mean magnitude is 15−2.5 log10(0.25)=16.505149978 mag, and 100/0.25=400 equivalent pixels cover the footprint.
Total magnitude, mean surface brightness and mean pixel magnitude refer to different areas of the same entered light.
Quick guide
How to use this calculator
- Use an integrated magnitude and footprint that describe the same measured region.
- Enter area directly or compute it from radius, full ellipse axes or rectangle sides.
- Choose the angular unit before entering dimensions; the result normalizes area to square arcseconds.
- If pixel scales are known, inspect the equivalent pixel count and average pixel magnitude without interpreting them as a measured central brightness.
Calculation method
Calculation and interpretation
Distinguish the total light inside an aperture from its average light per square arcsecond.
μ=m+2.5 log₁₀(A/arcsec²); m=μ−2.5 log₁₀(A/arcsec²); mpixel=μ−2.5 log₁₀(Apixel/arcsec²).
Worked example
Magnitude 10 over 100 square arcseconds
Total magnitude, mean surface brightness and mean pixel magnitude refer to different areas of the same entered light.
μ=m+2.5 log₁₀(A/arcsec²); m=μ−2.5 log₁₀(A/arcsec²); mpixel=μ−2.5 log₁₀(Apixel/arcsec²).
Supported inputs
Precision and limits
Mean within the stated region
No central brightness, half-light convention, radial profile, extinction correction or surface-brightness detection threshold is inferred.
Local angular area
Geometric shapes assume a locally planar angular map. Use a separately determined effective area for distortion, masking or large-field solid-angle work.
Continue calculating
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