What FWHM means
FWHM stands for full width at half maximum. For a star, it is the width of the star’s brightness profile measured at the height where the brightness has fallen to half of its peak value. A star that is a tiny point in reality is spread into a small blob by the atmosphere and the optics, and FWHM gives a single number for how big that blob is.
A smaller FWHM means tighter, sharper stars, which matter a great deal in astrophotography with a telescope. A larger FWHM means the stars are more blurred. It is usually quoted in arcseconds on the sky, or in pixels on the image.
FWHM in physics
In physics and optics, FWHM is a general way to describe the width of any peaked curve, such as a spectral line, a laser beam or a signal pulse. You find the maximum height, take half of it, and measure how wide the curve is at that level. Astronomy borrows the same idea for star images and spectral lines.
How to calculate FWHM
For a profile you can sample, the steps are:
- Find the peak value, after subtracting the background level.
- Calculate half of the peak.
- Find the two positions on either side of the peak where the profile crosses that half level, interpolating between pixels if needed.
- Subtract the positions to get the width.
In practice, software fits a model, often a Gaussian or Moffat profile, to many stars and reports the fitted width, which is more reliable than reading raw pixels.
Gaussian FWHM, sigma and the 1/e2 width
If a star profile is modelled as a Gaussian with standard deviation sigma, then:
- FWHM = 2 x sqrt(2 ln 2) x sigma, which is about 2.355 x sigma.
- The full width where intensity drops to 1/e2 (about 13.5% of the peak) is 4 x sigma.
So to convert a Gaussian FWHM to the 1/e2 width, multiply by 4 / 2.355, which is roughly 1.70. To go the other way, divide by about 1.70. The 1/e2 width is common for laser beams, while FWHM is more common in astronomy. These relations hold only for an ideal Gaussian; real stars have broader wings, so treat the conversion as approximate.
Pixels versus arcseconds
To convert a measurement in pixels to arcseconds, you need the image scale:
image scale (arcsec per pixel) = 206.265 x pixel size (micrometres) / focal length (mm) (see what focal length tells you)
FWHM in arcseconds is then FWHM in pixels multiplied by the image scale. For example, a 3.76 micrometre pixel at 800 mm focal length gives about 0.97 arcsec per pixel, so a 3 pixel FWHM would be roughly 2.9 arcsec.
What affects FWHM
- Seeing: atmospheric turbulence is often the largest contributor.
- Focus: a slightly off-focus star has a larger FWHM, which is why autofocus routines use star size.
- Guiding and tracking (telescope tracking explained): drift or periodic error elongates stars.
- Optics: aberrations, tilt, collimation and cooling can enlarge or distort stars.
- Wind and vibration, and local heat from buildings or the telescope itself.
- Altitude: low targets look through more air.
Image scale matters
A discussion on Ice in Space pointed out that a measured FWHM is hard to interpret without the image scale. If your sampling is coarse, you cannot measure small FWHM values accurately, because a star only covers a few pixels. A commonly quoted rule of thumb is to have roughly 2 to 3 pixels across the FWHM. So when comparing numbers with others, include your pixel scale, and ideally the target and altitude.
Measuring FWHM in practice
Most astrophotography software can report it. PixInsight offers tools such as SubframeSelector and FWHMEccentricity for batches of subframes, and capture programs show star size during focusing. Control suites such as EKOS typically use half-flux radius (HFR) for autofocus instead. Use the same tool and settings when comparing, since different programs fit stars in different ways and report slightly different numbers.
FWHM versus HFR
HFR (half-flux radius) measures the radius containing half of a star’s light. HFD is the corresponding diameter. For a well-behaved star the two are related to FWHM but not identical, so avoid converting between them casually. They are best used as relative measures within one program.
What is a good FWHM?
There is no universal answer. It depends on your site, equipment and image scale. Many amateurs see values of a few arcseconds, and values near or below 2 arcseconds typically need good seeing and a well-matched setup. Rather than chasing a number, track your own values over time and look for the best frames on a given night.
Using FWHM in processing
Imagers often use FWHM to rank subframes and reject the worst before stacking, and to compare focus runs. Tight stars also make later steps easier. If your stars are still bloated, our guide to star reduction in PixInsight shows ways to tone them down.