 # Grow Curve

Grow curve is derived as an approximation of the flux of stars in increasing apertures.

# The Definition

Lets I(r,φ) is a distribution of the intensity of a star. A flux inside radius R will be F(R):

F(R) =∫0R I(r,φ) dr dφ

The growth-curve is defined as the radial flux dependency with limit f(∞) = 1. The property F(∞) = F0f(∞) defines total flux of the star as F0. By another words, this is a flux in the infinite aperture without another stars (sources).

Lets observed intensity on CCD is Iij inside R aperture defines the empirical radial flux distribution

FR = ∑ij (Iij - Bij),

for √(i2 + j2) ≤ R The sum counts photons in radius around a centre of a star. The observed intensity contains photons from the star added to photons from background Bij which must be subtracted. When the value of background is poorly estimated, the flux is also affected.

# The Construction

An empirical growth curve fi at radii ri and areas Ai = π ri2 is

Fi = F0 fi + β Ai

Fi are measurements of fluxes at a set suitably distributed apertures. The effective half-radius (half of FWHM) can be used to estimate the aperture with minimal noise and background contamination: 2 ‒ 3 FWHM. For apertures smaller then the optimal, the growth-curve can be estimated as

fi = (Fi - β Ai) / (Fi+1 - β Ai+1) fi+1

fi = (Fi - β Ai) / F0.

The determination is choice with respect to minimise statistical errors. For proper estimate of the parameters, the use of bright stars is recommended.

# Properties

Grow curves are preferred against to pure aperture photometry:

• They gives total flux.
• They are averaged.

The total flux is more invariant quantity than pure flux in aperture because it is independent on actual shape of star image which is changed due to atmospheric conditions, telescope image deformation and specially on airmass. The measurements of extinction and absolute calibration requires the total flux.

The total flux is derived from more than one aperture, therefore the values are less affected by unexpected errors. The results has less noise. # Aperture Correction

Aperture correction is a procedure which converts value from a finite aperture to total flux. Growth curve can be used to derive the correction. Generally, the grow curve method superseded the aperture correction because use of more apertures together and correct estimate of background.