# 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) =∫*_{0}^{R} I(r,φ) dr dφ

The growth-curve is defined as the radial flux dependency with
limit *f(∞) = 1*. The property *F(∞) = F*_{0}f(∞)
defines total flux of the star as *F*_{0}.
By another words, this is a flux in the infinite aperture
without another stars (sources).

Lets observed intensity on CCD is *I*_{ij}
inside *R* aperture defines the empirical radial flux distribution

*
F*_{R} = ∑_{ij} (I_{ij} - B_{ij}),

for *√(i*^{2} + j^{2}) ≤ 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 *B*_{ij} which must be subtracted.
When the value of background is poorly estimated, the
flux is also affected.

# The Construction

An empirical growth curve *f*_{i} at radii
*r*_{i} and areas *A*_{i} = π r_{i}^{2}
is

*
F*_{i} = F_{0} f_{i} + β A_{i}

*F*_{i} 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

*
f*_{i} = (F_{i} - β A_{i}) /
(F_{i+1} - β A_{i+1}) f_{i+1}

and for larger radii as

*
f*_{i} = (F_{i} - β A_{i}) / F_{0}.

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Generally, the grow curve method superseded the aperture correction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.
## See Also

Manuals:
Aperture Photometry.
Data Formats:
Format of Processing File.