What is an isophote?¶

An isophote is a curve of constant surface brightness on the sky: every point on it is equally bright. For a smooth galaxy, the isophotes are nested closed curves, bright near the centre and fainter outwards.

In elliptical galaxies, lenticulars and bulges, the isophotes are very close to ellipses. The ellipses need not share one centre, one shape or one orientation. Measuring how they change with radius is one of the basic tools of galaxy structure. That is what ELLIPROF does: it describes the galaxy as a set of nested ellipses, one per radius, and measures each one.

In one sentence

ELLIPROF slices the galaxy into rings of equal brightness and records, for each ring, how bright it is, where its centre is, how flattened it is, which way it points, and how much it differs from a perfect ellipse.

The geometry of one isophote¶

Diagram of one ellipse. The centre is marked with a cross, and the
image's +x (columns) and +y (rows) axes are drawn as arrows. The
semi-major axis a runs from the centre to the upper left, and the shorter
semi-minor axis b runs to the lower left. An arc marks the angle alpha
between the +y axis and the major axis, counter-clockwise. Dots around
the ellipse mark sample points. — Schematic. The quantities ELLIPROF reports for each isophote. The dots are sample points at equal steps of the eccentric angle.

Quantity	Profile column	Meaning
semi-major axis \(a\)	`Rmaj`	half the longest diameter, in pixels. This is the "radius" of the isophote
semi-minor axis \(b\)	(from `ellip`)	half the shortest diameter
ellipticity	`ellip`	\(\epsilon = 1 - b/a\). 0 for a circle, larger for flatter ellipses
position angle	`alpha`	direction of the major axis, in degrees (0–180), counter-clockwise from the image +y axis
centre	`x0`, `y0`	the fitted centre of this isophote, in pixels
intensity	`I0`	brightness of the isophote, in image units per pixel, after the sky is subtracted

Position angle convention

ELLIPROF measures alpha counter-clockwise from the +y axis of the image (the pixel rows), not from north. Measured from +x, the major axis is at alpha + 90°. To get a sky position angle (east of north), you must also use the image orientation from its WCS. An angle of 0° and one of 180° are the same axis. For example, 3° and 177° differ by only 6°.

The eccentric angle¶

ELLIPROF measures positions around each ellipse with the eccentric angle \(\theta\), not the ordinary polar angle. A point on the ellipse is

\[ (x, y) = (a\cos\theta,\ b\sin\theta) \]

in the ellipse's own frame (major axis along \(x\)). Geometrically, \(\theta\) is the angle on the circle of radius \(a\) drawn around the ellipse.

Diagram of an ellipse inside its auxiliary circle of radius a. A point
on the circle at angle theta is projected vertically onto the ellipse,
giving the point (a cos theta, b sin theta). — Schematic. The eccentric angle θ. The harmonic phases A3 and A4 are measured in this angle, from the major axis.

The harmonic phases A3 and A4 (see Harmonic analysis) are angles of this kind. They are not position angles.

Surface brightness and the profile¶

The intensities I0 of the isophotes, as a function of semi-major axis, form the galaxy's surface-brightness profile. Together with how ellip, alpha and the centre change with radius, this is the profile that ELLIPROF writes. See The profile.

In magnitudes per square arcsecond, after a photometric calibration (Galaxy surface photometry):

\[ \mu = -2.5\log_{10}\!\left(\frac{I_0}{s^2}\right) + m_1 \]

where \(s\) is the pixel scale in arcseconds and \(m_1\) the magnitude of one unit of image flux.

Why isophotes change with radius¶

Real galaxies are not perfect nested ellipses, and the ways they depart from that are informative:

Ellipticity changes reflect the 3-D shape, or an embedded disk.
Position-angle twists (alpha changing with radius) can mean a triaxial galaxy seen in projection, or an interaction.
Centre shifts can mean asymmetry, dust or a disturbed galaxy. Fits near masked regions, or at very low surface brightness, also shift.
Boxy or disky isophotes (the 4th-order term) are linked to how the galaxy formed. See Harmonic analysis.

Isophote fitting in the literature

ELLIPROF's approach is to sample the image along each ellipse, expand the intensity in a Fourier series of the angle, and use the low-order terms to correct the ellipse. The same approach underlies other isophote-fitting programs, notably the method described by Jedrzejewski (1987, MNRAS 226, 747). ELLIPROF is an independent implementation with its own conventions. Check the conventions on How ELLIPROF fits a galaxy before comparing numbers between programs.

Next: How ELLIPROF fits a galaxy.