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Astron. Astrophys. 318, 797-804 (1997)

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2. The new method

2.1. The surface brightness method

The surface brightness method (Barnes & Evans, 1976), as used by Gieren et al. (1989), is based on the visual surface brightness [FORMULA], defined as the following equivalent relations:

[EQUATION]

[EQUATION]

where [FORMULA] is the angular diameter, and B.C. the bolometric corrections. Or, equivalently, the surface brightness parameter [FORMULA], given by

[EQUATION]

An empirical relation has been found by Barnes & Evans (1976), which correlates [FORMULA] to the Johnson [FORMULA] index, corrected for interstellar reddening:

[EQUATION]

where the angular coefficient m is in turn a function of the pulsational period (Moffett & Barnes, 1987):

[EQUATION]

The practical application starts from the [FORMULA] observations, which from (4) give [FORMULA], which in turn from (3) give [FORMULA], which in turn from (1) give [FORMULA]. Expressing the linear diameter [FORMULA] (r being the distance in parsec) as [FORMULA] ([FORMULA] being the instantaneous displacement) and performing a regression analysis of [FORMULA] against [FORMULA], one obtains the mean diameter [FORMULA], as well as the distance r.

2.2. The original CORS Method

The CORS method also starts from (1), but proceeds mathematically by differentiating it with respect to the phase, multiplying by the color index [FORMULA] and integrating over the whole cycle; after substituting:

[EQUATION]

it yields to the following equation, in which [FORMULA] is the phase

[EQUATION]

where [FORMULA],

[EQUATION]

[EQUATION]

P is the period, u the radial velocities and p is the radial velocity projection factor (Parsons, 1972; Gieren et al. 1989). The practical application of the CORS method starts from a fitting of data with respect to the phase [FORMULA] by means, e.g., of Fourier series. The data are given by the V magnitude, the [FORMULA] and [FORMULA] color indexes, and the radial velocities u.

The fit is easily obtained with an interactive procedure on a computer terminal with graphic capabilities; afterwards, the fitted curves are used to compute in an automated way the term B, the term [FORMULA] B, the derivatives and eventually to solve Eq. (7) to obtain [FORMULA], the radius at an arbitrary phase [FORMULA] ; [FORMULA] is usually taken at the minimum of the radial velocity curve, but its choice is inessential; the mean radius comes from integrating twice Eq. (6).

2.3. The modified CORS method

We will now show how the two methods can be used together, to obtain what we will call the modified CORS method.

From Eq. (4) we can obtain [FORMULA], and from it, we can obtain [FORMULA], using also Eq. (3) and (5):

[EQUATION]

The value of the constant is of no importance in our case, since we are interested in computing [FORMULA] ; this term, given by Eq. (9), is the area of the loop described by the star in the plane [FORMULA] vs [FORMULA]. If we make the transformation of variable from [FORMULA] to [FORMULA], given by Eq. (10), then:

[EQUATION]

Since the derivative of [FORMULA] with respect to the phase is, in first approximation, the same as the one of [FORMULA], we get from Eq. (11), with c=3.7-0.04 [FORMULA]:

[EQUATION]

Let us pose [FORMULA], which represents the area of the loop described by the star in the plane [FORMULA] vs [FORMULA], we get to the final formulation, given the following observations:

[EQUATION]

and the quantities, all based on the above observations:

[EQUATION]

The radius [FORMULA] at an arbitrary phase is obtained from the equation:

[EQUATION]

where [FORMULA] and [FORMULA], if we follow Gieren et al. (1989), or c=3.7 and p=1.36 in our approximation.

The mean radius is obtained from double integrating the radial velocity curve, since the first integration gives the radius curve and the second integration its mean value. Eq. (13) has to be solved by numerical methods, but this is easily accomplished with any computer, which nowadays is a common tool for any astronomer.

The assumptions and approximations which are behind this formulation are the following:

  1. the observational data give a good coverage of the pulsational period, and do not show big noise, so that the Fourier fitting (or any other fitting which guarantees the periodicity of the data) is a good approximation to the observations.
  2. the photometric and spectroscopic observations are simultaneous, or at least separated by few tens of pulsational cycles, so that no phase-shift is present in the terms of Eq. (13)
  3. the correlation of Eq. (5), given by Moffett & Barnes (1987) represents a good approximation to the data.
  4. the proportionality between [FORMULA] and C is valid; this was already proved in general terms by Onnembo et al., (1985), and therefore what we found here is just a confirmation, particularly valid for the colors [FORMULA] and [FORMULA].
  5. the colors are not affected by other contributions, e.g. the presence of companions to the cepheid (see Russo et al. 1981).
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© European Southern Observatory (ESO) 1997

Online publication: July 3, 1998
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