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Astron. Astrophys. 318, 416-428 (1997)

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A. Appendix

The uncertainties and biases affecting the distance determination by different methods for outer disc cepheids are estimated here. The estimations and conclusions are only valid for highly reddened cepheids of uncertain metallicity.

[EQUATION]

[EQUATION]

Let us define for a given real cepheid the following quantities:


[TABLE]



Simple arithmetics shows that the resulting uncertainties on the distance modulus depend of the above quantities in the following way:


[TABLE]


First, we note that because [FORMULA] is by far the largest term, (1) and (3) should be preferred to (2) and (5) respectively.

We can estimate the importance of the different terms for (1), (3) and (4):
[FORMULA] (the CC and PLC relations are practically exact).
[FORMULA], [FORMULA], [FORMULA], [FORMULA].
Then


[TABLE]

[FORMULA] is the largest factor, but is unbiased by definition. [FORMULA] and [FORMULA] can be large since the corresponding metallicity corrections are difficult to estimate. They are also very vulnerable to bias, which makes relation (1) dangerous to use for metal-deficient cepheids. [FORMULA] and [FORMULA] can also be large, but can be checked with Magellanic Clouds data. [FORMULA] and [FORMULA] are small.

In summary, for highly reddened cepheids with an uncertain metal deficiency, as in the outer disc, the PC/PL(I) combination is best, with a [FORMULA] scatter and small possible errors on metallicity corrections. The PC/PL(V) combination has a larger intrinsic scatter and metallicity correction, and the CC/PLC combination, while ideally exact, is open to significant metal-dependent biases.

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© European Southern Observatory (ESO) 1997

Online publication: July 8, 1998
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