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The above is the equation used in the Baade-Wesselink method to independently get distances to Cepheid variables (in the process of deriving Period-Luminosity relation).

According to the paper New Baade-Wesselink distances and radii for four metal-rich Galactic Cepheids, δθ in the equation accounts for a phase shift between the radial velocity curve and the angular diameter changes measured via surface-brightness relation. My question is how can I calculate this δθ value given that I have lists of radial velocity values and V-R values with the observed HJD dates (which is possible to be converted in terms of phase).

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It is admirable that you are tackling this type of problem as a high school student, but I really think that some of these topics are out of your grasp currently, partly because of the highly technical nature, but also because of your lack of access to the necessary data.

For one thing, you say you have V-R colors as a function of time, but all these equations were calibrated to use V-K colors. From your linked paper, you're going to need equation 2 for your process to calculate $p$ and equation 3 for getting $\theta_0$. Unfortunately, these equations assume V-K colors. You can redo the analysis done in this paper to calculate $p$ and this paper to get $\theta_0$ for V-R colors, but it's not going to be easy.

For another, you're going to need measurements of the angular size as a function of time (and your cited paper suggests that only exists for a handful of Cepheids and was measured using the VLTI) as well as the RV as a function of time. The $\delta \theta$ that you reference is described exactly in the paper as "a phase shift between the radial velocity curve and the angular diameter changes measured either interferometrically or via the SB relation." This means if you have $RV(t)$ and $\Delta \theta(t)$, you can determine a phase shift between the two curves, presumably by fitting some equation of the form $f(t) = a\ \mathrm{sin}(bt+c)$ to both equations and saying that $\delta \theta = c_{RV} - c_{\Delta \theta}$.

The analysis you're trying to do requires very specific data and it seems that the relevant papers on this subject have already done the analysis on all stars in which all necessary data existed. You cannot easily apply this process to any Cepheid without significant work.

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  • $\begingroup$ Thanks for your thoughtful and detailed comments. I was actually feeling lots of difficulties and frustration. As you say, it seems like it is almost impossible to derive p-p relation and θ-VR relation by myself in addition to my previous plan, which was relating linear radius variation, (integrated from radial velocity) and angular velocity(derived from θ-VR relation) to derive the distance. So what I thought was just referencing those p-p relation and θ-VR from the literature work, focusing on the later work. $\endgroup$
    – 7_G.S.N
    Commented Aug 11, 2016 at 18:02
  • $\begingroup$ The data I might use for the radial velocity is in this link and for VR colors in this database. But again, there's some problem. The data from the database one is quite outdated and I've seen from the other paper that the accuracy falls as the observed time difference gets bigger. So I need to find the lists of VR color values from the other site. I tried to used ESO archive but I couldn't open the file I downloaded from there. (I don't know why) :( $\endgroup$
    – 7_G.S.N
    Commented Aug 11, 2016 at 18:06
  • $\begingroup$ Like this, I'm facing lots and lots of difficulties not only these things I told you. And I don't know anybody who is professional in this topic. Can I personally contact you in this sense? Asking questions via this stackexchange or any other way? Thank you. $\endgroup$
    – 7_G.S.N
    Commented Aug 11, 2016 at 18:08
  • $\begingroup$ I cannot tag you. Can you please read my previous comments? $\endgroup$
    – 7_G.S.N
    Commented Aug 11, 2016 at 18:09
  • $\begingroup$ @7_G.S.N I can't say I'm an expert on this particular method. I just have good technical knowledge of astronomy. I think one major hurdle of your process, like I said, will be that you know V-R colors but all equations referenced are for V-K colors. I'd suggest you try and reproduce the results in the original papers first, for stars already done. It is a viable option to email the authors (most papers list the first author's email) asking for clarifications and/or raw data, but be extremely clear, precise, and respectful if you do, and don't expect an immediate answer (if any). $\endgroup$
    – zephyr
    Commented Aug 11, 2016 at 18:19

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