# How do I get the change in angular diameter value?

Given that I got the equation relating the angular diameter(θ) and $V$–$K$ band value, which is known as $\logθ_0=(0.262±0.004)(V−K)_0+(0.547±0.006)$ from this paper, how do I get the 'change' in angular diameter value from that equation?

1) Can I just get the max and mix value of the $V$–$K$ value and insert each into the equation and get max θ and min θ and get the difference?

2) Or I need to the whole lists of angular diameter value over each period and insert the lists of value into the equation and plot the graph and then get the difference?

3) Or is there any other method to do this?

• Question is very unclear. Why is $\theta$ varying? Are you talking about pulsating stars? – ProfRob Aug 22 '16 at 20:15
• @RobJeffries Yes, this user has been asking numerous questions about Cepheids. This equation and the paper cited specifically relate to a method of measuring Cepheid distance, independent of the standard period-luminosity relation. One component of that process is to measure the angular size variation over time due to the pulsations. – zephyr Aug 22 '16 at 20:24

You can see from the paper you linked that they followed the procedure you outlined in option (2). Figure 4 of that paper shows the $\theta (t)$ for a few stars. They specifically state towards the end of section 6

Figure 4 shows in the top panel the relation between the angular diameter predicted by the SB-relation versus radial variation, and in the bottom panel the angular diameter as a function of phase.

The SB-relation they're talking about is the one you cite in your question. So your process should be to take your list of $V-K$ colors over time and convert them to angular size using the surface brightness relation, then fit a function to that (via cubic spline, least squares, etc.) and determine the total angular variation over time.

I will note also that the $\theta$ vs $V-K$ relation you've cited here is equation 10 in the linked paper which is not a result of that paper. Their result is listed in equation 9 with equations 10 and 11 being results from other, previous work for the reader to compare. You should sure of which equation you want to use and cite the appropriate source.

• Thank you! One more, how can I be sure that which method of fitting the curve is the most reliable one among cubic spline fitting, least squares, etc? And is it okay to vary that method by different Cepheids? – 7_G.S.N Aug 27 '16 at 11:58
• @7_G.S.N It depends on the state of your data and what you're trying to achieve with your fit. A cubic spline though doesn't sound like what you'd want as that fits your data directly and is meant for interpolation generally. Least squares will allow you to fit a specific function to your data and is closer in line with what you want, but can be unreliable. A more robust method might be an MCMC algorithm or something line the Levenberg-Marquardt algorithm, but those can both be rather advanced for a new user. In general, you should be consistent with your analysis. – zephyr Aug 29 '16 at 15:19