Why can't you calculate the absolute magnitude of Cepheid variables from apparent magnitude?

Question

I'm currently working on an assignment related to Cepheid variable stars, and I feel like I've basically gotten all the relevant concepts down pat. However, in the course of my work, I've discovered that there are two separate ways to calculate absolute magnitude - one related to distance and apparent magnitude (which is for stellar objects in general)...

$M = m - 5(\log D - 1)$

...and the other related to the period of the star (which is unique to Cepheids - this equation, specifically, to classical ones):

$M = -2.43 (\log P-1) - (4.05)$

When I calculated the $M$ values separately according to these two methods, it turned out that the $M$ calculated from apparent magnitude were incorrect, whereas those calculated from the Cepheid period were always correct (according to the SIMBAD database). Just to make 100% sure that my first method was right, I checked with some non-Cepheid stars, and the results were perfect, so it's only the Cepheids that are being problematic.

So why is it that you can't use the "classic" formula for Cepheids? My current theory is that it has to do with the fact that the light curve is asymmetrical so the "average" apparent magnitude number is actually not representative of the real curve, but I'd like to make sure that this is actually the case!

Thank you!

EDIT: In response to a request for my working, I’ll use my data on Eta Aquilae as an example:

Method 1: Distance is 423.73 parsecs. Maximum m is 3.5 and min m is 4.3, avg. is 3.9. We thus plug into the formula: $M = 3.9 - 5((\log427.73) - 1). M = -4.24$, which, according to SIMBAD, is wrong. Using the similar parallax equation provides the same result.

Method 2: Period is 7.18 days. Plugging in gives: $M = -2.43 ((\log7.18)-1) - (4.05). M = -3.68$, which, according to SIMBAD, is correct.

Hope that helps!

Answer 1

You can do this if you know the distance.

And that is important because it is how you calibrate the period-luminosity relationship in the first place.

But the main point here is that we use Cepheid variable stars to measure distance (as one step on the distance ladder), meaning that we don't know that distance. By using

1. Groups of Cepheid variables that all have about the same distance (in the original measurement these are stars in the Magelenic clouds) to learn that longer periods are associated with brighter stars (and to fit a mathematical relationship between period and magnitude).

2. The small number of Cepheid variables close enough to range by direct parallax to calibrate the observation into absolute rather than apparent magnitude.

we make these stars into a ruler, and use them to measure the distance to nearby galaxies.

Answer 2

There is no "right" absolute magnitude or "right" distance. We don't know what the absolute magnitude of eta Aquila is or its distance, so if two methods of estimating the absolute magnitude disagree, that is not necessarily problematic.

We DO know its apparent magnitude, which varies, its pulsation period, and we have a fairly rough measurement of its trigonometric parallax.

First, using the period-luminosity relation (your second method). This relationship comes from Benedict et al. (2007) and is the relationship between mean V-band magnitude and pulsation period. It has a scatter (standard deviation) of around 0.1 mag. Thus I would say the absolute magnitude from this relationship will be $-3.68 \pm 0.10$ mag.

Second: Since your question was first asked the Gaia satellite has provided a new parallax for this star of $3.67 \pm 0.19$ milliarcsec. Taking the reciprocal (roughly ok for a parallax of this precision) gives a distance of $272^{+15 }_{-13}$ pc. Using this in your first equation, with a mean apparent magnitude of 3.80 gives me an absolute magnitude of $-3.37^{+0.11}_{-0.13}$.

The difference from your own calculation is entirely explained by the very different distance I've used. Your distance appears to come from the Hipparcos parallax of $2.36$ millarcseconds quoted by van Leeuwen (2007). But the error bar on that parallax is $\pm 1.04$ milliarcseconds, which you omitted to include and would mean that my (much more precise) result was consistent with yours within a single error bar and that your original method 1 calculation of the absolute magnitude encompassed the value obtained more precisely from method 2 within its own error bar. So there was no discrepancy at all.

The remaining $0.31$ mag difference (with an error bar of $\sim 0.2$ mag), I can't get too excited about. It is barely of any significance. However, another issue to look at would be the source of the mean apparent magnitude and whether the apparent magnitude quoted in SIMBAD is actually the mean magnitude.