RASC Calgary Centre - Cepheid Distance
Cepheid Distance
by Jason Nishyama
Page last updated November 5, 2018
So how far away is that star?
One of the ways we determine the distance to
objects is through the use of standard candles. A standard candle being an object the intrinsic
brightness of which we know through some means independent of its distance. One of the most
important standard candles is the Cepheid variable. These are older stars that have become
unstable and are pulsing. As they pulse, their brightness changes. In the early 1900's Henrietta
Leavitt was cataloguing these stars in the Large Magellanic Cloud (LMC). Since the stars in the
LMC are essentially the same distance from us any variation is due to their intrinsic brightness
and not their distance. This allowed Leavitt to discover that their maximum brightness was related
to their period, that is the time it take to go from maximum brightness to maximum brightness.
This has become known as the Cepheid Period Luminosity (P-L) relationship and is shown
mathematically by Formula 1 where Mv is the absolute visual magnitude, P is the period in days
and a and b are constants.
Mv = a + b log P
To determine the constants in the P-L relationship you need two things, the absolute magnitude of
some Cepheid variables as well as their periods. Since we can only measure apparent magnitude we
will also need the distance to our collection of Cepheid variables as determined by some means
independent of the P-L relationship. Fortunately the European Space Agency Hipparcos mission
provided accurate parallax data on thousands of stars out to distances of over 300 pc. Within that
range there are many Cepheid variables that can be used to determine the P-L relationship. So for
this armchair observing project you will use Hipparcos parallax data combined with data from other
sources to determine values for the two constants in Formula 1.
So for this project you will need a computer with access to the internet. A spreadsheet program
will be useful, but the calculations can be done with a calculator (or log tables for people who
really want to do it the hard way) and the graphing by hand. To collect the data needed for this
you will need to use the SIMBAD astronomical database located at
http://simbad.u-strasbg.fr/simbad/.
Since there are thousands of Cepheid variables in the
database and you only want to work on classical (Type I) Cepheids and of these only those with
good parallax data from Hipparcos this needs to be pared down a bit. To help a list of 20
classical Cepheids is available as a PDF from
http://www.evilscientist.ca/filemgmt/visit.php?lid=144.
To find the parallax use the query by
identifier link from the main page. To get the period (in days) and maximum apparent visual
magnitude you'll need to look at the General Catalogue of Variable Stars at
http://www.sai.msu.su/gcvs/gcvs/
and use the GCVS query form for each star.
At this point you should have three pieces of information on each of the 20 stars, its parallax,
its maximum visual magnitude and its period. The next thing you need to do is calculate the
distance to each star (in parsecs) from its reported parallax. For this you need Formula 2. In
this case r is the distance in parsecs (pc), pi is not 3.14... but the parallax in arc
seconds.
r = π -1
The parallax given by SIMBAD is in mas or milli-arc seconds. This needs to be converted to arc
seconds before it can be used in Formula 2. To do this is simple, just divide the parallax in mas
by 1000, so 3.2 mas becomes 0.0032 arc seconds. Divide 1 by the arc seconds gives the distance in
parsecs , so 0.0032 arc seconds means a distance of 312.5 pc.
Now that you have the distance to the star in parsecs, you can work out the star's absolute visual
magnitude. This is a measure of the star's intrinsic brightness as opposed to the apparent visual
magnitude which is a measure of how bright the star appears here on Earth. Formula 3 is used to
compute the absolute magnitude, Mv, from the apparent magnitude, mv, given a distance in
parsecs, r.
Mv = mv - (5 log r - 5)
You now have all the information needed to work out the Cepheid P-L relationship as shown in
Formula 1. There are a couple of ways to do this. The easiest is to perform a linear regression on
the data, specifically using log(P) for x and M for y. This can be done with a spreadsheet, some
graphing calculators can also do this. This will get you a and b from Formula 1 directly (a simple
tutorial on this rather powerful statistical method along with a calculator can be found at
http://easycalculation.com/statistics/learn-regression.php
and at other places on the internet if
you want to do this manually). Another method is to use plotting software such as Grace to graph
the points and then perform a curve fit. Finally you could graph the data manually and eyeball a
best fit line, but this would not be very accurate.
You now have a rough Cepheid variable period luminosity relationship (for classical Cepheids!).
You can test out your formula by using the periods you have, substituting them into Formula 1 with
the a and b you calculated. This will give you an absolute magnitude. Solving Formula 3 for r and
using the absolute and apparent magnitudes you can come up with a distance for comparison. It
won't be bang on, first since there is a fair amount of scatter in the data and second there isn't
any compensation for interstellar extinction due to dust between the star and the Earth. Still, as
a first order approximation what you have computed will provide you with the ability to measure
distances upwards of 24 Mpc from data available on the internet.
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