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Tutorial: determine H0 using SQL queries to LEDA

Determination of the Hubble constant: H0

The Hubble constant, H0, is the expansion rate of the Universe, it is counted in km/s/Mpc. To compute it, we divise recession velocity (in km/s) by the distance, measured with a distance estimator as the Tully-Fisher relation. The determination for several individual galaxies can then be averaged.

In the example below, we are applying the sosies method (Paturel, 1984).

In this page, requests will be issued into the box in the bottom frame. You may modify them and you will have to press the submit button to execute them.

The sosies method

We select in the database all galaxies with known velocity and magnitude and similar to a given template for which we know the distance, the magnitude and the radial velocity. Because they are similar, the selected galaxies and the template are believed to have the same intrinsic luminosity (ie. the luminosity of the template, that we know), and we can therefore derive the distance for all these galaxies.

For all the selected galaxies, we have the distance and the velocity so we can compute the Hubble constant:

    H0 = cz / distance
How do we choose sosie galaxies? We simply have to choose a property which does not depend on the distance, as for example the rotation velocity (log(vrot)) and select the galaxies having log(vrot) in a very narrow range around the value for the template galaxy.

How do we choose the template? We choose a galaxy whose distance is known from primary calibrators, for example from Cepheids, tip of the giant branch or surface bightness fluctuation.

Suppose that we choose M31 as template galaxy. The distance modulus of M31 is about 24.3 [mag], its B-band magnitude (btc) is: 3.3 [mag] and the logarithm of its rotation velocity is 2.389.

Execute this request to select the sosies of M31 and compute the mean H0. Note that we excluded the galaxies with v less than 500 km/s which are not usable for H0 determination because the velocity is too strongly affected by individual motions of the galaxies.

The SQL request take the average (function avg) of the ratio velocity to distance for sosie galaxies.
The sosie galaxies are selected as: abs(log(vrot)-2.389)<0.01, ie. the galaxies whose logarithm of rotation velocity log(vrot) differs from the template by less than 0.01.
The distance in Mpc is computed as: 10^((24.3+btc-3.3-25)/5)). Where 24.3 is the distance modulus of M31 and 3.3 its apparent magnitude.

The distance modulus is defined as: dmod = 5 log (distance) + 25 where distance is the distance in Mpc. The distance modulus for any of the selected galaxy (dmod_S)is: dmod_S = dmod_T + (B_S - B_T) Where dmod_T is the distance modulus for the template (here 24.3), B_S the apparent magnitude of the selected galaxy (here btc), and B_T the apparent magnitude of the template (here 3.3.

The standard value of the Hubble constant is about 70 km/s. The previous request give a significantly higher value. Why? The next section give the answer: this is due the the Malmquist bias.

The Malmquist bias

Actually, the parameter used to select sosies, log(vrot) have an intrinsic (or cosmic) dispersion. It means that galaxies with identical log(vrot) may have slightly different intrinsic luminosity. This would not have any effect if the database would be complete, the galaxies with higher luminosity would compensate those with lower luminosity at a given log(vrot).

But the database is not complete! Basically, the completeness is a function of the apparent magnitude: the database is fairly complete at low btc, ie. for bight galaxies, and completeness gradually decreases when galaxies go fainter (many faint galaxies are missing, or more precisely some data, as the radial velocity or the velocity of rotation, are missing for faint galaxies and we cannot use them to apply the sosies method.

The result of the incompleteness is that at larger distance and at a given log(vrot) we see only the brightest objects and the sosies method will underestimate their distance. Hence, at larger distance larger values of H0 are found.

If you execute this request you will see this effect. It computes all the individual values of H0 sorted according to the radial velocity vvir.

How to estimate an unbiased value? As we told, the incompleteness increase with the distance and at short distance the database is complete. So selecting only the nearest objects will give an unbiased value. For example, if you execute this request you will find a reasonable result. To find the range of velocity (v) you can try empirically different values to see when the bias start to be seen. If you select only the most nearby galaxies, the number of galaxies will be small and the statistical noise important... so try to find the larger distance for which the bias does not seem important.

To count the selected galaxies, execute this request.
It is of course possible to make more rigorous correction of the bias...
The sosies method is not the only one to determine distances, but its advantage is that it does not rely on any scaling relation. The most popular and precise method for Spiral galaxies is the Tully-Fisher relation.


Do the same exercice with M33 as template (btc=5.73, distance modulus = 24.5 and log(vrot) = 2.016). Compare with the results you got for M31. Why does the bias appear at smaller distances?

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