by Surjit Wadhwa

In this the first instalment of the Armchair Astrophysicist I propose to develop a continuously updating cosmological clock to measure the age of the Universe hence the title “How old is the Universe Today”. In order to calculate the age of the universe one needs to know the value of one number – the Hubble Constant. Calculating the Hubble constant is easier said than done. I propose to outline a way of utilizing the vast amount of data that is available to us online to develop a spreadsheet that can be quickly updated as soon as new data becomes available to generate an up to date value (as per our methods – see below) of the Hubble constant and hence the age of the universe. It must be noted that the methods I describe are not scientifically rigorous and prone to many many errors and would not be acceptable for a “scientific” publication however the sheer sample size I propose to use will, I believe, have the effect of canceling many of the errors to yield a very good approximation.

In the 1920's, Edwin P. Hubble discovered that distant galaxies were all moving away from the Milky Way. Not only that, the farther away he observed, the faster the galaxies were receding. He found the relationship that is now known as Hubble's Law: the recessional velocity of a galaxy is proportional to its distance from us. Mathematically the relationship can be written as:

Where **v** is the galaxy's velocity (in km/sec), **d** is the distance to the galaxy (in *megaparsecs*; 1 Mpc = 1 million parsecs), and **H**is the proportionality constant, called "The Hubble constant." This equation is telling us that a galaxy moving away from us twice as fast as another galaxy will be twice as far away, a galaxy moving away from us three times as fast will be three times farther away, and so on.

From the above relationship it is clear that H = v/d with units of km/sec/Mpc. Now 1 Megaparsec is actually a distance of 3.09 x 10^{19 }km so the units of distance (km) cancel each other leaving us the units per second (/sec). So the Hubble constant with the units “per second” and represents a “rate of change” of something. Without going into the physics it can be shown that H is actually the rate of expansion of the universe. In its simplest form we could simply reverse the rate of expansion to arrive at the age of universe, that is, age of the universe equals 1/H.

This simplification however assumes that the rate of expansion has remained constant since the big bang which is not the case. It is unlikely the Hubble constant has been constant over the lifetime of the universe. The universe has a lot of mass, and gravity tries to pull all that mass together. Gravity slows the expansion, just as a ball thrown vertically upwards decelerates from the gravitational pull of the earth. The age of the universe computed above corresponds to an "empty universe", one in which the average density of the universe (ρ) is zero. The rate at which the expansion has slowed over time depends on the average density of the universe, higher the density, the greater the deceleration. If the density is lower than some critical value we call ρ_{c}, the universe will continue to expand forever. If the density is higher than ρ_{c} the gravitational attraction will eventually halt the expansion, then begin a contraction toward what has been dubbed the "Big Crunch". If ρ is equal to ρ_{c}the expansion will slow to a stop, but only after an infinite amount of time has passed.

Calculating the density of the universe is no simple task. In this exercise we will make the assumption that the density of the universe is equal to the critical density and again without going into the details it can be shown that in such a case the age of the universe can be estimated to be 0.667/H.

Now that we know how to find the age of the universe lets get the velocities and distances of some galaxies and get on with it. Not quite. Getting the recessional velocity is quite simple using redshift. Getting an accurate distance to a galaxy is another matter. I will leave it up the readers to look up details regarding redshifts but briefly it is known that for objects moving away from the observer the light coming from the object is shifted towards the longer wavelength (or red) part of the spectrum. The amount of deviation towards the red side is known as the redshift represented by the symbol z. As with the Hubble constant if we choose our units carefully we can show the recessional velocity (v) is equal to the redshift (z) multiplied by the speed of light (c) or v=cz.

Finding the distance to a galaxy is not as easy as it might seem. The best measure of distance we have is light. We know that light dims in proportion to the square of the distance such that a light source of a known brightness would be 4 times fainter if it was twice as far. This is known as the inverse square law of light.

We need to have a “standard candle” or some celestial phenomenon whose exact brightness (or its absolute magnitude - M) we know. When we observe such a phenomenon in a far away galaxy it will appear faint (apparent magnitude – m). We can use the inverse square law of light and the difference in the absolute and apparent magnitude to calculate its distance. There are not many celestial phenomenon that occur on a regular basis, are visible and so homogenous that all of them can be thought of being exactly the same no matter where they occur. The Cepheid variables are an example of a standard candle. In this group of stars the period of the variability is related to the absolute magnitudes of the stars so by observing the variability of Cepheids in other galaxies we can calculate the difference between the absolute (which we know from the period) and the observed (apparent) magnitude and hence the distance. Unfortunately there are not many galaxies where we can observe individual stars and even with the best instruments we can only observe Cepheids in galaxies that are relatively close to us.

Recently it has been shown that Type 1a supernovae represent good standard candles. These explosive events are not only almost all equally bright (absolute magnitude ~ -19.5) and follow a predictable light curve. In addition they have spectral characteristics that seem to develop in a predictable way over a number of weeks from well before maximum light to months after maximum light. Typically if you can find the maximum brightness (apparent magnitude) of a type 1a supernova and the redshift of a galaxy you can calculate the Hubble constant. Unfortunately, type1a events are not very common and if one is detected (usually near maximum light), by the time it is reported and telescope time allocated a considerable amount of the light curve is lost. As such a large number of detected events go completely unanalyzed.

In this exercise I describe a method by which such detected but unanalyzed events can be used to calculate the Hubble constant and hence the age of the Universe.

Based on the properties of type 1a supernovae it is possible to construct a template light curve for a typical event. In figure 1 I have plotted a published template which can be used to predict the absolute magnitude of all other “normal” type 1a events. For ease of use I fitted a 4th order polynomial to the data so that the function could be used directly into a spreadsheet format. Next we need some data on magnitudes and radial velocities. The ASNSW subscribes to the IAU circulars which report most events and these can be used as the basic source of data.

For this project to get a flying start I decided to go back and look at type 1a events from 2000 onwards. The following site at Rochester (http://www.supernovae.net/) keeps an archive of IAU data on supernovae including type 1a events. I will go through the process by way of an example of how to find the distance and velocity that we can use.

In this example I will go through the steps for finding the distance module and radial velocity of SN2000B.

From the Rochester site we see that SN2000B in NGC 2320 was discovered on 11/1/2000 and reported in IAUC 7347. From a link on the same site to IAUC 7351 we see that a spectrum of supernova was obtained on Jan 22.8 and this confirms that SN 2000B is a type 1a and most importantly from the spectral features it is possible to say that around Jan 23 the supernova was near maximum light. The site also has a link to all observations (including amateur) for the supernova and from the reported observations we see that from Jan 22 to Jan 26 the supernova was in the range 15.8 to 16.0 with observations before and after this time being fainter. In this easy example we can safely assume that the maximum brightness of the supernova was about magnitude 15.9. Now going to the NED site (also linked from the Rochester site) we see that the recession velocity of NGC 2320 is 5944km/sec and as the supernova was in this galaxy it must also have the same recession velocity.

Now we know from our template that at maximum light type 1a supernovae have an absolute magnitude (M) of -19.5 and our observed apparent (m) maximum is magnitude is 15.9 therefore the distance module (m - M) is 35.4. Without going into detail the distance module can be converted into Megaparsecs (Mpc) as follows:

Distance (in Mpc) = (10^((5+D)/5))/10^6 where D is the distance module (m - M).

So for SN 2000B the distance in MPc is calculated as approximately 121 MPc. So the Hubble constant for SN 2000B is 5499/121 = about 45 km/sec/MPc.

Now not all observations are that easy. Take for example SN 2000cf which was found on 8/5/2000 and a spectrum taken on May 12 indicated that it was day 6 post maximum. Looking at the observations the most reliable estimate appears to be on May 11 at magnitude 17.1. We do not have an observed maximum. All is not lost however; we know from the spectrum that on May 12 the supernova was day 6 post maximum therefore on May 11, the date of our observation, the supernova must have been day 5 post maximum. From our template and the fitted curve we can estimate the absolute magnitude of the supernova on day 5 post maximum. The estimated absolute magnitude from our template for a type 1a supernova at day 5 is -19.32 therefore our distance module for SN 2000cf is 17.1 – (-19.32) = 36.42 = 192MPc. With the recession velocity of 10920km/sec yields a Hubble constant value of about 56 km/sec/MPc.

It is not necessary to calculate the Hubble constant for each event. A better option is to plot the distance against the recession velocity for a large number of events and use linear regression to find the line of best fit. This can be done quickly on a spreadsheet such that the only inputs required are the observed magnitude and the “days” pre or post maximum for the observation. So theoretically it is possible to record any supernova 1a event for which there is one reliable magnitude estimate and an accurate spectrum which records days before or after maximum. I have done this for all type 1a events (111 events) from 2000 to 2003 (inclusive) and the fitted graph (figure 2) known as the Hubble diagram is illustrated below. The fitted curve yields a slope of about 49.5 which is the Hubble constant for the sample.

The age of the universe based on methods described in the theory section with a H of 49.5 is approximately 13.3 billion years which is consistent with other estimations. Readers may note that the Hubble constant based on “high z” value supernovae 1A is about 70 and controversy continues as to the true value. The near (low z) advocate a value between 55 and 63 which again is higher than our estimate however there are many probable errors in our sample and our sample purely from the faint nature of these objects is biased towards nearby and hence very low z value galaxies. It is well known that galaxies with very low z values have radial velocities influenced by local gravitational effects with the classical example being M31 (Andromeda Galaxy) which actually has a blue shift and is actually moving towards us rather than away. As such it is possible that the estimated radial velocities are somewhat erroneous.

- At present I have used the observations from year 2000 to 2003 however to bring the data up to date and increase our sample size we need to collate all observations and than continually update the clock as new discoveries are made. This may seem hard but is actually not that difficult. If a dedicated group of say 5 members decide to take on the task of collating 1 observation a day (all up this would take about 5-10 minutes) than within weeks we would be up to date and thereafter would only need to update the clock with new observations. I am happy to meet with a small group and practically go through the exercise if anyone is interested.
- Make observations of new discoveries: Keep an eye on the IAU circulars and if a supernova is discovered and within the reach your telescopes take a digital photo. The photo doesn’t have to be pretty just useable. I know there are a number of members who already do digital photometry so getting magnitude estimates will not be hard. You don’t have to have filters to make useful observations, if however you can afford to purchase a “V” band filter it would yield the most useable results.

Figure 1 – Type 1a Supernovae absolute magnitude template with fitted polynomial

**Age of Universe: 13.25505757 billion years**

Figure 2: Plot of Velocity vs Distance for our sample size yielding a Hubble Constant value of 49.2.

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