Distance moduli

Life is a narrow vale between the cold and barren peaks of two eternities. We strive in vain to look beyond the heights. 

We cry aloud, and the only answer is the echo of our wailing cry. From the voiceless lips of the unreplying dead there comes no word but in the night of death hope sees a star, and listening love can hear the rustle of a wing. 

Bob bangs on the carbon doors that separate the museum corridor from Miss Muxdroozol s living quarters. Humanity entered the Carbon Age years ago, after soccer ball shaped cages of carbon atoms led to extraordinary new materials. The new materials are a hundred times stronger than any metal and weigh less than ordinary plastics. Unfortunately for Bob, they also block sound very efficiently. 

A few seconds later, Miss Muxdroozol opens her door. Bob, hi, what s up She has a cup of coffee in one hand and a blow dryer in the other. Currently, her hair is almost shoulder length, fashionably cut, with wonderful crimson streaks in it. A few strands of hair wriggle as if they have a life of their own, more like liquid solder than anything biological. Her cheekbones are high and pronounced, her mouth tenderly curved, her eyes skillfully shadowed. The only imperfection she has (though Bob thinks it only adds to her allure) is a F-shaped scar near her left eyebrow—a memento of a childhood accident. She seems self-conscious about it. 

On a low wooden desk is a collection of furculas, V-shaped bones formed by the fusion of the clavicles at the breastbone. Apparently it is customary for Miss Muxdroozol and her kin to collect the bones of their long-dead ancestors. 

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The interstellar extinction has a great effect on distance determination for stars. The B/V index derived in Chapter 2 will be distorted by the presence of interstellar dust, with an amount of radiation in the blue part of the spectrum removed. The difference between the observed colour index and the colour index on which it should have based its temperature is called the colour excess. We defined m to be the measured apparent magnitude, which must now be corrected by an amount Av and added to the distance modulus equation ... 

The reduction of these colours and the V magnitude to physical quantities is of necessity rather approximate and has been described in detail elsewhere (Dopita et al.). Suffice it to say that the Bolometric luminosities are obtained on the assumption that the apparent distance modulus of the supernova is 18.8, corresponding to a visual absorption Av = 0.44, and using the Bolometric Corrections of Carney (1980). The derived Bolometric luminosity is given in Figure 1. 

Fig. 4 - Visual light curve during the first two days. Time zero is defined by the Kamiokande - IMB neutrino signal. Light curves from four models (Table 1) are shown as solid lines. A distance modulus of 18.5 and visual extinction of 0.5m have been adopted. Shown for comparison axe observational data points.

Since no astronomical standard candle is known - all proposed objects have been shown to be essentially non-uniform in one way or another - we nowadays have to calculate and plot the distance modulus for the objects. The scatter around the linear expansion line is less than 0.2 magnitudes or 20% Tonry et al. 2003. Independent of our ignorance of the exact explosion mechanism or the radiation transport in the explosions this proves that SNe la can reliably be used as a distance indicator in the local universe. This situation is very much comparable to the Cepheid stars, where the period-luminosity relation is based on empirical data of objects in the Magellanic Clouds. 

Superb Now the last thing I want to teach you about today is the distance modulus. Bob says the phrase distance modulus slowly so as to heighten his friends suspense and increase their appreciation of his scientific prowess. 

The distance modulus is the difference between the apparent magnitude (m) and the absolute magnitude (M). Bob takes out a business card from his pocket and hands it to Miss Muxdroozol. The card reads ... 

Bob, Distance Modulus Expert Call me some time. 

Here d is the distance from Earth in parsecs, and the distance modulus is simply m-M. Sometimes you see this expressed in a related way M=m + 5-5 log d. Astronomers can measure the apparent magnitude m. Sometimes we can estimate a star s absolute magnitude M from its stellar spectrum, and therefore we can compute the star s distance from us. This can be useful if we want to estimate how far away a distant star is. Remember that stellar parallax is difficult to use when measuring distances to faraway stars. ... 

Miss Muxdroozol, could you retrieve my card with the distance modulus formula from your pocket. I gave it to you before when we were in your living quarters. ... 

In this chapter we also discussed the Cepheid variables, stars whose periods (the time for one cycle of dimness and brightness) are proportional to the stars luminosity. Using the distance modulus formula, this luminosity can be used to estimate interstellar and intergalactic distances. The American astronomer Henrietta Leavitt (1868-1921) discovered the relationship between period and luminosity in Cepheid variables (figure 7.9). I think of her on par with the... 

Neither distance nor speed of recession is directly measurable and it has become standard procedure to convert measurable parameters, such as apparent luminosity, into a distance measure, often relying on vulnerable assumptions. The familiar concept of distance modulus, n = m — M, describes the difference between apparent and absolute magnitude for an object of luminosity L ... 

In the case of Cepheids the period of variability is the measurable property. In order to derive a distance modulus, using (8.4), the period is converted into an absolute magnitude, specified by two arbitrary constants ... 

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