estimating galaxy ages from PNe ?

[Widdekind]

The number of stars, of mass [math]M[/math], decreases with increasing mass, according to a power-law, [math]N(M) \propto M^{-\alpha}[/math], where [math]\alpha \approx 2[/math]. And, the lifetime of stars, defined by the stars' ratio of mass to luminosity, decreases with increasing mass, according to a power-law, [math]\tau(M) \propto M / L(M) \propto M^{-\beta}[/math], where [math]\beta \approx 2[/math]. (Note that the mass-to-light ratio, observed by astronomers, is a direct measures, of the effective age, of the stellar population.)

 

Er go, the number of stars, of lifetime [math]\tau[/math], can be calculated

 

[math]N(\tau)d\tau = N(M)dM[/math]

 

[math]N(\tau) = N(M) \frac{dM}{d\tau} \approx M^{-2} \left( \frac{d\tau}{dM} \right)^{-1}[/math]

 

[math]\propto M^{-2} \left( M^{-3} \right)^{-1} \propto M[/math]

 

[math]\therefore N(\tau) \propto \tau^{-\frac{1}{2}}[/math]

To normalize that lifetime distribution density

 

[math]N_{tot} \equiv \int_{\approx 0}^{\tau_{max}} C \tau^{-\frac{1}{2}} d\tau \approx 2 C \sqrt{\tau_{max}}[/math]

 

[math]\therefore C = \frac{1}{2} \frac{N_{tot}}{\sqrt{\tau_{max}}}[/math]

 

[math]\therefore N(\tau) = \frac{1}{2} \frac{N_{tot}}{\tau_{max}} \left( \frac{\tau}{\tau_{max}} \right)^{-\frac{1}{2}}[/math]

Now, as a star population ages, all stars with lifetimes shorter than that age, will already have evolved off of the MS. So, at age [math]\tau_0[/math], [math]N(\tau_0)[/math] is the number of stars, then evolving off of the MS. Now, for "young" stellar populations, dominated by big, bright, blue OB stars, that rate will equal the number of SNe, generated by those OB stars, then evolving off of the MS, and undergoing SNe. And, for "old" stellar populations, dominated by sun-like stars, that rate will equal the number of PNe, generated by those G-like stars, then evolving off of the MS.

 

Qualitatively, since [math]N(\tau) \approx \frac{1}{2} \frac{N_{tot}}{\tau_{max}} x^{-\frac{1}{2}}[/math], younger star populations will have many more stars leaving the MS per year, cp. "star-burst galaxies"; and older star populations will have gradually ever fewer stars leaving the MS per year. Quantitatively, our galactic disk contains [math]\approx 400 \times 10^9[/math] stars; is [math]\tau \approx 10 Gyr[/math] in age; and the maximum stellar lifetime, of minimum mass stars [math]M_{min} \approx 0.1 M_{\odot}[/math], is [math]\tau_{max} \approx 1000 \; Gyr[/math]. Thus [math]N(\tau) \approx 1/yr[/math], i.e. this simple picture, of a single-aged stellar population, predicts the formation of approximately 1 PN / yr. Now, PN persist for [math]\approx 10^4 yr[/math]; and there are [math]\approx 10^4[/math] PN in our galaxy. Assuming equilibrium, that implies a rate of formation, of PNe, of approximately 1 PN / yr.

 

Yet, billions of years ago, that rate of PNe generation, would plausibly have been higher. Are PNe more common, in galaxies, observed at high redshift ?? Given the number of stars in some population [math]N_{tot}[/math]; and assuming a minimum star-mass, i.e. maximum star-lifetime [math]\tau_{max}[/math] then the number of PNe (assuming equilibrium, and a 'known' lifetime for PNe, e.g. 10⁴yr) implies the age of the star population.

Edited January 13, 2012 by Widdekind

[pantheory]

We can only see PNe (planetary nebula) in our own galaxy and maybe in a few of our peripheral galaxies, and possibly three of our neighboring galaxies like Andromeda. Even individual stars are difficult to observe in galaxies beyond these distances other than the biggest, brightest stars, or supernova, all of which would not have observable PNe. In our own galaxy when looking at stars with planetary nebula, we can only guess whether it is a first generation star, second generation star or even older.

 

We can certainly make a good guess how long ago a planetary nebula was created since it only lasts thousands of years compared to most stellar lifetimes of billions of years. Therefore galaxy age-dating by this mechanism seemingly could not be done.

 

(quote from link)

A planetary nebula is an emission nebula consisting of an expanding glowing shell of ionized gas ejected during the asymptotic giant branch phase of certain types of stars late in their life. This name originated with their first discovery in the 18th century because of their similarity in appearance to giant planets when viewed through small optical telescopes, and is otherwise unrelated to the planets of the solar system. They are a relatively short-lived phenomenon, lasting a few tens of thousands of years, compared to a typical stellar lifetime of several billion years.

(bold added)

 

http://en.wikipedia....lanetary_nebula

Edited January 17, 2012 by pantheory