Dark Matter and the Initial Mass Function

[Widdekind]

The Mass Distribution of stellar (star) populations is described by the Initial Mass Function (IMF), denoted N(M) dM, which describes the number of stars between masses M & M+dM, in some specified volume of space. The IMF is typically modeled as a Power Law, with exponent [math]\alpha[/math] around -7/3:

[math]N(m) dm \propto M^{-\alpha}[/math]

Now, Stellar Luminosities scale as L_* ~ M_*⁴(Bowers & Deeming. Astrophysics I: Stars, pg. 28). So, from the IMF, we can calculate the expected overall Mass-to-Light Ratio of that population of stars:

[math]\Upsilon \equiv \frac{M_{tot}}{L_{tot}} = \frac{\int M \times N(M) dM}{\int L(M) \times N(M) dM} \approx \frac{\int M \times M^{-\alpha} dM}{\int L_{\odot} \times \left(\frac{M}{M_{\odot}}\right)^{4} \times M^{-\alpha} dM} = \left(\frac{M_{\odot}^{4}}{L_{\odot}}\right) \frac{\int M^{1 - \alpha} dM}{\int M^{4 - \alpha} dM}[/math]

[math] = \Upsilon_{\odot} \; M_{\odot}^{3} \left( \frac{5 - \alpha}{2 - \alpha} \right) \frac{ M^{2 - \alpha} |_{M_{min}}^{M_{max}} }{M^{5 - \alpha} |_{M_{min}}^{M_{max}} }[/math]

[math] = \Upsilon_{\odot} \left( \frac{5 - \alpha}{2 - \alpha} \right) \frac{ \mu^{2 - \alpha} |_{\mu_{min}}^{\mu_{max}} }{\mu^{5 - \alpha} |_{\mu_{min}}^{\mu_{max}} }[/math]

where [math]\mu \equiv M / M_{\odot}[/math]. Plugging in approximate values for the IMF exponent ([math]\alpha \approx 7/3[/math]), as well as the minimum & maximum star masses ([math]\mu_{min} \approx 0.1, \mu_{max} \approx 100[/math]), we obtain:

[math] \frac{\Upsilon}{\Upsilon_{\odot}} \approx 7.2 \times 10^{-5}[/math]

However, the normalized Mass-to-Light Ratios ([math]\frac{\Upsilon}{\Upsilon_{\odot}}[/math]) of most Galaxies range from 2 - 10, some 5 Orders-of-Magnitude greater.

 

CONCLUSION: Standard stellar populations cannot account for the observed Galactic Mass-to-Light Ratios. This could be consistent with the inference of copious quantities of Dark Matter in the same.

[granpa]

so even though high mass stars are less numerous they are so much brighter that most of the light we see from other galaxies is coming from a relatively small population of very high mass stars.

 

so the more low mass stars or dark matter there is per high mass star the higher the mass to light ratio.

 

 

dust in that galaxy blocking the light from reacing us would also raise the mass to light ratio. but i dont think that there is a lot of dust in giant elliptical galaxies. I assume that they also fit the data given

[Widdekind]

For a power-law IMF, the you must increase the exponent [math]\alpha[math] above about 4.5, before you start getting M-L Ratios of order 10^0-1.

 

Note, in particular, that you must increase the IMF exponent above four (4), from the Luminosity-Mass estimate (L_* ~ M_*⁴). Without doing so, although the bigger brighter stars are less frequent (M_*^-2.3), they still produce much more radiation (M_*^2.3 x M_*⁺⁴ = M_*^+1.7). Thus, in sum, you get such huge quantities of radiation, even from a few such stars, that it depresses the M-L Ratio.

 

Only if bigger & brighter stars were incredibly infrequent ([math]\alpha[/math] ~ 4.5+), would even their incredible brilliances be offset by so many smaller dimmer stars as to darken the whole population.

[Widdekind]

The Mass Distribution of stellar (star) populations is described by the Initial Mass Function (IMF), denoted N(M) dM, which describes the number of stars between masses M & M+dM, in some specified volume of space. The IMF is typically modeled as a Power Law, with exponent [math]\alpha[/math] around -7/3:

[math]N(m) dm \propto M^{-\alpha}[/math]

Now, Stellar Luminosities scale as L_* ~ M_*⁴(Bowers & Deeming. Astrophysics I: Stars, pg. 28). So, from the IMF, we can calculate the expected overall Mass-to-Light Ratio of that population of stars:

[math]\Upsilon \equiv \frac{M_{tot}}{L_{tot}} = \frac{\int M \times N(M) dM}{\int L(M) \times N(M) dM} \approx \frac{\int M \times M^{-\alpha} dM}{\int L_{\odot} \times \left(\frac{M}{M_{\odot}}\right)^{4} \times M^{-\alpha} dM}[/math]

[math] = \left(\frac{M_{\odot}^{4}}{L_{\odot}}\right) \frac{\int M^{1 - \alpha} dM}{\int M^{4 - \alpha} dM}[/math]

so:

[math] = \Upsilon_{\odot} \; M_{\odot}^{3} \left( \frac{5 - \alpha}{2 - \alpha} \right) \frac{ M^{2 - \alpha} |_{M_{min}}^{M_{max}} }{M^{5 - \alpha} |_{M_{min}}^{M_{max}} }[/math]

[math] = \Upsilon_{\odot} \left( \frac{5 - \alpha}{2 - \alpha} \right) \frac{ \mu^{2 - \alpha} |_{\mu_{min}}^{\mu_{max}} }{\mu^{5 - \alpha} |_{\mu_{min}}^{\mu_{max}} }[/math]

where [math]\mu \equiv M / M_{\odot}[/math]. Plugging in approximate values for the IMF exponent ([math]\alpha \approx 7/3[/math]), as well as the minimum & maximum star masses ([math]\mu_{min} \approx 0.1, \mu_{max} \approx 100[/math]), we obtain:

[math] \frac{\Upsilon}{\Upsilon_{\odot}} \approx 7.2 \times 10^{-5}[/math]

However, the normalized Mass-to-Light Ratios ([math]\frac{\Upsilon}{\Upsilon_{\odot}}[/math]) of most Galaxies range from 2 - 10, some 5 Orders-of-Magnitude greater.

 

CONCLUSION: Standard stellar populations cannot account for the observed Galactic Mass-to-Light Ratios. This could be consistent with the inference of copious quantities of Dark Matter in the same.

 

For [math]\mu_{min} = 0.06, \mu_{max} = 150[/math], choosing [math]\alpha \approx 13/3[/math] yields a Mass-to-Light ratio of roughly 7. Is it possible, that the IMF for stars is significantly "steeper" than usually estimated ?? If "Brown Dwarfs", being dim, were often overlooked, wouldn't that artificially make the IMF look far "flatter", reducing the "apparent index" down to [math]\alpha \approx 7/3[/math] ??

---

Merged post follows:

Consecutive posts merged

Can the Hubble (Ultra, Double) Deep Fields detect Globular Clusters ?? Could there be sub-galactic scale structures, along the lines of Globular Clusters, in copious quantities across the Cosmos (enough to matter, for the Cosmic matter density) ??

[Widdekind]

For a power-law IMF, the you must increase the exponent [math]alpha[/math] above about 4.5, before you start getting M-L Ratios of order 10^0-1.

 

 

Note, in particular, that you must increase the IMF exponent above four (4), from the Luminosity-Mass estimate (L_* ~ M_*⁴). Without doing so, although the bigger brighter stars are less frequent (M_*^-2.3), they still produce much more radiation (M_*^2.3 x M_*⁺⁴ = M_*^+1.7). Thus, in sum, you get such huge quantities of radiation, even from a few such stars, that it depresses the M-L Ratio.

 

 

Only if bigger & brighter stars were incredibly infrequent ([math]alpha[/math] ~ 4.5+), would even their incredible brilliances be offset by so many smaller dimmer stars as to darken the whole population.

 

quick correction, sorry, thanks

 

---

 

Using a more realistic M-L fxn, w.h.t.:

 

The actual coefficients, in the last two formula, are [math]\sqrt{2}[/math], and [math]5.1 \times 10^4[/math], respectively.

 

Again, the Mass Distribution of stellar (star) populations is described by the Initial Mass Function (IMF), denoted N(M) dM, which describes the number of stars between masses M & M+dM, in some specified volume of space. The IMF is typically modeled as a Power Law, with exponent [math]\alpha[/math] around -7/3:

 

[math]N(m) dm \propto M^{-\alpha}[/math]

And, now more generally, Stellar Luminosities scale as L_* ~ M_*^b. And so, from the now-multi-faceted IMF, we can calculate the expected overall Mass-to-Light Ratio of that population of stars. Piece-by-piece, w.h.t.:

 

[math]\Upsilon \equiv \frac{M_{tot}}{L_{tot}} = \frac{\int M \times N(M) dM}{\int L(M) \times N(M) dM} \approx \frac{\int M \times M^{-\alpha} dM}{\sum \int L_{\odot} \times k \times \left(\frac{M}{M_{\odot}}\right)^{\beta} \times M^{-\alpha} dM} = \left( \frac{1}{L_{\odot}} \right) \frac{\int M^{1 - \alpha} dM}{\sum \frac{k}{M_{\odot}^{\beta}} \int M^{\beta - \alpha} dM}[/math]

 

[math] = \Upsilon_{\odot} \frac{ M^{2 - \alpha} |_{M_{min}}^{M_{max}} }{2-\alpha} \frac{1}{\sum \frac{k}{M_{\odot}^{\beta - 1}} \frac{ M^{1 +\beta - \alpha} |_{M_{min}}^{M_{max}} }{1+\beta-\alpha}}[/math]

 

[math] = \Upsilon_{\odot} \frac{ \mu^{2 - \alpha} |_{\mu_{min}}^{\mu_{max}} }{2-\alpha} \frac{1}{\sum k \frac{ \mu^{1 +\beta - \alpha} |_{\mu_{min}}^{\mu_{max}} }{1+\beta-\alpha}}[/math]

 

[math]\approx 5 \times 10^{-4}[/math]

 

where [math]\mu \equiv M / M_{\odot}, \alpha \approx 7/3[/math], as per previous; and, [math]\mu_{min} \approx 0.01, \mu_{max} \approx 150[/math], accounting for Brown Dwarfs. This is an order-of-magnitude greater than the simpler model above (b/c this more accurate model dramatically dampens the light loosed by the biggest & brightest stars, radically reducing their Luminosity exponent from 4 to 3.5, and then to 1). Yet, with the normalized Mass-to-Light Ratios ([math]\frac{\Upsilon}{\Upsilon_{\odot}}[/math]) of most Galaxies ranging from 2 - 10, some 4 Orders-of-Magnitude more. Such still cannot account, then, for observations. As before, most of the light comes from the larger stars. The luminosities, by bin, are:

 

mass_range  total_luminosity
0.01-0.43   0.11
0.43-2      2.3
2-20        430
20-150      27,000

Qualitatively, then, all of the mass resides in dim, dark stars; whereas, all of the light is loosed from the few furious big, bright stars. Such is characteristic, of the Dark Matter "where is everything" problem.

 

Note, that by eliminating both bins of super-solar stars ([math]M > 2 M_{\odot}[/math]), M/L ~ 5. Thus, you could account for observed M/L, by assuming that the stellar populations are truncated, lacking larger stars (cf. Globular Clusters w/ Main-Sequence Turnoff). High M/L seemingly suggests relic residual remnant post-Star-Formation populations.

Edited May 29, 2011 by Widdekind

[Widdekind]

According to this analysis of the diffuse, panoramic, Extra-Galactic Light (EBL), the cosmic-average space density, of White Dwarfs, is ~10% of the total stellar space density; and, generates ~20 nW / m² / ster worth of diffuse background radiation. Now, Brown Dwarfs are ~100x less luminous, than White Dwarfs; and, are 20-100x less massive. Thus, Brown Dwarfs have M/L ratios comparable to (x1-5) those of White Dwarfs. And, there may be as much as ~40 nW / m² / ster of observed, but currently-un-accounted-for, EBL. So, if all of that discrepancy shortfall were attributed to scattered, field Brown Dwarfs, then their inferred space mass density would be (40/20) x 1-5 = 2-10x that of White Dwarfs; or, 20-100% that of all currently-visible stars.

 

This is completely consistent, with the currently comprehended IMF ([math]\alpha \approx 7/3[/math]), extended down through these substar Brown Dwarf masses ([math]0.01-0.08 M_{\odot}[/math]). For, roughly half of all star mass may reside in such substars:

 

[math]\frac{M_{bd}}{M_{*}} = \frac{\mu_{bd}^{2-\alpha} |_{0.01}^{0.08}}{\mu_{*}^{2-\alpha} |_{0.01}^{100}} \approx 0.4-0.5[/math]

depending slightly on the choices for the mass-limits defining Brown Dwarfs. Does this not imply the (remote) possibility, that 'HDF-caliber' deep exposures, of the IGM, may one day reveal a vast population of Brown Dwarfs, embodying a total mass comparable to that currently visible ? However, even doubling the total space star density, to include Brown Dwarfs, would not account for current comprehensions of the extreme degree of DM present in space.

[Widdekind]

For most MS stars, [math]L \propto M^{3.5}[/math]:

 

(

utk

)

Accordingly, one can calculate the (theoretical) Mass-to-Light ratio, of a population of stars, described by a given IMF exponent [math]7/3 < \alpha < 4[/math]; and, by a given maximum mass [math]0.7 < M < 2[/math], i.e. MS "turn off" mass (per OP, but with L(M) exponent 4 --> 3.5):

 

(calculations & visualization

via

SciLab)

Even ancient stellar populations, with [math]M_{to} < 1 M_{\odot}[/math]; and, characterized by very steep, "bottom heavy" IMF, with [math]\alpha \rightarrow 4[/math]; can only generate [math]\Upsilon \approx 50[/math]. Thus, observation of stellar populations, with larger Mass-to-Light ratios, may demand the presence of large "reservoirs" of "dim-or-dark matter", e.g. the now-dim remnants, of dead stars, long long ago aged off the MS; or, non-luminous planetary bodies, etc.

 

Note, the generated Mass-to-Light ratios, are not very sensitive, to retention, by the still-surviving stellar population, of some fraction, of the post-MS stars' masses, e.g. [math]f_{remnant} \approx 1/2[/math]:

 

Edited December 16, 2011 by Widdekind

[Widdekind]

"target" Mass-to-Light ratio ?? [math]\left( \Upsilon_{cosmos} \approx 150 \right)[/math]

 

Large galaxy clusters are thought to be representative samples, of matter, in regions of "structure", in our universe, i.e. the 'CW'. And, galaxy clusters evidence Mass-to-Light ratios [math]\Upsilon \approx 150[/math], e.g. Coma Cluster. And, our universe, as a whole, evidences [math]\Upsilon \approx 600[/math], i.e. 4x larger than that of "structured" regions of space (Ryden & Peterson. Foundations of Astrophysics, p.514-525). And, that is easily explicable, as deriving from the fact that non-luminous, i.e. "dark", "voids" comprise [math]\approx 3/4[/math] of all matter-and-energy-and-volume of our universe (Wiltshire 2011). Therefore, simplistically stated, our universe may be an inhomogeneous, "two phase" medium, composed of [math]\approx 1/4[/math] "structures" [math]\left( \Upsilon_{cosmos} \approx 150 \right)[/math]; and, of [math]\approx 3/4[/math] "voids" [math]\left( \Upsilon_{cosmos} \approx \infty \right)[/math]; generating an over-all [math]\left( \Upsilon_{cosmos} \approx 600 \right)[/math], on "large" scales >100 Mpc, on which our universe is quasi-homogenous (Maoz. Astrophysics in a Nutshell).

 

What could account, for such high Mass-to-Light ratios ?? For example, high-mass stars form relatively infrequently; and, account for a small fraction, of over-all, aggregate, star mass, even when they are "whole" on the MS. Thus, the post-stellar remnants, of "dead" high-mass stars, account for an even smaller fraction, of aggregate star mass. So, inclusion of post-stellar remnants does not appear to be able to generate, i.e. explain, observed high [math]\Upsilon \approx 150[/math]. If so, then "some other mechanism", i.e. "some other reservoir of dim-or-dark matter" must "buoy" observed [math]\Upsilon[/math]. Stellar IMFs can only generate higher [math]\Upsilon[/math] by mathematically including more, smaller, dimmer star-like objects, i.e. "Red Dwarves" (via "steeper" IMFs); or, "Brown Dwarves" (via lower mass-cutoff threshold). The latter is less inconsistent, with current observations, of star-formation, in our galactic neighborhood, i.e. "simply extend the observed IMF into lower (sub-stellar, BD) masses".

 

 

 

"Red Dwarves" -- steeper IMFs ??

 

"Bottom heavy", i.e. "steep", IMFs, with [math]\alpha \approx 5[/math], generate [math]\Upsilon \approx 150[/math]:

 

 

 

 

"Brown Dwarves" -- include sub-stars ?? [math]\left( 0.013 < \frac{M_{BD}}{M_{\odot}} < 0.080 \right)[/math]

 

"Conventional" IMFs, characteristic of our local galactic neighborhood, can account for observed high Mass-to-Light ratios, if non-luminous, sub-stellar, 'Brown Dwarves' (BDs) are "added to the mix" of stars, i.e. BDs are generated according to the same IMF [math]\left( \alpha \approx 5/2 \right)[/math], but add only mass, not light. The following figures are similar, except that the latter "zooms" in on regions of lower IMF exponents, i.e. focuses on "flatter" IMFs. Inclusion of hypothesized populations, of BDs, enables the generation of "arbitrarily large" [math]\Upsilon[/math]:

 

 

[imatfaal]

!

Moderator Note

 

DeputyDaves

 

I have hidden your post. Do not post the same message in multiple threads merely because there is a passing relevance. Additionally please note that questions in the main forums regarding dark matter should not be answered with speculative notions.

 

[Leif]

Circular arguments using algorithms has nothing of value in them intrinsically.

 

Edited August 22, 2013 by Leif