Could some sort of "primordial turbulence" have imparted motion, to collapsing cloud 'clumps', so that their collisions could have triggered star-formation bursts (w/o need for as much DM) ??

could 'MACHOs' account for halo DM distributions ?

 

Assuming a power-law IMF, for the 'particles', w.h.t.:

 

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

 

[math]M = \int_{m_0}^{\infty} C m^{1-\alpha} dm = - C m_0^{2-\alpha} \frac{1}{2-\alpha}[/math]

 

[math]\therefore C = ( \alpha - 2 ) \, m_0^{\alpha - 2} \, M[/math]

By the process of "mass segregation", in a 'virialized' or 'relaxed' cluster, all more massive 'particles' will 'sink' center-wards; and, all less massive 'particles' will 'float' periphery-wards. Thus, w.h.t.:

 

[math]M_{<r} = M_{>m®} = \int_{m®}^{\infty} C m^{1-\alpha} dm = M \left( \frac{m®}{m_0} \right)^{2-\alpha}[/math]

By the process of "energy equipartition", in a 'virialized' or 'relaxed' cluster, all 'particles' will have equal energies:

 

[math]\frac{1}{2} m v^2 \equiv K[/math]

 

[math]-\frac{G M_{<r} m}{r} \equiv U[/math]

 

[math]K + U \equiv \bar{E} = \frac{U}{2} = -|K|[/math]

Thus, also equating the centripetal & gravitational forces acting on the 'particles', w.h.t.:

 

[math]\frac{v^2}{r} = \frac{G M_{<r}}{r^2}[/math]

 

[math]\frac{2 |\bar{E}|}{m®} = \frac{G M_{>m®}}{r}[/math]

 

[math]2 |\bar{E}| r = G M m_0 \left( \frac{m®}{m_0} \right)^{3-\alpha}[/math]

Then, defining:

 

[math]\mu \equiv \frac{m®}{m_0}[/math]

 

[math]x \equiv \frac{r}{R}[/math]

 

[math]R \equiv \frac{2 |\bar{E}|}{G M m_0}[/math]

w.h.t.:

 

[math]x = \mu^{3 - \alpha}[/math]

Now, to determine the radial density profile, w.h.t.:

 

[math]4 \pi r^2 dr \rho® = C m®^{1-\alpha} dm[/math]

 

[math]4 \pi R^3 x^2 dx \rho(x) = ( \alpha - 2 ) M \mu^{1- \alpha } d\mu[/math]

Defining:

 

[math]\bar{\rho} \equiv \frac{M}{4 \pi R^3}[/math]

and using:

 

[math]dx = (3 - \alpha) \mu^{2 - \alpha} d\mu[/math]

then, w.h.t.:

 

[math]\rho(x) = \frac{\alpha - 2}{\alpha - 3} \; \bar{\rho} \; \mu^{2 \alpha - 6} \mu^{1- \alpha } \mu^{\alpha - 2}[/math]

 

[math]\; = \frac{\alpha - 2}{\alpha - 3} \; \bar{\rho} \; \mu^{2 \alpha - 7}[/math]

 

[math]\; = \frac{\alpha - 2}{\alpha - 3} \; \bar{\rho} \; x^{\frac{2 \alpha - 7}{3 - \alpha}}[/math]

Or, restated, w.h.t.:

 

[math]\rho(x) = \frac{\alpha - 2}{\alpha - 3} \; \bar{\rho} \; x^{- \left( {\frac{2 \alpha - 7}{\alpha - 3}} \right) }[/math]

According to these calculations, 'flat' IMF, having [math]\alpha < 7/2[/math], are not stable against dispersion, generating radial densities that increase outwards. Thus, only 'steep' IMF, having [math]\alpha > 7/2[/math], generate centrally concentrated radial density profiles. Observations, and numerical simulations, suggest that [math]\rho \propto x^{-2}[/math], which corresponds, according to these calculations, to the limit [math]\alpha \rightarrow \infty[/math]. If so, then this analysis indicates, that DM halos can be mathematically pictured, as populations of 'particles', having 'ultra steep' IMFs.

 

For example, if [math]\alpha \approx 7/2[/math], then:

 

[math]\rho(x) = 3 \; \bar{\rho}[/math]

Or, if [math]\alpha \approx 4[/math], then:

 

[math]\rho(x) = 2 \; \bar{\rho} \; x^{-1}[/math]

Actual DM radial density profiles might be more centrally concentrated, than this analysis indicates. For, according to these calculations, 'particles' of mass m always reside at radius r(m). However, actual DM 'particles' would probably possess complicated, ellipsoidal, center-trafficking orbits, which would 'statistically concentrate' more mass towards the cluster core.

Edited October 9, 201114 yr by Widdekind