**Background:**

As introduced in the previous contribution to *The Dark Matter Crisis*, Question A: Galaxies do not work in LCDM, sociology and majority views, PK was recently contacted by a few people, and here are excerpts from some of the questions asked and the replies. These help to illustrate some of the issues at hand. The questions are

**A)** So the LCDM model fails on scales smaller than about 8 Mpc?

**B2)** What is a galaxy? (Addendum on the relaxation time) (this contribution)

**C)** What are the three best reasons for the failure of the LCDM model?

I: Incompatibility with observations

III: Fundamental theoretical problems

**D)** What about the Bullet cluster? And what about the Train-Wreck cluster Abell 520?

**E)** Why is the main stream community so reluctant to go along with accepting the failure of LCDM?

This contribution deals with Question B2, while upcoming contributions will concentrate on the remaining questions.

**Addendum on the relaxation time: **

In the previous contribution “Question B: What is a galaxy?” was raised, asking readers to cast a vote after reading the Forbes & Kroupa (2011) paper on this problem.

One of the criteria suggested originially by Kroupa (1998) is to define a galaxy as a stellar-dynamical system of stars which has a *two-body relaxation time* (TREL) that is longer than a Hubble time. A star cluster would be a system with TREL being shorter than the age of the universe.

**To add a few clarifying insights on this proposed definition: **

This definition implies that a galaxy does not evolve noticeably by energy re-distribution within it over the age of the universe.

**A simple well-known example:**

Take a box, subdivide it into two equal volumes. Fill one volume with hot gas and the other with cold gas, and then remove the wall between the two volumes. One can measure the time it takes for both sub-volumes of the now one large volume to evolve towards equal temperature. The atoms of the gas bounce off each other thus exchanging their kinetic energy such that after some time the whole combined volume is at the same temperature. The gas has achieved a state of

energy equipartition, and the time it takes for the temperature to change significantly in either sub-volume is theenergy-equipartition time-scale, or in short TREL.

**Now take a stellar-dynamical system (e.g. a star cluster or a galaxy). **

Here there are stars moving rapidly and stars moving slowly. They can exchange energy through distant weak gravitational encounters (stars very very rarely collide physically because their separations are vast). So the fast stars typically slow down, while the slows stars typically speed up, just like the gas atoms in the box. But, in a gravitationally self-bound system the slowed-down stars fall towards the system center (a stone also falls towards the center of the Earht unless it is thrown with very high speed). It speeds up as it falls, and is now faster again allowing it to shed its surplus energy again (a dropped stone gets ever faster until it can do some real damage). This goes on and on, such that a stellar-dynamical system cannot achieve a stable configuration; it cannot achieve a state of energy equipartition – there is an instability which stellar-dynamicists refer to as a

“gravo-thermal catastrophe”. The slow stars, on the other hand, typically gain speed and climb outwards thereby becoming slower (a stone thrown upwards also slows down). So they can again acquire kinetic enegy if they encounter a faster star. Thus, slow stars typically move outwards, and at some point they can be lost from the system. Note though that all of this is a statistical process: a fast star can become a slow star and vice versa. Mathematically, one treats these processes through studying the rate of energy diffusion.A stellar-dynamical system such as a galaxy or a star-cluster thus looses stars through the process of energy equipartition. This happens on a TREL time scale.

Thus, over the age of the universe, a “galaxy” does not loose many stars though this process, while a star cluster does.

**Mathematically**,

when stellar-dynamicists study the evolution of a stellar-dynamical system, it is necessary to calculate every single gravitational encounter between two distant stars,

if TREL is short.This requires extremely sophisticated and highly complex mathematical and algorithmic approaches and is highly computationally demanding.Such techniques have been under development over decades leading to sophisticated but non-trivial computer programmes, notably the Nbody4, Nbody5, Nbody6 codes by Dr. Sverre Aarseth in Cambridge. Thus, to compute the evolution of a small globular cluster with some 100,000 stars today requires the fastest super-computers. A large globular cluster with more stars is beyond current computational means. A star cluster with a million stars requires a computer a thousand times faster to do the calculation in the same human time as for a cluster with 100,000 stars!

Thus, galaxies with many more times a million stars cannot be computed with this direct-N-body technique using current technology.

But, because they have a **TREL longer than the age of the universe**, astrophysicists can use a **different mathematical approach altogether to compute galaxies**. Instead of solving the *6N* coupled first-order differential equations they can solve the *Collisionless *Boltzmann Equation. Here *N* is the number of stars in the system, and “collisionless” means that the gravitational encounters between stars are not important, which is the case when TREL is very long.

By being able to ignore the many distant weak gravitational encountes between two stars, astrophysicists can treat each star as moving in the average smooth gravitational field generated by all the other stars. This is like calculating the trajectory of a cannon ball in Earth’s gravitational field. So the computational problem becomes significantly easier and astrophysicists can calculate the evolution and the interactions of galaxies with modern computers.

In fact, some of the mathematical methods being used, e.g. in the Stellar-Populations and -Dynamics Research Group in Bonn are so fast that present-day laptops can be used to compute the evolution of a few galaxies over the time of the universe, taking no longer than a few hours, if at all, of human time.

But this is only possible for systems of stars with TREL longer than the age of the universe.

Now, what happens **when the universe is twice its age** (about 28 Gyr)**?** Not much: what we call a galaxy today will still be a galaxy then, except that the transition mass where TREL is about the age of the universe, will have shifted by a factor of four.

What happens **when the universe is ten times older?** Well, then the objects we today call transition-type objects or ultra compact dwarf (UCD) galxies will have evolved partially through the energy equipartition process, similarly to the present-day massive globular clusters.

By Pavel Kroupa and Marcel Pawlowski (23.01.2011): “Question B2: What is a galaxy? (Addendum on the relaxation time)” on SciLogs. See the overview of topics in *The Dark Matter Crisis*.