We have covered all of the major phenomena associated with the attractor and our exam- ination of it is almost at an end. To close out we will examine how a system will move through the parameter space to arrive at the attractor. Firstly we investigate whether the large scale behaviour of our simulations is mirrored by the behaviour of a single mass bin. For this section we only use the results from our Newtonian simulations as the MOND simulations will potentially introduce some amount of systematic error when comparing masses and radii.
In Fig. 6.14 we overlay the successive positions of the 20thbin in each of our Newto-
nian simulations and note that they all follow the same path towards the attractor. The 20th bin was not chosen for any special reason and this plot could have been produced
with any of the bins in our data. However, the 20thbin was positioned nicely in the middle
of the available parameter space and was thus chosen for aesthetic reasons. Generally the poorest agreement is found in the furthest bins as the low densities at large radii lend them a greater uncertainty in the range of radii they will cover and the innermost bins barely evolve from their initial conditions.
Figure 6.14:The position of the 20th bin from the 6 Newtonian simulations thus far. Note
that all of the bins travel along fairly similar paths as they converge towards the attractor.
material should move if it is perturbed through our parameter space. By combining all the bins and measuring the normalised change in position between kicks we can create a vec- tor at each point that represents how a bin placed at that location will move if perturbed. Dropping a population of test bins into this vector field allows us to measure and plot it.
Fig. 6.15 is the best overall visualisation of the attractor. It neatly summarises the universality of the attractor and provides a predictive framework for the characteristics of partially converged systems. The need for discretised test particles and a lack of sys- tems that tested certain regions of the parameter space creates some blank areas around
β =−0.5, γ+κ=−5. This is an unavoidable compromise between a scarcity of very tan- gentially anisotropic data points and a very large number of radially anisotropic points.
6.5
Summary
Over the course of the last two chapters we have covered a lot of information on this attractor effect. We have found the two primary conditions that the attractor requires to function as well proving that radial infall and radial orbit instability are not the driving mechanisms. Specifically, we demonstrated that energy exchange and phase mixing in a
Figure 6.15:A representation of the attractor in terms of a vector field. Hotter colours rep- resent larger rates of change of phase-space coordinates between kicks plotted as the total displacement per kickp
∆2β+ ∆2(κ+γ)/∆t.
dynamical potential are necessary conditions for the appearance of the attractor. Since ear- lier studies have also indicated that those two are sufficient conditions (Sparre & Hansen, 2012a,b), we believe we have established the basic physics underlying the attractor.
As noted in Hansen et al. (2010) it is possible to defy the attractor with specifically biased perturbations. However, since both necessary effects are present during structure formation, in particular through violent relaxation of mergers, we are confident this shows that the attractor is relevant for the fully equilibrated part of cosmological structures. Why the attractor is the exact relation that it is is still an open question as nothing in our results has shown that the attractor solution is special. Indeed, the only special thing about the attractor solution itself is the fact that it appears to be an attractor.
So the story of the attractor ends here for now, as does our look into how to tackle a problem with numerical simulations. In the next, and final chapter, we will bring all our discussions to a close with a final look at how to best communicate our results to the next generations of people who will study them.
7
Epilogue – communication
“The effort to understand the universe is one of the very few things that lifts human life a little above the level of farce, and gives it some of the grace of tragedy.”
— Steven Weinberg
For this, the final chapter, we will discuss the field of science teaching and communi- cation as it is an area that I have always had a personal interest in. It also feels fitting to end this thesis with a discussion on how best to communicate scientific ideas to others. As this section is based more on personal, subjective experience than previous chapters I feel it is appropriate to use a more informal style than I have been. In fact, informality is exactly what I would like to talk about.
As anyone who has been taught by me will probably attest, I have something of a tendency towards informal ways of describing and explaining concepts. It has been a longstanding concern of mine that this may be in some way harmful, but I could never
reconcile it with my instinct that it was an acceptable, if not preferable, way to convey certain kinds of information. Over the course of this chapter I aim to use tools developed by educational psychology to examine which of these points of view is correct.
For the sake of clarity, I’m going to define my usage of ‘formal’ against ‘informal’. By formal teaching I mean teaching physics as if there is an absolute etiquette describing how this must be done primarily via a preference for technical language over more colloquial terms and by having a reserved, unemotional, often authoritarian demeanor. Informal lecturing would by contrast use more colloquialisms and relate more to familiar words and phrases while coming across as more friendly and relaxed in how the material is delivered.
It is important to stress that this isnotan argument about mathematical formality or anything associated with the technical content of conveyed information. It is, of course, never recommended to be lax when it comes to mathematics and physical ideas. The point of this chapter is to discuss how best to package scientific information so that it is understood and retained by the student. In particular, the biggest problem I have with overly formal teaching methods is the distance I feel it creates between the teacher and the student. In my experience as an undergraduate, the lecturers, even when taking small groups, would project a sense of authority, impersonally hand out facts and figures, and only require that we could say them back correctly.
7.1
The power of analogy
Firstly, I am not going to argue that there is any problem with a lecturer projecting a sense of authority per se as of course that is important for maintaining basic classroom control. The problems begin when that authority goes beyond a simple awareness of who is teaching who and turns into teaching via argument from authority i.e. ”The lecturer said something so it must be true because the lecturer said it and for no other reason”. Information learnt this way is almost exclusively acquired as surface knowledge and lead to minimal critical analysis performed by the student as they feel unable to question or investigate what has been said. This turns the act of ‘doing science’ into a mystical art with strange mathematical incantations and arcane rituals with no apparent rhyme or reason besides the fact that doing them ‘gets the right answer’.
This issue is well-understood as a problem of how students build their mental models (Redish, 1994) and is known to be exacerbated by factors such as modular curricula (Fry et al., 2009, chapter 9). Of concern to me is that it is also perpetuated by over-formality. One of the best lecturers I ever had was a very personable man. He was chatty, always had a smile, and had plenty of little funny stories to break up the duller sections and maintain audience engagement. He seemed to make an active effort not to stand on ceremony and presented physics as being something very down to earth and practical, even when discussing the more mathematical aspects. Crucially, he did all of this without compromising the integrity of the syllabus he was teaching.
The fact that this style is more engaging is well known (Brown, 1987; Fry et al., 2009, chapter 9) and a happy, interested lecturer with an infectious enthusiasm is clearly a boon to the class, but there seems to be a good reason why that style would also be good for conveying information clearly and effectively. That reason is analogy.
Analogy is a powerful tool and one that is well-motivated for teaching physics (Redish, 1994). According to Vygotsky (1978) the most efficient learning takes place through the addition and modification of prior knowledge in a way that is neither too alien nor too familiar. An appropriately chosen analogy employed to illustrate a point can take a familiar part of the student’s everyday experience and force them to re-examine or challenge that experience in such a way that new information can be naturally incorporated.
The connection I find between this and informality is actually quite simple. If a teacher makes the effort to present physics in the most relatable way they can by choosing exam- ples that fit well with the student’s prior, informal, everyday life, then the information will be absorbed as something relevant, tangible, and important rather than a list of arbitrary, ritualised facts needed to pass an exam. This is equivalent to the ‘personal experience’ section of the Kolb cycle of learning that relates knowledge to personal experience. This is much harder to do with a more arm’s length and distanced approach that is couched only in technical language that is unfamiliar to the student.
For example, Sobel (2010) is an interesting note. It is a short response to a paper by Art Hobson about how to teach the scientific method. In his letter, Sobel makes much the same point as I am. He argues that teaching something like the scientific method as a series of inscrutable yet compulsory steps can lead to students having no deep understanding of
how or why they are doing things. He concludes that it is much more important to simply teach good practice in a practical and ‘ordinary way’ that people can relate to.
The argument can, of course, be made, that using everyday examples to demonstrate physical processes is not quite the same thing as being informal in teaching them, a point raised by Hobson in response to Sobel’s comments. I would argue that that is not quite the case. Yes, one could choose to teach a course in a very distanced, formalised way and still make reference to everyday examples in passing but I feel that that is missing the point. I prefer to see it the other way around and ask whyshouldit be formal in the first place? If all the good analogies and relatable paths to learning come from common experience and wewant students to understand that then why raise barriers by putting the subject on a pedestal to begin with?