A Model for the Dynamics of Space -Expedition to the Early Universe Hans-Otto Carmesin

A Model for the Dynamics of Space -Expedition to the Early Universe Hans-Otto Carmesin To cite this version: Hans-Otto Carmesin. A Model for the Dynamics of Space -Expedition to the Early Universe. PhyDid B, FU Berlin, 2018. hal-02077596 HAL Id: hal-02077596 https://hal.archives-ouvertes.fr/hal-02077596 Submitted on 23 Mar 2019 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Didaktik der Physik Frühjahrstagung – Würzburg 2018 A Model for the Dynamics of Space - Expedition to the Early Universe - Hans-Otto Carmesin* *Gymnasium Athenaeum Stade, Harsefelder Straße 40, 21680 Stade, Studienseminar Stade, Bahnhofstraße 5, 21682 Stade, Fachbereich Physik, Universität Bremen, 28334 Bremen [email protected] Abstract The expansion of our universe is fascinating. Additionally there are exciting mysteries about the early universe: How is causality achieved? Why is quantum gravity essential? How can the problems of reheating and big bang singularity be solved? Here I present a teaching unit that I developed and tested in a research club at school. The ‘process related competences’ of ‘modeling’ and ‘problem solving’ are applied: Observations are modeled systematically. Problems are utilized to improve the models progressively. Additionally models developed by the teacher are available for critical tests and cooperative research projects. 1. Introduction Hubble law d ~ z (Hubble, 1929). The proportionali- The formation of models by students is an essential ty factor z/d is named Hubble constant H0. topic at school (Kultusministerium, 2017). Accord- ingly we start with observational data about the expanding and about the early universe and we form corresponding models. Thereby we start with simple models and improve these when necessary. In order to improve a model we establish a cognitive conflict first. This conflict establishes a problem to be solved by the students with help of the teacher. Such problem solving establishes a particularly efficient method for learning (Hattie, 2009). As a result we develop a progressive sequence of models, all related to observations and solving problems that are inherent to the previous model. In that process a model first establishes a hypothesis about a solution of a problem and this hypothesis is critically tested by the Figure 1: Hubble diagram: redshifts observed by students students themselves. Such a learning by forming at the school observatory using a telescope with the aper- hypotheses and testing these actively is a particularly ture 11’’ (Helmcke u. a. 2018). efficient method for learning (Hattie, 2009). Alto- gether I present a teaching unit about the early uni- 3. First model: discovery of age of our universe verse. Additionally I report about experiences with In order to develop a first model for the observed teaching this sequence in a research club at school. Hubble law, the students model a radial velocity v The students are from classes 9 to 12. Each of the for distant galaxies. Corresponding lessons are pre- following sections describes the content of a lesson. sented in Carmesin (2014). Thereby we apply the Thereby the activity of the students is emphasized Doppler effect to the emitted wavelength λe = c∙T by italic letters. Performing research in a research and obtain the observed wavelength λobs as follows: club one might arrive at a novel theory. This is pre- λ = λ + v∙T {1} sented at the end of this report. obs e We solve for the velocity and obtain: 2. Discovery of the Hubble law v = c∙z with z = (λobs – λe)/λe {2} The students observed distant galaxies with help of So we obtain a radial velocity v for each galaxy and our school observatory (Carmesin, 2012). They we can determine from our data when the distance d determined the redshifts z and they either estimated of that galaxy was zero. Here we discover that the the distance similarly as Carl Wirtz did it (Wirtz, distance was zero 14 billion years ago for each dis- 1922) or they applied data about the distance d from tant galaxy. This establishes our first concept of the the literature (see figure 1). As a result we obtain the big bang. Carmesin In the framework of the first model and the Hubble constant H0 the students realize that the rate v/a is essential since it corresponds to H0. More generally this rate may vary with time and is named Hubble parameter H correspondingly: H = v/a {4} Accordingly equation {3} is solved for the Hubble Figure 2: Quasar APM 8279+5255 at redshift z = 3.9 parameter: observed by students at the school observatory using a H2 = 8πG∙ρ/3 – k∙c2/a2 {5} telescope with the aperture 11’’. Thereby we abbreviate the scaled mechanical energy 4. Observation of a very distant quasar - 2E/(m∙c2) by a parameter k. Altogether the students In our school observatory our students observed derive the dynamics (equation {5}) with little help. very distant galaxies or quasars and took pictures. That dynamics presents the well - established For instance they made a picture of the quasar APM Friedmann Lemaitre equation FLE (Unsöld, 1999). 8279+5255 (see figure 2). That quasar has the red- Here the students discover that the rate of expansion shift z = 3.9. According to equation {2} the velocity H is not fixed like in figure 1, but it increases with would take the value v = 3.9c. This contradicts other the density. Moreover that rate H depends on the observations of our students according to which the scaled mechanical energy k. velocity of an object in space cannot exceed the That scaled energy k is interpreted further as fol- velocity of light c (Carmesin, 2006). So the student’s lows: Some students know already that k additional- own observations give rise to a cognitive conflict or ly describes the curvature of space and that it is a falsification of our first model at large z (Popper approximately zero (Carmesin, 2014). While the full 1974). Accordingly our task or problem is to estab- analysis is presented in (Carmesin, 2018a) we con- lish an improved model. sider k = 0 here, since it corresponds to observations (Planck Collaboration 2016). 7. Discovery of the big bang singularity In this lesson the students perform a computer simulation with equation {5} which establishes a differ- ential equation. With it the students determine the time development of the radius a(t) with a computer Figure 3: Second model: Prototypical ball of the universe simulation based on the Euler method. When they with a radius a, volume V, density ρ mass M = ρ∙V and a investigate the radius a(t) for the past then they ob- small probing mass m. In the early universe the mass is serve that the radius a(t) tends to zero. Correspond- present in terms of the equivalent energy E = M∙c2 or ingly the density ρ tends to infinity. A diverging E = m∙c2. density is not realistic. It is named the big bang singularity (Kiefer, 2008). With it the students expe- 5. Second model: expansion of space rience the next cognitive conflict and falsify our The Hubble law or the increase of the redshift z second model as well as the Friedmann Lemaitre proportional to the distance d might be explained by equation at large density. So we look forward to an expansion of space. Next we test this hypothesis. develop a model for very high densities. For it we For it the students perform a model experiment: they develop a model for a white dwarf first. draw a wave onto a balloon and measure how it grows with increasing radius. The result is in rough 8. A model for a white dwarf accordance with the Hubble law. In this lesson the students model a white dwarf in Progressively the students develop a dynamical order to investigate states at high density (Sexl, model that provides the fully relativistic Friedmann 1975). For it the students apply their knowledge – Lemaitre equation FLE (Friedmann, 1922; Le- about quantum physics to a white dwarf. In such a maitre, 1927; Unsöld, 1999). From a didactic point star the electrons move freely at high density. The of view they apply the hypothetic deductive method students realize that the required space is limited by (Kircher, 2001). For it we model a mass m on a the Heisenberg uncertainty relation, including the sphere with radius a and mass M (see figure 3). Planck constant h as well as the reduced Planck constant h = h/(2π) (Ballentine, 1998): 6. Rate of expansion Δp∙Δx ≥ ½ ∙ h {6} With the model (see figure 3) the students analyze The students model the wave function as follows. It the dynamics in a next lesson. The students establish is plausible that the extension of electrons at high the energy equation of the probing mass: density might be modeled by Gaussian wave packets 2 E = ½ ∙ m ∙ v – G∙M∙m/a {3} ψ. Correspondingly the students learned some basics about Gaussian wave packets in an extra lesson A model for the dynamics of space about mathematical tools. In particular we may re- r ≥ 2G∙E/c4 and r ≥ ½ ∙ h ∙ c/E {14} place the inequality in equation {6} by the equality These conditions are combined graphically in figure for the case of such wave packets.

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