Pub Date : 2020-12-18DOI: 10.32374/aej.2021.1.1.003
Liam Nolan, M. Mechtley, R. Windhorst, K. Knierman, T. Ashcraft, S. Cohen, S. Tompkins, L. Will
We have developed a Java-based teaching tool, "Appreciating Hubble at Hyper-speed" ("AHaH"), intended for use by students and instructors in beginning astronomy and cosmology courses, which we have made available online. This tool lets the user hypothetically traverse the Hubble Ultra Deep Field (HUDF) in three dimensions at over 500x10^12 times the speed of light, from redshifts z=0 today to z=6, about 1 Gyr after the Big Bang. Users may also view the Universe in various cosmology configurations and two different geometry modes - standard geometry that includes expansion of the Universe, and a static pseudo-Euclidean geometry for comparison. In this paper we detail the mathematical formulae underlying the functions of this Java application, and provide justification for the use of these particular formulae. These include the manner in which the angular sizes of objects are calculated in various cosmologies, as well as how the application's coordinate system is defined in relativistically expanding cosmologies. We also briefly discuss the methods used to select and prepare the images in the application, the data used to measure the redshifts of the galaxies, and the qualitative implications of the visualization - that is, what exactly users see when they "move" the virtual telescope through the simulation. Finally, we conduct a study of the effectiveness in this teaching tool in the classroom, the results of which show the efficacy of the tool, with over 90% approval by students, and provide justification for its further use in a classroom setting.
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Kepler’s 2nd law, the law of the areas, is usually taught in passing, between the 1st and the 3rd laws, to be explained “later on” as a consequence of angular momentum conservation. The 1st and 3rd laws receive the bulk of attention; the 1st law because of the paradigm-shift significance in overhauling the previous circular models with epicycles of both Ptolemy and Copernicus, the 3rd because of its convenience to the standard curriculum in having a simple mathematical statement that allows for quantitative homework assignments and exams. In this work I advance a method for teaching the 2nd law that combines the paradigm-shift significance of the 1st and the mathematical proclivity of the 3rd. The approach is rooted in the historical method, indeed, placed in its historical context, Kepler’s 2nd is as revolutionary as the 1st: as the 1st law does away with the epicycle, the 2nd law does away with the equant. This way of teaching the 2nd law also formulates the “time=area” statement quantitatively, in the way of Kepler’s equation, M = E – e sin E, (relating mean anomaly M, eccentric anomaly E, and eccentricity e), where the left-hand side is time and the right-hand side is area. In doing so, it naturally paves the way to finishing the module with an active learning computational exercise, for instance, to calculate the timing and location of Mars’ next opposition. This method is partially based on Kepler’s original thought, and should thus best be applied to research-oriented students, such as junior and senior physics/astronomy undergraduates, or graduate students.
开普勒的第二定律，面积定律，通常是在第一定律和第三定律之间，作为角动量守恒的结果，“稍后”解释。第一定律和第三定律得到了大量的关注;第一定律是因为它颠覆了托勒密和哥白尼之前的循环模型，具有范式转变的意义，第三定律是因为它便于标准课程，因为它有一个简单的数学表述，可以用于定量的家庭作业和考试。在这项工作中，我提出了一种教学第二定律的方法，该方法结合了第一定律的范式转换意义和第三定律的数学倾向。这种方法植根于历史方法，事实上，在它的历史背景下，开普勒第二定律和第一定律一样具有革命性:正如第一定律废除了本轮，第二定律废除了等量。这种第二定律的教学方式也定量地表述了“时间=面积”的表述，用开普勒方程M = E - E sin E(关于平均异常M、偏心异常E和偏心E)，其中左手边是时间，右手边是面积。在这样做的过程中，它自然地为完成一个主动学习计算练习铺平了道路，例如，计算火星下一次冲日的时间和位置。这种方法部分基于开普勒的原始思想，因此最好适用于研究型学生，如大三和大四的物理/天文学本科生或研究生。
{"title":"A Historical Method Approach to Teaching Kepler’s 2nd law","authors":"W. Lyra","doi":"10.35542/osf.io/a5bqn","DOIUrl":"https://doi.org/10.35542/osf.io/a5bqn","url":null,"abstract":"Kepler’s 2nd law, the law of the areas, is usually taught in passing, between the 1st and the 3rd laws, to be explained “later on” as a consequence of angular momentum conservation. The 1st and 3rd laws receive the bulk of attention; the 1st law because of the paradigm-shift significance in overhauling the previous circular models with epicycles of both Ptolemy and Copernicus, the 3rd because of its convenience to the standard curriculum in having a simple mathematical statement that allows for quantitative homework assignments and exams. In this work I advance a method for teaching the 2nd law that combines the paradigm-shift significance of the 1st and the mathematical proclivity of the 3rd. The approach is rooted in the historical method, indeed, placed in its historical context, Kepler’s 2nd is as revolutionary as the 1st: as the 1st law does away with the epicycle, the 2nd law does away with the equant. This way of teaching the 2nd law also formulates the “time=area” statement quantitatively, in the way of Kepler’s equation, M = E – e sin E, (relating mean anomaly M, eccentric anomaly E, and eccentricity e), where the left-hand side is time and the right-hand side is area. In doing so, it naturally paves the way to finishing the module with an active learning computational exercise, for instance, to calculate the timing and location of Mars’ next opposition. This method is partially based on Kepler’s original thought, and should thus best be applied to research-oriented students, such as junior and senior physics/astronomy undergraduates, or graduate students.","PeriodicalId":424141,"journal":{"name":"Astronomy Education Journal","volume":"10 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-11-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127697964","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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