### March** 2023 Issue,**

**Volume 91, No. 3**

An exploration of circumbinary systems using gravitational microlensing

Gravitational microlensing is one of the methods to detect exoplanets–planets outside our solar system. Here, we focus on the theoretical modeling of systems with three lensing objects and in particular circumbinary systems. Circumbinary systems include two stars and at least one planet and are estimated to represent a sizeable portion of all exoplanets. Extending a method developed for binary lenses to the three lens case, we explore the parameter space of circumbinary systems, producing exact magnification maps and light curves.

#### EDITORIAL

In this issue: March 2023 by John Essick, Harvey Gould, Adam Fritsch, Claire A. Marrache-Kikuchi, Beth Parks, B. Cameron Reed, Donald Salisbury and Jan Tobochnik. DOI: 10.1119/5.0142817

#### AWARDS

2022 AAPT award citations at the winter meeting in Portland, Oregon. DOI: 10.1119/5.0143043

#### PAPERS

Is contour integration essential? Alternatives for beginning physics students by Onuttom Narayan. DOI: 10.1119/5.0084475

We rely on contour integration in many physics derivations. But sometimes we want to teach these topics before students have studied complex variables. What to do? This paper shows alternate techniques to evaluate many standard integrals that are found in topics ranging from response functions to Fourier transforms of Gaussian functions. These techniques will be valuable to anyone seeking to expand the range of integrals they can present in lower-level courses.

A shorter path to some action variables by Juan F. Zanella Béguelin. DOI: 10.1119/5.0118683

In advanced dynamics, conjugate momenta are used to compute action variables, derivatives of which yield information on the periodic motions of a system. A classic application is the problem of Kepler orbits. Unfortunately, computing the action variables often involves complicated integrals that are conventionally treated by contour integration. This paper shows how these integrals can be more easily solved by first differentiating them with respect to a parameter of the problem, a technique developed by Leibniz and often exploited by Richard Feynman, hence becoming known as Feynman’s trick. Appropriate for teachers of intermediate and advanced-level dynamics.

An exploration of circumbinary systems using gravitational microlensing by Brett C. George, Eleni-Alexandra Kontou, Patrycja Przewoznik and Eleanor Turrell. DOI: 10.1119/5.0088604

This work provides an introduction to gravitational lensing by planets bound to a binary stellar system. Instructors of introductory astrophysics courses can use this introduction to allow undergraduates to simulate possible observations. The prospects for further exoplanet discovery are exciting.

Producing slow light in warm alkali vapor using electromagnetically induced transparency by Kenneth DeRose, Kefeng Jiang, Jianqiao Li, Macbeth Julius, Linzhao Zhuo, Scott Wenner and Samir Bali. DOI: 10.1119/5.0128967

In the past decades, atomic physicists have gained a new superpower: They are now routinely able to slow light down to a few hundred meters per second, a million times slower than the speed of light in vacuum, c. They do so by making polarized light go through alkali vapor, already excited by a stronger “pump” laser. Alkali atoms can be approximated by a three-level atom with two low energy states, ∣∣1〉 and ∣∣2〉, and a higher energy excited state, ∣∣3〉. The intense pump beam excites the transition from ∣∣2〉 to ∣∣3〉, whereas the weaker polarized probe beam is tuned to the ∣∣1〉 to ∣∣3〉 transition. It turns out that, when the two beams excite the atoms, a “dark state,” a combination of ∣∣1〉 and ∣∣2〉 , appears, which cannot absorb any photons. This transparency to light also comes with a group velocity that is much smaller than c. The alkali medium is then transparent to light but slows it down. How paradoxical! This paper gives a detailed description of the experimental setup needed to create such electromagnetically-induced transparency (EIT), appropriate for an undergraduate project or an advanced optics lab. EIT can also be used to store quantum information, so that this lab could be introduced as a practical example of quantum memory in an advanced course on quantum computing.

#### INSTRUCTIONAL LABORATORIES AND DEMONSTRATIONS

A tabletop experiment for speed of light measurement using a Red Pitaya STEMlab board by Che-Chung Chou, Shi-Yu Hsaio, Jun-Zhi Feng, Tyson Lin and Sheng-Hua Lu. DOI: 10.1119/5.0099720

A tabletop optical setup, which includes an affordable Red Pitaya STEMlab test and measurement board and a low-cost laser diode module, is used to accurately determine the speed of light. The experiment is carried out by measuring the frequency response of the phase shift between intensity-modulated light beams reflected by two mirrors separated by 50 cm. By using the STEMlab unit’s built-in Bode analyzer to automatically scan the modulation frequency over the range from 10 to 40 MHz, the frequency response of phase is measured and recorded. These phase shift data are then used to calculate the speed of light with an uncertainty of less than 0.5%. All of the required components are commercially available and no electronic construction work is necessary so the experiment can be implemented by students in plug-and-play manner.

#### ADVANCED TOPICS

Singular Lagrangians and the Dirac–Bergmann algorithm in classical mechanics by J. David Brown. DOI: 10.1119/5.0107540

A clear compact summary of the conventional Hamiltonian analysis of systems constrained to evolve on lower dimensional phase spaces enables applications of this formalism to masses attached to rods and springs. Instructors could introduce the constrained Hamiltonian dynamical formalism into upper-level undergraduate mechanics courses using several systems that are offered as examples.

#### COMPUTATIONAL PHYSICS

Particle-in-cell method for plasmas in the one-dimensional electrostatic limit by Sara Gomez, Jaime Humberto Hoyos and Juan Alejandro Valdivia. DOI: 10.1119/5.0135515

This paper provides an introduction to computer simulation methods that is accessible to early undergraduate physics majors and discusses the interesting and common phenomenon of electron-electron two stream instability. It illustrates how, even though most plasmas are not one dimensional, many of the key insights into their behavior can be obtained from one-dimensional models.

Continuous fractional component Gibbs ensemble Monte Carlo by Niklas Mayr, Michael Haring and Thomas Wallek. DOI: 10.1119/5.0135841

This paper discusses a clever modification of the usual algorithm for simulating vapor-liquid coexistence. The usual method requires a trial move of a particle from one phase to the other. Because most random moves to the liquid phase require too much energy and are not accepted, the authors discuss using fractional particles which require less energy. These particles gradually become complete particles as the surroundings adapt to the change. This modification is a good example of how physics motivates the choice of computer algorithms.

#### Note

From Sackur–Tetrode entropy to the ideal gas adiabatic equation in one step by P.-M. Binder and Ian R. Leigh. DOI: 10.1119/5.0139175

For an ideal gas, adiabatic processes are characterized by the relation between pressure and volume: 𝑃𝑉𝛾=constant (𝛾 is the adiabatic constant). Usually this equation is derived using thermodynamics principles. In this paper, an alternate derivation is proposed based on statistical mechanics, using the expression of entropy for an ideal gas, the so-called Sackur-Tetrode equation. It is breathtakingly simple, and illustrates how statistical mechanics is indeed consistent with thermodynamics. This derivation would make for a nice undergraduate statistical physics exercise.

#### NOTES AND DISCUSSIONS

Erratum: “Introducing simple models of social systems” [Am. J. Phys. 90, 462-468 (2022)] by Pablo Jensen. DOI: 10.1119/5.0134837

#### Additional Resources