The problem: This booklet responds to the question posed by many universities: "How can we introduce some astrophysics in our physics courses?" The question is often qualified: "We cannot teach a whole course in astrophysics."

This solution: This booklet presents an array of astrophysical problems, any one or a few of which can be selected and used within existing physics courses on elementary mechanics, or on heat and radiation, kinetic theory, electrical currents, and in some more advanced courses. Answers are provided to all problems.

   These astrophysics problems are designed to be an interesting and challenging extension of existing physics courses, to test the student's understanding of physics by testing it in new realms, and to stretch the student's imagination. The table of contents shows the relevant physics involved in the various problems. A brief tutorial on the astrophysics is provided with each problem, enough so that the physics professor can present the problem in class. The higher-level problems start with a brief introduction to the physics.

   All the problems seek compact algebraic and numerical solutions that can easily be translated into physics. For many problems, the solution is shorter than the statement of the problem. A few mathematical proofs appear in the Appendix.

The physics: The problems on mechanics (sections I to III) are nearly independent of each other, so that any one of them can be used in an appropriate introductory physics course. However, the seven problems in section I, "Mechanics - Orbits and Kepler's third law" can be used together as a mini-course on many of the interesting topics in modern astronomy and astrophysics, ranging from the solar system to black holes in galaxies.

   The problems on thermal radiation (section IV) can be selected independently of each other. The problems on the lives of stars (section V) are best done in sequence, starting with the introduction, but it is easy to stop at any point without reaching the (academically more challenging) end of the section. It is quite possible to create a mini-course on the Sun by selecting the solar problems in sections IV to VI.

   The problems on the cosmic magnetic fields and high-energy astrophysics (sections VI and VII) are academically more advanced. However, although their backgrounds involve Maxwell's equations and special relativity, the actual problems deal with quantities that are physically intuitive. A mini-course on neutron stars and pulsars can be constructed using problems in sections III, IV, VI, and VII.

The astrophysics: The format for all the problem is similar. Once the students have been introduced to the physics, the text outlines the astrophysical setting, which is normally to be given as part of lecture. Then time needs to be allowed for the physics problem to be done by the students. Finally, the interpretation of the results of the problem is to be given as part of lecture.

   Astrophysics is an attractive science not only because it stretches the imagination but also because it is highly interdisciplinary. Astrophysics involves atomic physics, nuclear physics, fluid and plasma physics, solid state physics, chaos theory, organic chemistry, special and general relativity, and more. But students are trained in solving specific problems, and they acquire a broad view of science largely through solving many kinds of specific problems. Thus, the problems in this booklet provide a focus for the students to which the broader astrophysical challenges can be tied. Most of the text provided with each problem is designed to high-light the broader questions and challenges, which are then crystallized in the given specific problems that are to be solved by the students.

   Astrophysics is a frontier science. Even students can formulate good questions suitable for research. Some observations made by the Hubble Space Telescope have been requested and will be investigated by high-school students. But the frontier nature of astrophysics makes teaching it difficult. Even the professional astrophysicist soon learns to admit to some students' questions: "I do not know", or preferably "I do not know, but perhaps we can try to figure it out together." Indeed, the problems in this booklet will be difficult to teach because students will inevitably ask questions that go well beyond the specific problem and the provided tutorial astrophysics. But the value of students formulating questions far exceeds the discomfort of the professor answering "I do not know". Many physics students merely memorize their physics. The astrophysics breaks them out of memorization and gets them to think independently. Students' questions are a sign of their progress.

Interpretation: Physics relies on a mixture of theory and experiments. In physics experiments, one controls the parameters such as temperature or imposed magnetic field. Astrophysics relies on observations that one cannot manipulate. Often the interesting astrophysics is based on observations that are just barely possible (though they might be easy to verify three years later with newer equipment). Observations must be qualified as to their accuracy and theories must be interpreted as to their plausibility.

   A by now classical example of astrophysical discovery and interpretation is provided by quasars: Observations of quasars indicate an astronomically enormous energy output from an astronomically tiny source at an astronomically enormous distance. Is the energy perhaps derived from one star per year falling into a black hole that has already consumed a hundred million stars? At first this answer seemed so many orders of magnitude different from anything we knew that it seemed most unlikely. Yet it was the only available answer, and the problem seemed so compelling that scientists pursued it. It took twenty years' observations and their interpretations, but now this kind of answer is generally accepted. It is not proven, but generally accepted.

   Astrophysics is often the first frontier science encountered by physics students. Therefore, physics students find it unsettling that one always needs to qualify astrophysical observations and theories. They must gradually learn the meaning of the various qualifying words. There is "compelling evidence for" Newton's laws of motion as observed in our daily lives (even though these laws are not correct in a highly relativistic setting). The source of quasar energy is "probably" gravitational; that means that major aspects of the observations fit, there is an approximate theory, but there are many observational and theoretical details unexplained. The direction of the rotational axis of Neptune "may" be due to a collision; that means there is no direct evidence for that collision (although major collisions clearly are and have been important in the solar system), and alternative explanations are likely (chaos theory?). The description of this manner of thinking, the interpretation and the judgments, the qualifiers such as probable, may be, or perhaps, these all are an essential part of teaching astrophysics, and of teaching any frontier science. The parts of the text labeled introduction and didactics aim to provide some of this description.

Didactics: How does one deal theoretically with a newly observed phenomenon? No, one does not start with a computer! One starts with the question: "What kinds of physics are relevant?" It is essential to select merely a very few physical parameters and construct a minimum of analytical equations that "contain the essential physics". These are often called "back of the envelope" calculations. In astrophysics, one first considers appropriate forms of energy without worrying, yet, about the detailed forces that lead to these energies. Are we dealing with gravitational, nuclear, kinetic, electromagnetic energies or some exchange between two of them? What are the main parameters such as size or mass of an object that influence these energies? Sometimes answers can be found by dimensional analysis. Never mind if the numerical coefficients in these estimates may be off by a factor of two or three. Several of the problems in this book emphasize this kind of analysis. In particular, some problems ask students to solve differential equations by a one-step integration, which explicitly brings out the main physical parameters.

   In all problems, the student is encouraged to think about the desired degree of accuracy. A few problems have as their main goal a discussion of likely errors of measurement. Any student who gives an answer to any problem accurate to four decimal places has totally missed the scientific process. Only after "back of the envelope" estimates make sense, and only after the relevant observational quantities are actually known to better than a factor of two or three, then one can think of solving a more complete problem either analytically or, more often these days, using a computer. Successful computing requires a thorough understanding of physics and astrophysics.

   In all the physical sciences, students trained to ask "what are the important physics" and "what are the appropriate magnitudes" gain a much wider view of science than the students who can work out prescribed equations to three-digit accuracy. In fact, such a wide view of science is needed for scientists to help the development of their country effectively. Hopefully, students' study of the astrophysics in this booklet helps them toward this lofty goal.

Group collaborative learning. Frontier science is a collaborative venture. Discussion is an integral part of learning and research in astrophysics. If necessary, the problems in this book can be presented and solved as part of a lecture, but they are selected and written so that they can be discussed and solved by small groups of students, preferably during a class period. Groups of 2, 3, or 4 students work well, depending in part on the physical limits of the seating arrangement.

   The professor should introduce the physics and the astrophysical problem. Then the student groups should work on the problem, while the professor wanders about the classroom to check on the progress of the groups and asks helpful questions when students are stuck. When most groups are finished, one group should be asked to report the group's answer. Other groups may report that they have a different answer, which leads to discussion. Quite possibly the groups have selected alternative satisfactory paths to the solution. If necessary, finally, the professor should identify either the correct answer, or a reasonable answer for problems where students must estimate input quantities. This answer should then be interpreted in terms of astrophysics and the uncertainties in observational and theoretical quantities.

   If students are not used to working in groups, perhaps the problem must be divided into smaller parts. The professor seeks a group's report and discussion of each part after most groups have finished that part. This also gives more groups a chance to contribute their reports. However, not every group needs to report for any given problem. A way to make sure that all students within a group participate is to identify one student who writes down the group answer, and another student who is ready to report to the class. Every class member should have these jobs at one time or another.

   Students working in groups take much time. The professor can lecture about three problems in the time that student groups need to do one. Compared to pure lecture courses, some topics of the course must be omitted because the time is no longer available. But assuredly the students will understand the one problem they solved, and the professor will have evidence of it. This is much more useful to the student, in the long run, than some additional material stored incompletely in the students' memory.

Astronomical numbers: It is amazing that our tiny brains comprehend objects such as stars, galaxies, and the cosmos. But "astronomical numbers" can be rather intimidating. Our understanding requires time! We all, including physics students, need time to assimilate astronomical distances and energies. Mere memorization does not help much. It is necessary to think about the astronomical objects for a while, to become familiar with them.

   Most people assimilate astronomical distances best by progressing step by step, from objects we know to new (further or larger) objects; once we know them, we can progress to still newer (further or larger) objects. Therefore, the problems in section I, involving Kepler's third law, follow a progression of ever larger distances, and the problem in sections II and III follow such a progression at least approximately. The computational units in this book are mks units. But various other units appear because they are convenient, because they help to understand the physics without drowning in huge numerical exponents. Thus students must become used to units of km/s, Astronomical Units, light years, and the solar mass, radius and luminosity. A table of frequently used values follows. For most problems, only the first significant digits will be needed.

Acknowledgment:

   I acknowledge extensive and detailed constructive comments on a draft of this booklet by Prof. Francesco Zaratti, Planetario Max Schreier, La Paz, Bolivia, and by Prof. John Wang, University of Maryland, USA.

Numerical values

G = 6.7x10-11 N m2 kg-2		gravitational constant
k = 1.4x10-23 j/K		Boltzmann constant
c = 3.0x108 m/s		speed of light
h = 6.6x10-34 j s		Planck constant
= 5.7x10-8 w m-2 K-4		Stefan-Boltzmann constant
T = 6.7x10-29 m		Thomson cross section (sometimes s = collision cross section)
mp = 1.7x10-27 kg		mass of proton
me = 0.9x10-30 kg		mass of electron
A.U. = 1.5x1011 m		Astronomical Unit = mean Sun-Earth distance
yr = 3.2x107 s		one year = period of Earth's orbit around Sun seen from afar
ly = 9.5x1015 m		Light year = distance traveled in one year at speed c
Mo = 2.0x1030 kg		mass of Sun
Lo = 3.9x1026 w		luminosity of Sun
Ro = 7.0x108 m		radius of Sun
ME = 6.0x1024 kg		mass of Earth
RE =6.4x106 m		radius of Earth
1 radian = 2.1x105 arcsec		units of angle
= 4 x 10-7		vacuum magnetic permeability
eV = 1.6x10-19 j		electron Volt (KeV = 103 eV, MeV = 106 eV, GeV = 109 eV)