The final project and presentation will count for 25% of the final grade, to be broken down as follows:
- 5%: Providing a clearly written description of the problem itself (e.g., motivation & background) that goes beyond the material presented in class, with an eye on proper grammar, spelling, and citations to relevant references.
- 10%: Applying (a) both analytic and numerical methods, and (b) two or more distinct mathematical techniques from the topics covered in the course. In many cases, analytic solutions are useful for validating the results of your numerical code(s).
- 5%: Clearly presenting your results via equations, plots, tables, drawings, animations, or other means. It's all about choosing the most informative ways to convey the information.
- 5%: Providing a professional and concise written discussion of those results, and summarize what they tell us about the physical system that you chose to study.
The project will enable you to explore a chosen topic in a bit more detail and gain some extra experience with scientific writing and coding. In your responses to Homework 4, you will begin to select the specific problem or application that you will investigate, but we can work together to refine the concept to something achievable in the specified time-frame. Below are some example topics and resources for scientific writing.
Note that I didn't say much about the length of your final document. I suppose that a bare minimum would be roughly 1 page for the motivation/background, 1 to 2 pages for the discussion of the methods to be applied, a few pages for the presentation of results, and 1 page for the discussion of the results. However, it's really all about first figuring out what you want to convey to the reader, then letting that dictate the amount of material that you prepare.
The due-date is Wednesday, December 7, 2022 (the last day of class). There will be an assignment on Canvas that will allow you to upload a document (preferrably PDF, but other formats are okay).
Example Topics:
-
When stars or planets are orbiting one another in very close proximity,
there are tidal forces that distort their shapes away from exact spheres.
Can you develop ways to compute (and understand conceptually!)
the shapes of these
Roche lobes
for observers at various orientations?
Can one go further and compute how these shapes are modified when
other
forces are present, or if
gravity
itself is modified?
-
Use orthogonal transformations (e.g., Euler and other matrices)
given in lecture set 2(d) to write your own CGI "rendering" code.
In other words, if you specify the 3D positions of various objects,
can you show how an observer would see them (projected on a 2D "viewscreen")
at various finite distances and orientations?
Can one use such a code to assess the accuracy of the various
perspective
techniques developed by Renaissance painters?
-
In lecture set 4 (d.3), I introduced
North's
(1975) energy-balance model of the Earth's climate.
In that model, we ignored both diurnal (day/night) and annual (seasonal)
time-variability.
If you put those back in, you will have turned the ODE into a PDE.
With this, you can determine why the hottest part of the day is usually
a few hours after noon, and why the hottest part of the year is a few
months after the summer solstice.
It's also possible to put in longer-term variability, from, say,
anthropogenic CO2.
How much does one have to vary the parameters of these models before the
Earth turns into either a frozen ice-ball, or a runaway hot-house?
-
If you're also taking AMP (ASTR-5110), you will have seen many different
ways that the Schrödinger equation
is relevant to physics and astronomy.
Why not apply the many ODE and PDE techniques that you're learning here
to find useful solutions to the Schrödinger equation for, e.g.,
traveling particles, particles trapped in potential wells,
particles encountering potential barriers, and so on?
The sky is the limit! (Well, ℏ is really the limit...)
-
We have examined examples of the diffraction of light around solid
obstacles in several places (e.g., lecture set 3 (c.5), and
lecture set 4 (d.3)).
In real telescopes, though, the shapes of the open apertures aren't
simple circles or rectangles.
Telescopes often have central mirrors (which make for donut shaped
apertures) or "spider-leg" struts that block out even more of the
apertures.
Linear ("knife-edge") occulters are sometimes replaced by "serrated"
(sawtoothy) occulters.
Can you compute the diffraction patterns generated by these complications,
and thus investigate ways of minimizing the spread of diffracted light?
-
Any of the suggested projects in Koonin & Meredith's textbook
Computational Physics
would be fine for this course.
-
Lastly, here's an example PDE project idea from Brad Hindman:
Resources for Scientific Writing:
-
Astonomer Lynn Hillenbrand teaches a course titled
Writing
in Astronomy, whose web page contains many links to helpful
resources (scroll down in the above link).
-
Each journal has its own "author instructions" for preparing papers.
The one for the Astrophysical Journal is close to being a
community standard for astronomers; find it
HERE.
The AIP also has a good online Author Resource Center
HERE.
-
Other possibly useful articles include
The
Science of Scientific Writing
(Gopen & Swan 1990; American Scientist)
and
English
Communication for Scientists
(Doumont et al. 2014; Nature ebook)
and
The
Art of Writing Science (Plaxco 2010; Protein Science).
- A brief and fun editorial on how to properly include equations in scientific text.
If you have any questions about any aspect of the final project, please let me know.