Local Warming

Using 55 years of daily average temperatures from a local weather station, I made a least-absolute-deviations (LAD) regression model that accounts for three effects: seasonal variations, the 11-year solar cycle, and a linear trend. The model was formulated as a linear programming problem and solved using widely available optimization software. The solution indicates that temperatures have gone up by about 2 degrees Fahrenheit over the 55 years covered by the data. It also correctly identifies the known phase of the solar cycle; i.e., the date of the last solar minimum. It turns out that the maximum slope of the solar cycle sinusoid in the regression model is about the same size as the slope produced by the linear trend. The fact that the solar cycle was correctly extracted by the model is a strong indicator that effects of this size, in particular the slope of the linear trend, can be accurately determined from the 55 years of data analyzed. The main purpose for doing this analysis is to demonstrate that it is easy to find and analyze archived temperature data for oneself. In particular, this problem makes a good class project for upper-level undergraduate courses in optimization or in statistics. It is worth noting that a similar least-squares model failed to characterize the solar cycle correctly and hence even though it too indicates that temperatures have been rising locally, one can be less confident in this result. The paper ends with a section presenting similar results from a few thousand sites distributed world-wide, some results from a modification of the model that includes both temperature and humidity, as well as a number of suggestions for future work and/or ideas for enhancements that could be used as classroom projects.Comment: 12 pages, 5 figures, to appear in SIAM Revie

Splitting dense columns in sparse linear systems

AbstractWe consider systems of equations of the form AATx = b, where A is a sparse matrix having a small number of columns which are much denser than the other columns. These dense columns in A cause AAT to be very (or even completely) dense, which greatly limits the effectiveness of sparse-matrix techniques for directly solving the above system of equations. In the literature on interior-point methods for linear programming, the usual technique for dealing with this problem is to split A into a sparse part S and a dense part D, A = [S D], and to solve systems involving AAT in terms of the solution of systems involving SST using either conjugate-gradient techniques or the Sherman-Morrison-Woodbury formula. This approach has the difficulty that SST is often rank-deficient even when AAT has full rank. In this paper we propose an alternative method which avoids the rank-deficiency problem and at the same time allows for the effective use of sparse-matrix techniques. The resulting algorithm is both efficient and robust

Two-Mirror Apodization for High-Contrast Imaging

Direct detection of extrasolar planets will require imaging systems capable of unprecedented contrast. Apodized pupils provide an attractive way to achieve such contrast but they are difficult, perhaps impossible, to manufacture to the required tolerance and they absorb about 90% of the light in order to create the apodization, which of course lengthens the exposure times needed for planet detection. A recently proposed alternative is to use two mirrors to accomplish the apodization. With such a system, no light is lost. In this paper, we provide a careful mathematical analysis, using one dimensional mirrors, of the on-axis and off-axis performance of such a two-mirror apodization system. There appear to be advantages and disadvantages to this approach. In addition to not losing any light, we show that the nonuniformity of the apodization implies an extra magnification of off-axis sources and thereby makes it possible to build a real system with about half the aperture that one would otherwise require or, equivalently, resolve planets at about half the angular separation as one can achieve with standard apodization. More specifically, ignoring pointing error and stellar disk size, a planet at $1.7 \lambda/D$ ought to be at the edge of detectability. However, we show that the non-zero size of a stellar disk pushes the threshold for high-contrast so that a planet must be at least $2.5 \lambda/D$ from its star to be detectable. The off-axis analysis of two-dimensional mirrors is left for future study.Comment: 21 pages, 7 figures. For author's webpage version see http://www.orfe.princeton.edu/~rvdb/tex/piaa/ms.pdf This version has improved figures and addresses comments of a refere

New Orbits for the n-Body Problem

In this paper, we consider minimizing the action functional as a method for numerically discovering periodic solutions to the $n$-body problem. With this method, we can find a large number of choreographies and other more general solutions. We show that most of the solutions found, including all but one of the choreographies, are unstable. It appears to be much easier to find unstable solutions to the $n$-body problem than stable ones. Simpler solutions are more likely to be stable than exotic ones.Comment: 16 pages, 2 tables, 6 figure

Rectangular-Mask Coronagraphs for High-Contrast Imaging

We present yet another new family of masks for high-contrast imaging as required for the to-be-built terrestrial planet finder space telescope. The ``best'' design involves a square entrance pupil having a 4-vane spider, a square image-plane mask containing a plus-sign shaped occulter to block the starlight inside 0.6 lambda/D, and a Lyot-plane mask consisting of a rectangular array of rectangular opennings. Using Fraunhofer analysis, we show that the optical system can image a planet 10^{-10} times as bright as an on-axis star in four rectangular regions given by {(xi,zeta): 1.4 < | xi | < 20, 1.4 < | zeta | < 20}. Since the design involves an image plane mask, pointing error is an issue. We show that the design can tolerate pointing errors of about 0.05 lambda/D. The inclusion of a 4-vane spider in the entrance pupil provides the possibility to build a mirror-only on-axis system thereby greatly reducing the negative effects of polarization. Each of the masks can be realized as two masks consisting of stripes of opaque material with the stripes oriented at right angles to each other. We call these striped masks barcode masks. We show that it is sufficient for the barcode masks by themselves to provide 10^{-5} contrast. This then guarantees that the full system will provide the required 10^{-10} contrast.Comment: 12 pages, 5 figure

Diffraction Analysis of 2-D Pupil Mapping for High-Contrast Imaging

Pupil-mapping is a technique whereby a uniformly-illuminated input pupil, such as from starlight, can be mapped into a non-uniformly illuminated exit pupil, such that the image formed from this pupil will have suppressed sidelobes, many orders of magnitude weaker than classical Airy ring intensities. Pupil mapping is therefore a candidate technique for coronagraphic imaging of extrasolar planets around nearby stars. Unlike most other high-contrast imaging techniques, pupil mapping is lossless and preserves the full angular resolution of the collecting telescope. So, it could possibly give the highest signal-to-noise ratio of any proposed single-telescope system for detecting extrasolar planets. Prior analyses based on pupil-to-pupil ray-tracing indicate that a planet fainter than 10^{-10} times its parent star, and as close as about 2 lambda/D, should be detectable. In this paper, we describe the results of careful diffraction analysis of pupil mapping systems. These results reveal a serious unresolved issue. Namely, high-contrast pupil mappings distribute light from very near the edge of the first pupil to a broad area of the second pupil and this dramatically amplifies diffraction-based edge effects resulting in a limiting attainable contrast of about 10^{-5}. We hope that by identifying this problem others will provide a solution.Comment: 23 pages, 13 figures, also posted to http://www.orfe.princeton.edu/~rvdb/tex/piaaFresnel/ms.pd