Computing an infinite sum of matrix powers seems a bit nasty, but actually reformulating the problem as a transfer matrix (where legendary -> legendary with probability 1) let's you express the state of the system after N loops simply as $A^N x_0 = x_n$. Then, A^N can be readily computed by diagonalizing the matrix so that $A = V diag(\lambda_1, ...) V^T$. The matrix exponential means that different eigenvalues decay at different rates. The largest should be lambda_1 = 1 which will survive in the infinite N limit, and tells you the steady state solution. This machinery is a bit more general than the expectation based equations that the author solves with Gaussian elimination, and is a nice application of SVD/matrix diagonalization.
No, because it would create a contradiction. If a "perfect, endless repeat of pi" were eventually found (say, starting at the nth digit), then you can construct a rational number (a fraction with an integer numerator and denominator) that precisely matches it. However, pi is provably irrational, meaning no such pair of integers exists. That produces a contradiction, so the initial assumption that a "perfect, endless repeat of pi" exists cannot be true.
Is there actually a balance needed between pressure and collapse? Radiation pressure presumably doesn’t do anything to the constituent dm particles. Similarly, wouldn’t the particles in the star be on various elliptical trajectories and not collapse?
The precession of mercury has a few contributions. The largest is the tug of planets (500 arc seconds/century), the next largest is from general relativity (50 arc seconds /century). There are other, smaller contributions stemming from the sun being oblate. Knowledge about the anomaly in the precession was partly what motivated Einstein, and was one of the early predictions of the theory.
Wow! It's amazing that I can run python in my browser using an IDE that feels like my daily driver. Even the little keyboard shortcuts, ctrl+} to indent a line of code worked as expected. I became so immersed I accidentally used alt+f4 to close a terminal window, and instead closed my browser!
Geosynchronous orbit is at 37,000km (well above ISS which iirc is at 400km), so by most definitions that is in space. Often that threshold is put as low as ~100km, when the atmosphere becomes too thin to support winged flight.
The parent post said that only observations at dusk and dawn are affected. I'm pointing out that, even at astronomical midnight, large fractions of the sky aren't in earth's shadow. For other wavelengths the shadow is irrelevant, since satellites contaminate data even without the sun's direct illumination.
The idea of assembling large space telescopes is very exciting, although there are obviously many engineering challenges to solve. What that doesn't yet address is large, ground based radio telescopes. Consider the square kilometer array, which will have one million square meters of observing surface. How many launches would it take to get something comparable into orbit? Oddly, the bandwidth requirements (about one terabyte/second) for the SKA might be the hardest thing to meet in space.
Professional astronomers observe during the daytime too, using radio or microwave. Mega-constellations impinge on their ability to do science, especially for techniques like VLBI, where there's no obvious way to remove satellites from the data.
Are you sure this is true? You can frequently see satellites in the dead of night. Further, they show up quite plainly in non-visible wavelengths like radio or infrared even during the day.