So, the final formula is: where: p - Pogson's ratio [] (dimensionless) m - magnitude [] (dimensionless) F - flux [J / (s * cm² * Hz)] c - speed of light [cm / s] λ - wavelength [cm] r - Earth's radius [cm] π - pi [] (dimensionless) Checking units: Power = [J / (s * cm² * Hz)] * [1 / s] * [cm²] Power = [J / (s * cm² * Hz)] * [Hz] * [cm²] Power = [J / s] * [(Hz * cm²) / (Hz * cm²)] Power = [J / s] = [W] No problems here. Using our values: p = 2.512 m = 4 F = 3.64E-27 [J / (cm² * Hz * s)] c = 3E10 [cm / s] λ = 5.5E-5 [cm] r = 6E8 [cm] pi = 3.14 we obtain: Power = 8.94E7 [W] So… pretty close and the difference comes down mainly to rounding. Other than that, under your assumptions, I see no problems with reasoning or method. Sorry for taking so long to respond, though. You need to double it, that's when Centaurs would get your message. Power = (p ^ m) * F * (c / λ) * π * r² Power = ([] ^ []) * [J / (s * cm² * Hz)] * [cm / s] * [1 / cm] * [] * [cm²] Power = (2.512 ^ 4) * 3.64E-27 * (3E10 / 5.5E-5) * 3.14 * (6E8)²Hey, what're you up to just after January 28th of 2024? Asking for a friend.
Thanks for the unicode formatting and generalization! I was in a hurry to get a quick response in. I'm happy to see we agree within about 10%, but if there's no calculus involved, is it even math at all?? :/
It's a good estimate of the lowermost power consumption, and you've done all the legwork. Kudos! Might be a base for fun Fermi problems or a reference point to some other interstellar communication discussions. Does it even need calculus? Beam dispersion would probably be some Gaussian bundle, dissipation is most likely adequately described as a sum of elements in form of whatnot_optical_coefficient * distance, error correction is a bunch of algebra, and involving any fancy astrophysics is just going to make us look desperate. Also, it's been a while since I had to use actual numbers to solve a problem. :Pbut if there's no calculus involved, is it even math at all?? :/
No no, I mean, when I do arithmetic or algebra only, it feels like I've cheated and made unrealistic assumptions. I'm almost inclined to go looking for an integral or something, but it's like you said, there's not a point, here. Maybe I am desperate! Us chump experimentalists use real numbers all the time. Come join us on the dark side, Devac :D. edit: But be careful out there! I've already written a satirical public service announcement skit that culminates with my pilot friend, wearing full captain's attire in the cockpit of the jets he flies, looking deadpan into the camera and saying, "Remember: If it's math, on a plane, it's a bomb."Does it even need calculus?
I just handed over my homework which concluded that the derivation of thermodynamic parameters in our model is A-OK because second derivatives are finite everywhere and third derivatives only tend to infinities when T = 0, which is, like, acceptable in critical systems under fluctuation/perturbation regime. We're all chumps, experimentalists just get swaggerific toys. Can't believe I forgot about that incident. Also, would love to see that PSA.when I do arithmetic or algebra only, it feels like I've cheated and made unrealistic assumptions.
Us chump experimentalists use real numbers all the time.
sigh NO, you say "damn, dude, are you seriously doing math for fun?" and then listen to what he says. The best conversations I've ever had have been interacting with people doing weird shit on planes. Last flight I saw someone writing up an inspection report on a '75 Targa and told him not to hate me because I drive a 996. I proceeded to have a 2-hour conversation with one of the world's foremost Porsche experts, a guy who gets paid by Larry Ellison to fly around buying Porsches for him, a guy who has bought and sold Porsches since he was sixteen years old, a guy who had so many stories about nameless rich people that I sat through four bourbons just goading him on. ALWAYS ask an expert about their expertise. You will learn shit you didn't even know you'd find interesting.
On the way back from Santa Fe, I sat next to a woman around my age, and I was working out some unit conversions in my notebook, pen and paper, just numbers and abbreviated units. She asks, "Looks like physics?", and I say, "Yeah! That's right, how'd you know?". And then she mentioned minoring in chemistry or something, but I forgot, because just a few exchanges later, we both went back to our own little worlds. Not even any awkwardness, just two people having some quality alone time on a plane. Which is about the most precious thing in the world after sharing a room to cut costs at a week-long conference.
Sorry Klein, bad joke about economics having basic logarithms and economic theory being a threat to national security. Of course I would at least ask. I've read enough of those papers to make me want to cry in my salad though. On the way back from Rwanda I actually met a dude who was pretty high up in the U.S. military. He served on peacekeeping missions in the Congo and stuff, and you bet he was opinionated on Trump. He knew a bunch of former colleagues that ended up promoted in the Department of State because they basically played the game. Shifted their plans in support of the new leader. It was an interesting perspective into how the inside of the government actually works and how the chain of command functions.
It's a much lower number than I expected, probably because in sci fi it's never a signalling laser, it's always a launch laser and that's a whole 'nuther animal. FUN FACT - disturbing portions of the historical record thinks Sirius was red. There are two real ways to resolve this: (1) presume that the historical record, as evidenced time and time again, is faulty (2) presume that the Sirians were pointing a launch/signalling laser at us for a few hundred years back in antiquity. (2) is a lot more fun and could be the impetus for a pretty fun sci fi conspiracy tale, or so I've heard. Or, if you're Larry Niven and Jerry Pournelle, you take the idea and throw it in the distant future at a completely different part of the galaxy so that your social commentary doesn't have to include people. 100MW is chump change. I've worked with powerplants that big. You can buy them on Alibaba. Which leads me to believe that divergence is more important than we're accounting for but I'm too lazy to do more than throw some numbers at an online calculator and watch it choke on the light years.
Do remember that we're discussing a case of illuminating Earth (and only Earth) with some dispersionless, cylinder-like, 100% efficient laser beam with perfect accuracy. Even then, with those idealisations, power scales with the square of the radius of the thing we want to illuminate. Accuracy is also fun: in our case, it's like pinpointing something roughly the size of a credit card on the surface of our Moon, but without the joys of 4.3 years worth of one-way delay or tracking a moving object. Also, I didn't say that divergence isn't significant. Just that it likely won't involve higher maths to find an approximation, which is semi-true. Had to do a double integral over a disk to get from intensity [W/m²] to power [W]. Here's how we can calculate the power delivered by a Gaussian beam, and it's ripe for plugging numbers in. I took the formulae and symbols from the article. There's also a calculation of how narrow the beam would have to be at its narrowest point, which turned out to be essentially zero (which I, perhaps mistakingly, interpreted as equivalent to a point source). Pinging am_Unition for peer review and help in moving it forward. It's not pretty, though. My initial intensity assumption goes asymptotically to infinity the narrower the beam, so there's possibly a problem/fuckup. I absolutely encourage everyone to play around with the numbers. Maybe it could work for other wavelengths?