Today is Einstein's birthday. If he were still alive he'd be 131. Those of you who have been reading here for a long time know that Einstein was (and is) one of my "culture heroes." When I was a kid I sent him birthday cards (yes, I'm that old) and when he died made a scrap book filled with news clippings. One of the great loves of my younger life gave me an Einstein bust as a present and it still sits on my desk, more than 40 years later (she reads the blog from across the ocean, so I hope she sees this! Mrs. R. knows and likes her so this isn't a guilty secret). I also have first editions of his second and third published works and a fairly large library of books by and about him. Unlike quantum mechanics, relativity theory is essentially the achievement of a single person, Albert Einstein. Both are beautiful theories and quantum theory may be the most successful theory in the history of science. But relativity is no slouch, either, having been confirmed again as recently as this week. Not bad.

Given all this, it seems fitting to commemorate the occasion on the blog. Enjoy:

Happy birthday, Albert, hero of my youth. Still a hero.

Read the comments on this post...

It was already known that we were totally doomed, but now there is a new and exciting scenario. In this one, comets rain down on us from the Ooort Cloud, said comets loosened by contact and interaction with a star called Gliese 710.

Read the rest of this post... | Read the comments on this post...

I'm en route to the March Meeting in Portland, which involves a three-hour layover in Chicago, between two flights on Southwest, my preferred airline. I'm always impressed by how much more efficient Southwest seems that the other major airlines.

One weird manifestation of that efficiency is the flight plans that Southwest uses. Where most flights on other airlines seem to go back and forth between two cities over and over, Southwest's routes tend to roam all over the country. This morning's flight from Albany to Chicago continued on to San Antonio, TX, Phoenix, and San Jose. Another recent trip involved a flight from Albany to Baltimore, Kansas City, San Antonio, Las Vegas, and San Jose.

This always makes me wonder about the computation behind these flight plans. There's got to be some logic to it, that allows them to get all the planes they need to the places they need them, presumably as cheaply as they can manage. The result is most likely derived empirically from crunching some numbers on existing flight networks, then optimizing the results, but it's kind of amusing to imagine that the folks at Southwest are sitting on a solution to the Traveling Salesman Problem, and using it to gain an advantage over their competitors...

(It's also a good idea for a throw-away element of a science fiction story about Singularity-type stuff, but I'm not the guy to write that...)

Read the comments on this post...

Longtime readers of this blog may remember last year's orgy of pies on the run-up to Pi Day (March 14th, or 3-14). This March at Casa Free-Ride, there's been less pie making, in large part due to the fact that I'm no longer on sabbatical (either from my job or from coaching soccer).

But the bake-off is on again, so I figured that I needed to feed you all one really good pie (or pie recipe, anyway).

This pie melds three flavors that play very well together: rich chocolate, tart cherries, and almonds. As a bonus, it puts those flavors together in a pie that is rich but not heavy, one that doesn't lean on eggs, or cream cheese, or butter, or milk.

I make this pie with a food processor, but if you don't have one, you can manage with a blender, a heavy rolling pin, and a knife and cutting board.

Read the rest of this post... | Read the comments on this post...

The most basic explanation of Pi is that it is the ratio of the circumference to the diameter for a circle. That seems simple enough, but it turns out that Pi is an irrational number - so you can't just write it down. Oh, I know that you are an uber-geek and you could recite the first 80 digits of Pi. But the question is - how many digits are enough?

In this post, I am going to assume that we don't know the true value of Pi (which is essentially true). I can then use propagation of error techniques to see how dependent different calculations are on the value of Pi.

Super Brief Intro to Uncertainty

I still can't believe I haven't put a post together on the basics of measurement and uncertainty. Add that to the todo list. The most important idea in measurements is that they are not exact values. Let me start with my favorite example. Suppose I have a table that I want to know the area of. To do this, I measure the length and the width. The value I come up with for the length is 133.2 cm. But what does this mean? Is this the exact length of the table? No. Two problems.

• The table doesn't have an exact length. What does the length mean for a table? Is it a perfect rectangle? No. Is it even straight on the edges - probably not.
• Even if it were a perfect table, would my measurement be perfect? No.

Maybe I measured this length a whole bunch of times and at different locations. This would give me an estimate of how the measurements are spread out. If I do the same for the width, I might get something like:

This means that the length of the table is almost certainly between 133.0 cm and 133.4 cm. If a similar thing can be said about the width, then this diagram could represent the area.

The point I would like to make - since the width and the length have uncertainty, the calculated area would have uncertainty. How do you determine this calculated uncertainty? I have three ways:

• Use the extreme values of length and width to calculate the extreme values of the area (in this case the smallest area uses the smallest length and width). This is the method I use for my algebra-based physics labs.
• Assume the error is small, linear, and normally distributed. In this case, you can use the partial derivatives of the functions to determine the relationship of the uncertainty for the measured stuff on the calculated stuff. Here is wikipedia's page on this, but I am not really going to go into the details.
• Assume that if you measure the stuff a whole bunch of times, the data would be normally distributed. Write a program that generates normal data and use that to calculate tons of times the calculated value. Look at the spread of all these calculations to determine the uncertainty. I am not going to do this right now.

Back to Pi

Archimedes used 96 sided polygons to estimate the value of Pi. He showed that Pi was greater than 3 and 10/71 and less than 3 and 1/7th. This gives a decimal value from 3.14084507 to 3.142857143 (with no rounding). I could write this as an average and an uncertainty of about:

That is not too bad of a value. But what about pi = 3? Is that bad? First - according to Snopes, no state has ever proposed a law that would officially change Pi to 3. It is still a fun story. Anyway, in this case I could perhaps say:

I chose the uncertainty in this fictional Pi to be +/- 0.2 so that the range would cover the true value of Pi. Really, though you could in general write Pi as:

Where Delta pi is the uncertainty in pi.

Some uses of Pi

So what effect does the uncertainty in Pi have on different uses of Pi? Let me start with something practical - the speedometer in your car. Basically, your speedometer needs Pi to make the conversion between angular velocity and linear velocity using:

I know, there is no pi in that equation. But, how do you know the angular velocity (omega)? If this is measured in revolutions per second (or minute) then you have to convert units. Let me write this as:

Now, I will assume that omega, r, and pi all have uncertainty. Then the uncertainty in the velocity would be (using the max-min method from above for simplicity):

And I would do a similar thing for the minimum value. I could average the difference between average and the max and the average and the min. (I will put these calculations in a spreadsheet for you).

What about the volume of a sphere? This same thing is used for calculating things such as - the volume of the sun or the volume of a spherical cow. Here is the volume of a sphere:

These two uses of Pi seem boring - but really this is the basis for many applications of pi. There are tons of others, but they are maybe more abstract (but just as important). Now, on the to the spreadsheet. I will put in some values for the stuff, but you can change them if you like.

Note - I don't know how to change the number of digits presented in google docs. Also, I seem to have hit a creative wall with uses of pi. How about you list your favorite use of Pi in the comments?

Read the comments on this post...

Earlier this week, I wrote about an inclined treadmill, and talked about physical work. Physically, the amount of work that you do is equal to the amount of force that you exert in a certain direction multiplied by the distance you move in that direction.

If you walk up an incline, as opposed to moving on a level surface, you have to also fight the force of gravity to get up that hill, hence you have to do extra work.

I contended that, in order to walk up an inclined treadmill, you also have to do extra physical work, the same way you have to do extra physical work to walk up a real hill.

"Not so fast," said many of the commenters! Unlike a real hill, where you actually wind up at a higher elevation than you started at, you don't travel any real distance on a treadmill, and you certainly don't end at a different elevation than you started at! Therefore, the argument goes, you don't do any extra physical work.

So, I ask you, what would Einstein have to say about this? Einstein -- mastermind of the equivalence principle -- realized that as far as forces go, the only thing that you feel is acceleration. Your environment doesn't matter at all.

What's a good, analogous thought experiment for our treadmill/hill problem here? Imagine an escalator. Now, you're going to walk up this escalator, and you're going to walk up 50 steps on this escalator for three different cases, like Fred here.

If you walk up 50 steps, and each step is 0.2 meters high, then you will raise your elevation by 10 meters relative to not taking those 50 steps. In other words, it doesn't matter whether that escalator is moving up while you do it, in which case you might wind up raising your elevation by 20 meters total, or the escalator is moving down, in which case you might wind up not changing your elevation at all.

In either case, you have to do work against the force of gravity, regardless of what your velocity is. This even works in the case of a broken escalator, a.k.a. stairs.

And that's why, regardless of whether your elevation changes or not, you do work whenever you walk up an incline! Hope you've had a great week, and I wish you all a great weekend!

Read the comments on this post...

The APS March meeting is next week as 10000 physicists invade Portland, Oregon. I hope Powell's bookstore has stocked their science sections well! GQI, the topical group on quantum information, sponsors a good number of sessions at the meeting including sessions with invited talks, focus sessions, and general sessions. Below the fold I'm assembling a list of quantum computing sessions, but before the fold I'd like to point out the invited sessions, which have longer speaking slots where one can actually learn more than the speakers name and research project title, that are sponsored or cosponsored by GQI (also below note the Focus sessions listed below have invited speakers)

• Monday, March 15 8:00am-11:00am Session A8: Quantum Opto-Mechanics
  Room: Portland Ballroom 255
  (Jointly sponsored with DAMOP)
  Invited speakers: Jack Harris, Klemens Hammerer, Philipp Treutlein, Nathaniel Brahms, Keith Schwab
• Monday, March 15 11:15am-2:15pm Session B6: Controlling Dissipation in Quantum Systems
  Room: Portland Ballroom 253
  (Jointly sponsored with DAMOP)
  Invited Speakers: Frank Verstraete, Hans Peter Buechler, Matthias Lettner, Luis A. Orozco, Sergio Boixo
• Monday, March 15 2:30pm-5:30pm Session D4: Quantum Computer Science
  Room: Oregon Ballroom 204
  Invited Speakers: Graeme Smith, Aram Harrow, Ben Reichardt, Sandy Irani, Stephanie Wehner
• Thursday, March 18 11:15am-1:40pm Session W6: Superconducting Qubits
  Room: Portland Ballroom 253
  Invited Speakers: Radoslaw Bialczak, Franco Nori, Leonardo DiCarlo, Sahel Ashhab

Read the rest of this post... | Read the comments on this post...

In the Stealth in Space post earlier this week, we discussed the problem of detecting the thermal emission from a spacecraft. If the interior isn't generating a lot of power, there's not much thermal radiation being emitted, making it a tough job to detect.

But it was pointed out in the comments that the heat from the sun would itself warm the spacecraft exterior, increasing the thermal signature by potentially a large amount. Let's verify this. If you take a perfectly absorbing sphere and set it in orbit around the sun (say, at the distance of the Earth), it'll absorb all the light from the sun that intersects its cross-sectional area. But assuming it conducts heat well, it'll be radiating equally in all directions - the area of emission will be the whole surface of the sphere. Thus at equilibrium, the power in from the sun equals the power out via radiant heat.

The power in is the power-per-area from the sun (about 1300 W/m^2 at the distance of the earth) time the area of cross-section, and the power out will be equal to the thermal-power-per-area given by the Stefan-Boltzmann law times the whole area:

Where Ac is the area of cross section, At is the total surface area, capital phi is the flux from the sun, and sigma is the Stefan-Boltzmann constant. For a uniform sphere, we can plug in the area:

Solve for T:

Notice the r's have canceled; the final temperature is independent of the size of the sphere. With our numbers, the temperature is a toasty 275 K. Well, toasty with respect to space. You'd still want to wear a jacket.

But that's a pretty hefty temperature if you want to stay stealthy. In the comments I suggested making the spacecraft out of reflective or transparent materials, both of which have potentially serious problems. It would probably be a better bet to just tweak our formula above by simply giving our spacecraft a convenient shape. If for instance we shaped our spaceship like a pencil and kept either the point or the eraser aimed at the sun, there would only be a small fraction of our total radiating area available to absorb sunlight. I'll leave the equation-modification aside and quote some results: if there's 100 times as much radiating area as there is absorbing area, the temperature is down to 123 K. If there's 1000 times, it's down to 69 K. Returns are diminishing (it scales as the inverse fourth power of the ratio), but still significant. That combined with some judicious reflectors and I think you can get the hull temperature pretty low.

The problems the sun's heat poses are also strongly influenced by where you're trying to be stealthy. Around Mercury there's a lot more solar light to deal with. Around Saturn, much less. All other things being equal, your hull temperature scales with the inverse square root of your distance from the sun. At Saturn, for instance, our 100x area blackbody spacecraft would have a hull temp of a frigid 39 K.

So it's something to think about if people ever come to blows over mining Ceres or something.

UPDATE: In the previous post, commenter Anthony brings up something well worth discussing here:

...if a ship is being hit by X watts of sunlight, it's pretty much going to emit X watts worth of photons (unless it has somewhere to store the heat, and it usually only takes hours to days to overwhelm any practical heat sink), and tweaks to albedo and heat distribution across the surface just modify the spectrum and direction of the emissions.

Which is pretty important, lest we fall into the same trap I criticized in my previous post. In the end, total incoming = total outgoing. Lowering the temperature is good for reducing that part of the total emissions which are promiscuously broadcast in all directions. Hopefully you can catch most of the rest with a mirror and reflect it in a narrow, specific direction away from enemy sensors.

Read the comments on this post...

Lack of understanding [is] called stupidity; deficiency in the application of the faculty of reason to what is practical we [recognize] as foolishness; deficiency in power of judgement as silliness; finally, partial or even complete lack of memory as madness... That which is correctly known through the faculty of reason is truth. -From The World as Will and Representation, First Book, Section 6.

As moralizing as it sounds, Schopenhauer is not being so here. This is his way of defining the terms of discussion. As I continue reading, it is remarkable to notice how he uses the Principle of Sufficient Reason as a sort of philosophical bedrock to plant his philosophy and as a curious Occum's razor to untangle old epistemological knots. Einstein was greatly influenced by Schopenhauer's ideas of space, time and causality. Perhaps, one reason why he so stubbornly rejected the probabilistic interpretation of quantum mechanics. Read the comments on this post...

Leaving for Death Valley tomorrow - I'll be sure to take some pictures of Ubehebe Crater and the volcano at the Mirage. This will likely be the last new post until about a week from now, but look for the Erta'Ale Volcano Profile, maybe a new Mystery Volcano Photo and I'll leave a thread open for any new volcano news.

Colima in Mexico.

• Eruptions reader Tim Stone sent me this image from Japanese astronaut Soichi Noguchi's Twitpic feed - it is a stunner of the caldera on Jebel Marra in Sudan. The only known historical eruption for this volcano was ~2000 BC within the Deriba Caldera, but it has produced pyroclastic flows that travelled upwards of 30 km from the caldera.
• There are a couple new pictures from Colima in Mexico posted on the Colima Volcano Database. The picture show the growing phases of the dome since 2008 and the new lava flow on western side of the volcano.
• A number of Eruptions readers have mentioned an increase in activity at Costa Rica's Turrialba. OVSICORI said that it appears clear that new magma (spanish) is moving into the volcano based on the current seismicity and gas emissions at the volcano. Turrialba experienced its first phreatic eruption in the last 100 years in January, but new magma has yet to be seen at the summit crater.
• You can always count on great shots of volcanoes on the Kamchatka Peninsula from the NASA Earth Observatory team, and this new image of Klyuchevskaya (Kliuchevskoi). It shows the new dark grey ash from the current eruption on the snow the lines the flanks on the volcano. Some of the recent eruptions of Kliuchevskoi have produced plumes upwards of 6 km / 20,000 feet.
• Before I forget, here is this week's Smithsonian/USGS Weekly Volcano Activity Report.
• And for those of you not following the Eyjafjallajökull (Iceland) earthquake swarm thread, the swarm does continues to march on - the question still is, to what is it leading?
Read the comments on this post...

Nam et ipsa scientia potestas est (And thus knowledge itself is power)
-- Sir Francis Bacon.

The next edition of Scientia Pro Publica (Science for the People) is less than two weeks away and it is seeking submissions! Can you help by sending URLs for well-written science, medicine, and nature blog essays to me?

Read the rest of this post... | Read the comments on this post...

Hm, today we seem to have a posting from beyond the grave, arXiv:1003.2133:

Proof of the Ergodic Theorem and the H-Theorem in Quantum Mechanics
Authors: John von Neumann

Abstract: It is shown how to resolve the apparent contradiction between the macroscopic approach of phase space and the validity of the uncertainty relations. The main notions of statistical mechanics are re-interpreted in a quantum-mechanical way, the ergodic theorem and the H-theorem are formulated and proven (without "assumptions of disorder"), followed by a discussion of the physical meaning of the mathematical conditions characterizing their domain of validity.

Read the comments on this post...

A while ago, I complained that the people running the LHC did not have their act together when it came to managing and disseminating information for the interested public. I took a little flack for that (see comments) but I was right. And I'm still right. We (the interested public) were just recently given a very nice overview of the potential for the next several months of research. Then, today, we find out that the LHC is fundamentally busted and will be shut down for a significant rebuild. And part of that news is that this has been the plan for a long time. But I guess they forgot. Or something.

Can anyone explain to me what is going on?

Read the comments on this post...

Ever wanted to ask a Nobel Laureate in physics a question? Well here's your chance: check out this youtube page where you can upload your own questions to Albert Fert, 2007 Nobel prize winner for Giant Magnetoresistance.

Read the comments on this post...

In an American Chemical Society paper, "Forecasting World Crude Oil Production Using Multicyclic Hubbert Model" authors Ibrahim Sami Nashawi, Adel Malallah and Mohammed Al-Bisharah propose:

Even though forecasting should be handled with extreme caution, it is always desirable to look ahead as far as possible to make an intellectual judgment on the future supplies of crude oil. Over the years, accurate prediction of oil production was confronted by fluctuating ecological, economical, and political factors, which imposed many restrictions on its exploration, transportation, and supply and demand. The objective of this study is to develop a forecasting model to predict world crude oil supply with better accuracy than the existing models. Even though our approach originates from Hubbert model, it overcomes the limitations and restrictions associated with the original Hubbert model. As opposed to Hubbert single-cycle model, our model has more than one cycle depending on the historical oil production trend and known oil reserves. The presented method is a viable tool to predict the peak oil production rate and time. The model is simple, accurate, and totally data driven, which allows a continuous updating once new data are available. The analysis of 47 major oil producing countries estimates the world's ultimate crude oil reserve by 2140 BSTB and the remaining recoverable oil by 1161 BSTB. The world production is estimated to peak in 2014 at a rate of 79 MMSTB/D. OPEC has remaining reserve of 909 BSTB, which is about 78% of the world reserves. OPEC production is expected to peak in 2026 at a rate of 53 MMSTB/D. On the basis of 2005 world crude oil production and current recovery techniques, the world oil reserves are being depleted at an annual rate of 2.1%.

It looks like this is an interesting attempt to adapt Hubbert Linearization to other factors. It is interesting, and the major news sites seem to have taken notice, which is good. That said, they seem to be using high estimates for Kuwait, perhaps because the paper comes out of a Kuwaiti University. But I think what's important is the degree to which the paper validates Hubbert's methodology. You can quibble about the OPEC projections, or any given figure - since reserves are such a contentious subject, what I think is more important is that it is coming out of an OPEC country, with a peak date in the near future.

Sharon

Read the comments on this post...

Lots of good suggestions as to Portland activities for my trip to the March Meeting next week. There's a second, related problem that I also need help with: What should I do at the meeting itself?

My usual conference is DAMOP, which I'll be going to in May, so while DAMOP is a participating division, and offers some cool-sounding sessions, it seems a little silly to go to the March Meeting and go to DAMOP talks. The whole point of being at the gigantic meeting is to see different stuff than usual.

The problem is, the scientific program includes forty-odd parallel sessions in each time slot, most of those featuring a dozen or so 12-minute talks, which are generally incomprehensible if you're outside the field. The invited talks are longer, and often better, but still variable. And there are so many of them...

So, here's my question for readers who know stuff about non-AMO physics: What sessions should I be attending at the March Meeting? I'm interested in invited talks, ideally by people who are good speakers, that will be reasonably comprehensible to someone outside the field. There are a couple of things I've already identified, but the only block that is definitely out of the question is the Tuesday 11:15 block (J sessions), when I'm speaking.

If there's something at the March Meeting that you think of as an absolute must-see, leave a comment and let me know.

Read the comments on this post...

I already went over a Monte Carlo method for estimating Pi - you know, for Pi-Day (March 14). Well, here is a small addition. This is the same thing done in Scratch.

Click the image to run the scratch app. It is kind of fun to watch.

About Scratch

If you are not familiar with Scratch, basically it is a graphical programming language a lot like the stuff for the Lego robotics. It is free, runs on Windows and Mac OS X, and can be embedded in a webpage as a java applet. Sure, Scratch has some limitations, but it is great for kids. Here is what the code for the pi-estimation program looks like:

If you create an account on Scratch, you can download the code of any project. I like Scratch.

Read the comments on this post...

A bad day for your ego is a great day for your soul. -Jillian Michaels

One of the most popular exercises at the gym is the treadmill. And why wouldn't it be? Whether you're running or walking, it's a great way to get your heart rate up, get your body moving, and for many people, a great way to burn calories.

But however you use a treadmill, there's one extremely simple thing you can do to dramatically intensify your workout: incline it!

If you're an outdoor walker/runner, this is the equivalent of going uphill instead of over level ground. There are many physiological differences in walking along an incline versus on level ground, but what does physics have to say about it?

Normally, if you're on level ground (or a level treadmill), you stay at the same level in the Earth's gravitational field.

But if you walk uphill (or on an inclined treadmill), you not only need to move forward at whatever pace you were moving at, you also need to climb -- a little with every step -- out of the Earth's gravitational field!

The Earth's gravitational field is no slouch, either. I'm an 80 kg individual, and for me to raise my elevation by just 5.3 meters (about 17 feet) costs me 4,200 Joules of energy, also known as one food calorie.

Now, if I actually exercise, I burn significantly more than one calorie by raising myself those 5.3 meters. Why? The two most significant reasons are as follows:

1. I am not a perfect engine. This means, in order for me to do 4,200 Joules of physical work, I need to burn about three times as much in food energy in order to get that much useful energy out. Alas, our bodies are inefficient in that manner.
2. When you exercise and then stop, your body doesn't know that it's okay for your heart to slow down for quite some time. So spending an hour walking uphill will elevate my metabolic rate for a lot longer than an hour!

Ahh, the power of exercising. But I'm not a physiologist; I deal in terms of physical work alone. So, just looking at the extra amount of energy you'd have to spend to climb up an incline rather than level ground, what are we talking about?

Let's make a helpful table. We'll just look at the total distance you travel (e.g., if you walk at three miles-per-hour for one hour, you go three miles), put in the incline, and see how much extra physical work you need to do! Distance (miles) Distance (km) Incline (degrees) Extra Work (Calories) 1.0 mi 1.6 km 1 degree 5.3 Cals 1.0 mi 1.6 km 3 degrees 15.8 Cals 1.0 mi 1.6 km 5 degrees 26.3 Cals 1.0 mi 1.6 km 10 degrees 52.3 Cals 2.0 mi 3.2 km 1 degree 10.6 Cals 2.0 mi 3.2 km 3 degrees 30.6 Cals 2.0 mi 3.2 km 5 degrees 52.6 Cals 2.0 mi 3.2 km 10 degrees 104.6 Cals 3.0 mi 4.8 km 1 degree 15.9 Cals 3.0 mi 4.8 km 3 degrees 47.4 Cals 3.0 mi 4.8 km 5 degrees 78.9 Cals 3.0 mi 4.8 km 10 degrees 156.9 Cals 5.0 mi 8.0 km 1 degree 26.5 Cals 5.0 mi 8.0 km 3 degrees 79.0 Cals 5.0 mi 8.0 km 5 degrees 131.5 Cals 5.0 mi 8.0 km 10 degrees 261.5 Cals 10 mi 16 km 1 degree 53 Cals 10 mi 16 km 3 degrees 158 Cals 10 mi 16 km 5 degrees 263 Cals 10 mi 16 km 10 degrees 523 Cals

This is all for a person with a mass of 80 kg (about 176 pounds). Isn't that a spectacular difference? In other words, if you make a long-term change from walking on a flat ground (or treadmill) to walking up inclined ground (or an inclined treadmill), you burn extra energy with every step you take!

And what's with the Jillian Michaels quote? Well, I'm no longer the fittest guy on scienceblogs; say hello to Travis and Peter over at Obesity Panacea, our newest ScienceBlog! But whatever you're doing, don't forget to take the time to get out there and do something active; you'll feel better and you'll be healthier. And who doesn't want a higher quality of life?

Read the comments on this post...

Pi day is March 14th - get it? (3.14) I am a big fan of Pi. Here is my first post to celebrate the awesomeness of Pi (I know this is early, but I was too excited to wait).

How can you determine Pi?

Oh sure, tons of high schools do the classic experiment. Measure the circumference and diameter of as many round things as possible. Plot diameter vs. circumference. The slope will be Pi. Really, this is a great lab to do for all sorts of ages. The key thing is that students can see what Pi really means. I am not going to talk about this lab, I am going to do some thing cooler.

What if I had a 1 meter by 1 meter square taped out in the grass near the physics building and shot ping pong balls off the roof at this square? Ping pong balls are very difficult to aim, so you could easily imagine that they would fall in a random pattern in the square. I don't really want to set this up, so I am going to do it with a program instead. Here is a program that will generate random points in a 1 x 1 square.

This is what the output would look like:

Now, here is the key. What if I calculate the distance of each of these point from the origin? This would simply be:

Doing this, I can separate all the points into those that are more than 1 unit away from the origin and those that are less than 1 unit away. In this plot, (of 1000 points) the red dots are more than one unit away and the blue are 1 unit or less.

Note that this plot does not have exactly the same horizontal and vertical scale - no idea why it came out like that. However, this is enough dots that maybe you can see a pattern. The blue dots are filling up 1/4th of a circle. The full area of this circle would be:

Since I am a physicist, I have trouble leaving the area as unitless. What if I look at the ratio of blue dots to total dots? This should be the same as the ratio of the area of the quarter circle to the whole square, or:

Doing this for my last random number run, I get pi = 3.04. What happens as I increase the number of random dots? This a plot of the estimation of Pi for numbers of dots starting at 1000 up to 100,000 dots. I did this 5 times because each time is different.

So, you can see that as the number of random points increases, the estimated value of Pi is a little less spread out and closer to 3.14-something.

P.S. Thanks to Dave for this idea. You know who you are.

Update

I also made a Scratch program version of this calculation. (Scratch is a graphical programming language made at MIT - you know, for kids).

Read the comments on this post...

While doing some poking around online, I came across a website called Project Rho, which tries to provide some science background for science fiction writers who want some degree of technical accuracy in their imaginative work. Generally it looks like they're on the right track.

In their section on stealth in space, they explain with the weary air of repetition that there's no such thing. The flare of a rocket is bright enough to be seen from basically anywhere, and the thermal signature of even a spacecraft with rockets off is visible from clear across the solar system. The first I can believe (though of course it's worth checking), but the second sounds fishy. But so do lots of true things. Why not run the numbers ourselves? Before we do, let's see what they say:

"Well FINE!!", you say, "I'll turn off the engines and run silent like a submarine in a World War II movie. I'll be invisible." Unfortunately that won't work either. The life support for your crew emits enough heat to be detected at an exceedingly long range. The 285 Kelvin habitat module will stand out like a search-light against the three Kelvin background of outer space.

They go on later in the article:

The maximum range a ship running silent with engines shut down can be detected with current technology is:

Rd = 13.4 * sqrt(A) * T^2
where:
Rd = detection range (km)
A = spacecraft projected area (m^2)
T = surface temperature (Kelvin, room temperature is about 285-290 K)

If the ship is a convex shape, its projected area will be roughly one quarter of its surface area.

Example: A Russian Oscar submarine is a cylinder 154 meters long and has a beam of 18 meters, which would be a good ballpark estimate of the size of an interplanetary warship. If it was nose on to you the surface area would be 250 square meters. If it was broadside the surface area would be approximately 2770. So on average the projected area would be 1510 square meters ([250 + 2770] / 2).

If the Oscar's crew was shivering at the freezing point, the maximum detection range of the frigid submarine would be 13.4 * sqrt(1510) * 273^2 = 38,800,000 kilometers, about one hundred times the distance between the Earth and the Moon, or about 129 light-seconds. If the crew had a more comfortable room temperature, the Oscar could be seen from even farther away.

The equation given isn't derived. We have no idea where they're getting that 13.4 proportionality constant. Dimensionally it's correct, and it's pretty easy to derive the equation up to that constant which will depend on the sensitivity of the detector. That equation modulo some uncertainty with respect to that constant is accurate as far as it goes given a spacecraft of hull temperature T and cross-sectional area A.

I would take you through the steps of the derivation, but it would be pointless because the assumption that the hull temperature has anything to do with the interior temperature is simply flat wrong. We can prove this with a potato.

Switch your oven to the "Bake" setting at a temperature of 350 F. After preheating, put in the potato. The interior of the oven, and eventually the potato, are maintained at a constant temperature of 350 degrees. How hot is the exterior surface of the oven? Depends on how well insulated your oven is, but I can guarantee it's a lot less than 350 degrees.

The key is the understanding the relationship between heat and energy. Put hot coffee in a thermos - the hot coffee is hot because it contains thermal energy. If the energy can't leave, the coffee will stay hot because the energy stays inside the thermos. The outside of the thermos stays at the temperature of the surroundings. Now neither the thermos nor the oven is a perfect insulator. Some energy leaks out of the oven's interior, cooling it down. The oven thus has to pump energy into the heating elements to make up for this loss. Equilibrium is reached when the rate of energy being put into the oven equals the rate of loss through the insulation.

For a spacecraft in a vacuum, the pretty much the only way to lose energy from the interior is by radiant heat. The higher the temperature of the outside, the higher the rate of energy loss via radiation. But the temperature itself is irrelevant, since just like the oven and the thermos it's not necessarily related to the actual temperature inside the cabin at all. It is always and everywhere a function of the total power passing through the hull. If the temperature inside the cabin is constant, the power leaving the hull by radiation is exactly equal to the power being generated inside the hull.

So how far away can we detect a given amount of emitted power? According to Wikipedia, a telescope of 24" aperture can detect stars of magnitude 22 after a half-hour exposure. I think this is a pretty good realistic limit for detection with reasonable equipment in a reasonable time frame. Now we need to compare this magnitude to something of known power output. How about the Sun? The sun has magnitude -26.73 as seen from the Earth's surface (smaller magnitude is brighter), for a difference in magnitude of 48.73. The exponent used for magnitude is 2.512, so the difference in power per unit area of telescope is 2.512^48.73 = 3.1 x 1019. Since the Sun radiates about 1000 watts per square meter at the distance of the earth, the smallest radiant power we can reasonably detect in our telescope is about 3.123.1 x 10-17 watts per square meter.

Our hypothetical spacecraft is radiating that power into space, evenly distributed over the surface of a sphere of radius r, where r is the distance to the detector. When that power-per-area is the same as the limit of our telescopic capability, that gives us the maximum detection range. Mathematically,

Where rho is the sensitivity of our detector. Solve for r:

So what's the power? Well, each human on board is going to produce about 100 W just from basic bodily metabolism. Computers, life support, sanitation, and all the rest will contribute more. We might assume 10,000 watts total for a futuristic ship that's specifically designed to emit as little power as possible. It might well be significantly lower. Plugging in, I ger r = 5.98 x 109 meters. This is pretty far, but it's only around 4% of the distance from the earth to the sun. Practically nothing in terms of solar system distances. Even a ship dumping a megawatt of power should only be visible from a third of the earth-sun distance.

The reason for this divergence in our estimate versus the Project Rho estimate is that it takes a huge amount of energy to maintain a hull exterior at cabin temperatures. But insulation means that's not necessary, all that's necessary is that the power out equals the power generated in the interior.

Or at least this is my impression. I could be wrong. Thoughts?

Read the comments on this post...