Fun with numbers for Sep 9, 2019

So, this might become a thing, blogging augmented tweets.

Rotten Tomatoes is at it again

Dave Chappelle apparently has a Netflix special that has critics and audience at loggerheads. Being a little more quantitative, we can say that the critics are 862,712 times more likely to be using criteria opposite to those of the audience than the same criteria. The logic is in this post.

There are two differences between that blog post and this calculation that are worth mentioning:

1. Computing $c(12352,124)$ without loading special numerical packages that can handle large numbers is beyond the capabilities of most mathematical software, so we use a trick: as we're only interested in a likelihood ratio, and those combinations appear in the numerator and denominator, we know that in the end they'll cancel out, so we ignore them altogether.

2. Small probabilities raised to a large exponent quickly get to the precision limits of the floating point representations; to deal with that we make our calculations in log-space. So instead of computing $0.01^{124}$, which would be well below the 1E-99 ($10^{-99}$) limit for most numerical software, and be treated as zero, we compute $124 \times \log(0.01)$, do all the operations in this log space and at the end we exponentiate the result.

Used Apple Numbers (in lieu of RStudio) for this one, was surprised to learn that LOG is $\log_{10}(\cdot)$ despite Numbers also having a LOG10 function. Oh, well, no problem as long as one is careful:

Tesla bull tries to praise superchargers, arithmetic and hilarity ensue

Sooooo I tried the 250kW charger for the first time and I think I'm in loooooove 🥰 -- Went from 19% to 60% I kid you not in just 5 mins. Thanks @elonmusk @Tesla 🙏

Your battery capacity is 51 kWh?!

5 minutes = 300 s
250 kW * 300 s = 75 MJ
75 MJ = (60%-19%) * Capacity, or
Capacity = 183 MJ = 51 kWh

I thought TSLA batteries started at 75 kWh?! 🤔 Possible explanations:

1. Tesla bull is exaggerating; if it took 10 minutes, or the starting point was close to 40%, that would point to a 100 kWh battery.

2. Tesla software is lying to the car owner, making the numbers look rosier than they actually are.

3. Battery has lost capacity, which happens to batteries because of the underlying principles (two main chemical reactions, one exoelectric, one endoelectric; but secondary, parasitical reactions exist that lower the battery capacity over time). Even for a 75 kWh battery that would be a very big loss (1/3), unless his charging cycles are deep and irregular (that kills batteries faster).

Counting calories is like Enron accounting

Just thinking logically, here, if you were bailing water out of a boat, would you keep adding water in?

Okay, so if you're trying to lose weight by using body fat for energy, why would you eat carbs, whose sole nutrition value is as energy? Why eat when not hungry? *

(Most of the arguments I have about calories are with people who for some reason want others to eat carbs. Counting calories biases you towards choosing carbs over fat, since fat is more energy-dense.)

But more to the point, the whole foundation of calorie counting is Enron-like accounting, where some things are counted (more or less), some things are estimated, and many other things are sort-of, kind-of assumed away in some "basic metabolic energy needs" or other ways of saying "let's assume everyone has the same basic efficiency in chemical energy extraction and mechanical power production."

The lack of accounting for energy lost as heat, which anyone out of shape who's ever jogged with an athletic friend can tell you varies a lot with the person, is the most obvious Enron-like accounting.  Higher body effort for the same mechanical output is reflected in heat loss, and differences in that heat loss can be (as calculated in that figure) in the 100-200 kCal/hour range.

That's the same difference as the mechanical energy difference between jogging and walking.  One hour at the low end of that difference (remember, this is just heat, the mechanical energy is the same for both people) every other day is equivalent to 2 kg of fat extra per year if we believe in the basic model of calories-in calories-out.

Now,  how much difference can there be in unmeasured chemical energy output? Depends on the person and the diet, but note that on a 2500 kCal/day diet a systemic difference of 2%, that is 50 kCal/day, is equivalent to 2 kg of fat extra per year if we believe in the basic model of calories-in calories-out.

Can different people with the same general diet show a 2% difference? Yep. For example, a paper called "Energy content of stools in normal healthy controls and patients with cystic fibrosis," by Murphy, Wootton, Bond, and Jackson in Archives of Disease in Childhood (1991) [thanks PubMed], includes data about the controls' intake and stool. Here are the computations for the first 5 healthy controls:

Yeah, just like that, if CICO were true, these people, on the exact same diet, would show a 3 kg per year weight gain difference. 30 kg per decade.

So, whenever people start talking about calories, be aware that they might be looking for a way to say "you're overweight because of your moral failings; if only you were as virtuous as I am!"

- - - - -
* 1. Carbs are delicious, even addictive. Just be aware of the trade-off: they slow down body fat loss and make you hungrier faster. Because controlling appetite is key to fat loss, that second part is much more damaging than the first. Any "diet" that requires constant attention and self-control is going to fail for normal people with normal lives in normal society: just look around you.

2. There's a situation when I'll eat even though I'm not hungry: if I know I'll become hungry later when no high-protein food will be available and the hunger will be inconvenient or require iron will to avoid eating institutional carbs-and-fat food. Usually this situation can be avoided by taking high-protein foods like Biltong (no, it's not jerky; yes, it's worth the price) or hard-boiled eggs with you, but there are situations when that's socially unacceptable.

Nerding out with science fiction

The book is Dream of the Iron Dragon by Robert Kroese. Highly recommended science fiction.

At $c/3$, each kg of mass in the ship has kinetic energy of 607 TJ, the equivalent of a large tactical nuclear weapon (145 kiloton TNT), or about nine times the Hiroshima explosion. The relativistic increase in mass in small (around 6%, of course), but that velocity-squared, that's the big deal. (At these speeds we have to use the relativistic formula for KE, the one with $mc^2$ in the numerator.)

A table of temporal dilation (it's a highly non-linear transformation):

At 99.95% of the speed of the light, one hour of ship time would be 31 hours, 37 minutes, and 48 seconds in the resting frame. At that speed, each kilogram of mass in the ship would have 2.75 exajoule of kinetic energy or, in big boom terms, about 13 times the energy of the largest hydrogen bomb explosion (the Tsar Bomba at 210 PJ or 50 MtTNT).

Another excerpt of the same book, non-numeric, but very dear to anyone who's ever worked in a large bureaucratic organization:

#NerdWhoMe

Interstellar delivers truth bombs

Early on in the movie Interstellar there are two important lessons about what makes a society fail (or succeed), both delivered in the parent-teacher conference that Cooper attends.

Lesson one: don't underestimate the power of engineering (and science)

Lesson two: beware of those who would rewrite the truth

(Excerpts from the novelization of the movie by Greg Keyes. No, I'm not a nerd. Ok, I am.)

Andrew Rader points out some problems with the movie:

The main problem was also pointed out by Kip Thorne in The Science of Interstellar: that fighting the blight on Earth would make a lot more sense than going to a different planet.

Thorne also raises the problem of orbital mechanics in chapter 7 of the book:

and proposes a few speculative mechanisms to get the necessary changes in velocity from gravity assists. Note that there are two decelerations one of $c/3$ and one of $c/4$ for a total speed change of  $7c/12$ or $1.75\times 10^{8}$ m/s. Returning to the Endurance requires an increase in speed of $1.75\times 10^{8}$ m/s as well.

To see the size of the problem, let's say they take 500 seconds (8 minutes and 20 seconds) to do each maneuver (while the rest of the Universe ages significantly) and the Ranger's mass is 2 metric tons (for simplicity, we'll assume that the water taken in on the planet makes up for the loss of Dr. Doyle to stupidity, indiscipline, and lack of planning). If we assume constant thrust for simplicity, assume away all friction and ignore the propellant mass loss (yay, infinite specific impulse!), the thrust needed for each maneuver is $7 \times 10^8$ Newton or about the same as 1077 SpaceX Merlin engines (averaging their atmosphere and vacuum thrust to 650 kN). Since there's propellant mass loss, let's say we "only" need the equivalent of 900 Merlin engines. So, yes, only a gravity assist would do.

Yes, it's an oversimplification, but didn't feel like solving the Tsiolkovsky equation. Hence the drop from 1077 to 900 engines. (That's still equivalent to 100 Falcon 9 rockets.) By the way, Thorne appears unconvinced of the feasibility of those gravity assists and hence of the feasibility of whole expedition to Miller's planet. But at least they tried to be accurate with some science in the movie.

Oh, and speaking of nerds: