007: GoldenEye gets an orbit right
I was reading the book 007: GoldenEye and noticed that Xenia Onatopp's description doesn't match Famke Janssen's looks; oh, and also this:
At first glance, the book appears to be playing fast and loose with orbits; after all, the ISS, which orbits around 400 km, is also on a roughly 90-minute orbit. So, let us check the numbers.
The first step is computing the acceleration of gravity $g_{100}$ at 100 km altitude. Using Newton's formula we can compute it from first principles (radius and mass of the Earth, gravitational constant... too many things to look up), or we can use the precomputed $g=$ 9.8 m/s$^2$ and solve for the altitude using a ratio of two Newton's formulas at different radii (using 6370 km as the radius of the Earth):
$ g_{100} = 9.8 \times \left(\frac{6370}{6470}\right)^2 = 9.5$ m/s$^2$
This acceleration has to match the centripetal acceleration of a circle with radius 6470 km, $a = v^2/r = g_{100}$, yielding a orbital speed of 7.84 km/s.
The circumference of a great circle at 100 km altitude is $2 \times \pi \times 6470$ km = 40,652 km, giving a total orbit time of 5180 s, or 1 hour, 26 minutes, and 19 seconds. So close enough to ninety minutes for a general.
So, yes, GoldenEye's orbit makes sense (-ish). Even though it's much lower than that of the ISS, which also has around 90 minute orbital period (92 minutes, and it's on a very mildly elliptical orbit).
On the other hand, a 100 km orbit would graze the atmosphere (it's inside the thermosphere layer, near the bottom) and therefore lose energy over time, so not a great orbit to place an orbital weapon masquerading as a piece of space debris, because you can't boost up "space debris."
Here are the circular orbital times for different altitudes; because of the approximation of $g=9.8$ m/s$^2$ and radius of the Earth as 6370 km, there are increasing errors with altitude, which are obvious for the GEO orbit (in yellow), still not bad since GEO shows that errors will be less than 2 minutes 38 seconds on all the other orbits:
There's no True(x) function for the internet (or anywhere else)
(Ignore the bad grammar, it was a long day.)
What happens if we feed the [putative social media lie-detector] function $\mathrm{TRUE}(x)$ the statement $x=$"the set of all sets that don't contain themselves contains itself"?
Let's take a short detour to the beginning of the last century...
Most sets one encounters in everyday math don't contain themselves: the set of real numbers $\mathbb{R}$ doesn't contain itself, neither does the set $\{$chocolate, Graham cracker, marshmallow$\}$, for example. So one could collect all these sets that don't contain themselves into a set $S$, the set of all sets that don't contain themselves. So far so good, until we ask whether $S$ contains itself.
Well, one would reason, let's say $S$ doesn't contain itself; then $S$ is a set that doesn't contain itself, which means it's one of the sets in $S$. Oops.
Maybe if we start from the other side: say $S$ contains itself; but in that case $S$ is a set that contains itself, and doesn't belong in $S$.
This is Russell's set paradox and it shows that there are propositions for which there is no possible truth value.
On the price of micro-SD cards
Browsing Amazon for Black Friday deals (I saved 100% on Black Friday with coupon code #DontBuyUnnecessaryStuff and you can too), I saw these micro-SD cards:
Instead of buying them, I decided to analyze their prices, first computing the average cost per GB (as seen above) and then realizing that there's a fixed component to the price apart from the cost per GB, which a simple linear model captures:
All the electricity California needs is about 6 kilos of antimatter
Okay, antimatter is a bit dangerous, so how about we develop that cold fusion people keep talking about? Here:
(Divide that by an efficiency factor if you feel like it.)
Relativity misconceptions and the reason I restarted blogging
I was listening to a podcast with Hans G Schantz, author of the The Hidden Truth trilogy (so far… fans eagerly await the fourth installment; highly recommended) and he had to correct the podcast host on what I've noticed is a very common misconception: that "near" the speed of light relativistic effects are very large.
Which is true, for an appropriate understanding of "near."
Time dilation, space contraction, and mass increase are all regulated by a function $\gamma(v) = (1 -(v/c)^2)^{-1/2}$, a very non-linear function. For the type of effects that people typically think about, like tenfold increases, we're talking about speeds near $0.995 c$; for the type of effect that would be noticeable in small objects or short durations, one needs to go significantly above that:
Interestingly, the decision to restart blogging (first under the new name "Fun with numbers," then back to the admonition to keep one's thoughts to oneself by Boetius) was due to a number of calculations I had been tweeting regarding relativistic effects in the Torchship trilogy by Karl K Gallagher (highly recommended as well). Here are some examples, from Twitter:
And it's always heartwarming to see an author who keeps the science fiction human: that in a universe with mass-to-energy converters, wormhole travel, rampaging artificial intelligences, and AI-made trans-Oganesson-118 elements, there's a place for the problem-solving power of a wrench:
Computerphile has a simple data analysis course on YouTube using R
Link to the playlist here.
Download RStudio here.
Another promising lab rig that I hope will become a product at scale
The Phys.org article is here and the actual Science Advances paper is here.
Strictly speaking, what the paper describes is a successful laboratory test rig, but let's be generous and consider it a successful tech demo, also known in the low-tech world as a proof-of-concept. Note that though not all successful lab test rigs become successful tech demos, the ratio is much higher than the number of lab rigs (successful and otherwise) that become tech demos, so it's not that big a leap in the technology development process.
