The Q+A column in today’s New York Times is uncharacteristically weak. The question was “Why doesn't [the Sun] have a core of heavier elements that sank to the center?” The main problem with the column is that it doesn’t really answer the question. Here’s what I’d say.
Although the Sun is made mostly of hydrogen and helium, it does have trace amounts of heavy elements. These do not settle to the core but are distributed throughout the volume of the Sun. The reason is temperature: there is enough thermal energy in the solar interior to keep the heavy elements bouncing around. As an analogy, consider the Earth's atmosphere. Although carbon dioxide molecules are heavier than nitrogen molecules, there's plenty of thermal energy to keep the CO2 from settling at the bottom of the atmosphere.
The article also presents the impression that the Sun will produce its own heavy elements later in its life. The Sun will produce only helium, carbon, and oxygen. Elements heavier than oxygen are produced only in more massive stars.
I sent the Times a letter to this effect, but I don’t know if they’ll care.
Update: They didn’t.
I’m not going to write about politics in this blog — there’s plenty of that out there already. And I’m certainly not going to engage in any sort of political advocacy. But I thought of an interesting application of probability theory to the upcoming election, and I thought I’d summarize it here.
Remember back in 2004, when it seemed like every Democratic voter was basing his or her choice on which candidate was most “electable”? People’s gut feelings about this sort of thing are generally pretty unreliable, so it’s kind of interesting to look for a data-based answer to the electability question. It occurred to me recently that the various political futures markets provide a good way to answer that question for the upcoming election.
For those who don’t know about the political futures markets, they’re basically a way that people can bet on various political events, including the upcoming US Presidential race. Slate has a good description with lots of nice graphs. The odds on all of these bets can be interpreted as giving the probabilities of various outcomes in the race, as estimated by the community of bettors. These probabilities give enough raw data to measure each candidate’s “electability.”
By electability I mean the probability that a candidate will win in the general election, given that he (or she) gets the nomination. One of the futures markets (Intrade) lets people bet on both who will get the nomination and who will win the general election. The ratio of these for any given candidate is the electability. It’s just Bayes’s Theorem:
P(Hillary wins the presidency) =
P(Hillary gets the nomination) * P(Hillary wins the presidency | Hillary wins the nomination).
[In case it’s not familiar notation, P(y | x) means the probability that y occurs given that x occurs.]
The last factor on the right is Hillary’s electability. The futures market tells us the other two probabilities for each candidate. So we can find the electabilities of all the candidates by simple division. Before looking below, take a guess about which leading candidates in the two parties are most electable.
Continue reading Electability