Wikipedia:Reference desk/Archives/Mathematics/2010 June 1

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June 1

Find extremum

I remember from calculus (a looongg time ago ;) that to find if a point is an extremum of a function you take increasinly higher order derivatives until the result is a nonzero constant. I've come across this again, but I'm stuck. How do you do that with something like sin(2x)? That will never differentiate to a constant! Of course I know that from the basic properties of the sin function that the extrema are at 45, 135, etc., etc. in my example, but the problem I have to deal with may be much more complex than that.

The higher order derivatives don't have to be a nonzero constant. They just need to have a nonzero value at the point you are evaluating. For $\sin(2x)$ at $x_{0}=\pi /4$, differentiating once gives $2\cos(2x_{0})=0$; once more gives $-4\sin(2x_{0})=-4<0$, so this is a local maximum. -- Meni Rosenfeld (talk) 18:33, 1 June 2010 (UTC)

Our article is here: higher-order derivative test.—Emil J. 18:36, 1 June 2010 (UTC)

Crop circles of pi and Euler's identity

How does this crop circle work? Kittybrewster ☎ 20:00, 1 June 2010 (UTC)

What crop circle? -- Meni Rosenfeld (talk) 20:46, 1 June 2010 (UTC)

As shown in google search crop circle eulers identity Kittybrewster ☎ 21:26, 1 June 2010 (UTC)

explanation here [1]. why are aliens communicating in ASCII and not Unicode ? 87.102.77.88 (talk) 23:17, 1 June 2010 (UTC)

Thank you. Very comprehensive response. Kittybrewster ☎ 05:05, 2 June 2010 (UTC)

As shown in google search crop circle pi 2008 Wroughton. Kittybrewster ☎ 05:05, 2 June 2010 (UTC)

Found it. Kittybrewster ☎ 15:00, 3 June 2010 (UTC)

Maximum/Minimum Irradiance for "Human habitable" planet temperatures?

(Third time's a charm)
I keep trying to create a simple model for anyone who wants to speculate on a planet having
habitable temperatures for people (not extremophiles) based on Irradiance.
Take for example HD 38801 b:
Star Radius = 2.53 sol
Star Te = 5222 K
Stefan–Boltzmann constant, σ = 5.67051E-8
Semi-major axis = d, in this case 1.7
Eccentricity = e, in this case 0
Emissivity = ε, (Earth=0.62009)
Albedo = A, (Earth=0.3)

=((((R^2)*σ*(Te^4)*(1-A))/(4*ε*(d±(d*e))^2))^0.25)-273.15
So I can get a global annual average temperature for a planet,
If I assume a global average albedo and a global average emissivity

	Albedo
εmissivity	0.20	0.25	0.30	0.35	0.40	0.45
0.80	5°C	1°C	-4°C	-9°C	-14°C	-20°C
0.75	10°C	5°C	0.7°C	-4°C	-10°C	-15°C
0.70	15°C	10°C	5°C	0.3°C	-5°C	-10°C
0.65	20°C	16°C	11°C	5°C	-0.1°C	-6°C
0.60	26°C	21°C	16°C	11°C	5°C	0.5°C
0.55	33°C	28°C	23°C	17°C	11°C	5.4°C

Ignoring the perspective that single value Albedo and Emissivity are very
simplistic, since "Global Annual Average Temperature" is also but it exists:
A) at what levels of irradiance (max/min) does this become moot?
B) what combinations of Albedo & Emissivity are unrealistic or possible??

Just to be clear I am talking about people. The GAAT (14°C) is a surface temperature.24.78.167.139 (talk) 20:45, 1 June 2010 (UTC)

I came upon this chart in a search and realized this illustrates well the idea that there are a lot of possibilities but some are unrealistic. - 24.78.167.139 (talk) 20:45, 1 June 2010 (UTC)

This isn't really the right forum for that question. We cover mathematical physics but this seems more like general science or astronomy.--RDBury (talk) 00:45, 2 June 2010 (UTC)

"Astrophysics (...) is the branch of astronomy that deals with the physics of the universe, ..." 24.78.167.139 (talk) 03:33, 2 June 2010 (UTC)

Note that planets and moons are heated by many factors other than the light of their suns:

1) Tidal forces between planets, moons, and suns. Most significant if you are nearby a large body.

2) Residual heat from initial formation. This dissipates quickly for small bodies, but lasts for billions of years on larger planets.

3) Radioactive decay (fission).

4) Nuclear fusion. Only a factor on the largest planets, which are essentially "failed stars".

5) Bombardment by meteors, comets, solar wind, etc.

While we tend to think of solar heating as the most significant heat source, this is only true for the surfaces of planets near the host star. However, it's quite possible for there to be other scenarios where the heat is mostly provided by different sources, such as a moon around a massive planet far from it's sun. StuRat (talk) 04:17, 3 June 2010 (UTC)

Regardless of whether solar heating is the most significant or not that is my question, I am not asking about "other scenarios," I am asking about solar heating! Why do you people always have to change the question so that you can lecture on what makes you feel important to express?? 24.78.176.110 (talk) 08:56, 11 June 2010 (UTC)

There may be ice on Mercury. Dmcq (talk) 07:38, 3 June 2010 (UTC)

• "The mean surface temperature of Mercury is 442.5 K" (169.35°C), and it has almost no atmosphere, so these are weird responses considering I mention I was referring to "habitable (surface) temperatures" for people.
• Scientists and journalists are willing to speculate wildly that there may be life on this or that extrasolar planet, so it isn't all that odd to look at an extrasolar planet or moon as having habitable temperatures for people. I don't understand the lack of imagination on that possibility. 24.78.167.139 (talk) 20:35, 3 June 2010 (UTC)

As I said before the mean temperature is quite a bad guide to whether there are places where people could live. There are places on Mercury where there may be a quite reasonable temperature for people though you'd have to move around. For Venus floating around in a balloon might give a reasonable habitable temperature even though the surface temperature is hotter than the mean Mercury temperature. I don't see why that is lack of imagination. Dmcq (talk) 21:49, 3 June 2010 (UTC)

You seem to be unable to hear what I am asking over the importance of your opinion; I have not asked for the bad possibilities I have asked for the good ones. Venus is not considered in the habitable zone for a reason, it is ludicrous for you to suggest that Mercury is in the habitable zone.

• I repeat: Global Annual Average Surface Temperature:
  

A) at what levels of irradiance (max/min) does this become moot?
B) what combinations of Albedo & Emissivity are unrealistic or possible??
Anyone? - 24.78.176.110 (talk) 08:56, 11 June 2010 (UTC)