Even Screaming Chicken’s argument that a function must be in its simplest terms falls flat. Pi is no more a fundamental quantity than Pi/4. In fact, some people have suggested making 2Pi the constant (and calling it “Tau”) because it simplifies certain other formulas. Not that that helps here, but my point is that there is nothing special about Pi that privileges it over any other linear scaling of Pi. Hell, what if we lived in a world where the equivalent fundamental constant was called Fnord and had a value of approximately 0.785… would you then be arguing that area was a function of diameter but NOT a function of radius?!

Sorry, I know you are trying to save face after making a really stupid remark, but better now would be to simply say that you said something silly, you feel silly, and of course you realize that area is a function of diameter.

]]>Josh way back on #11 has it right, actually. While it is true that Europa is 7^4=2400 times brighter than Neptune, the images I was taking were well resolved (even Europa, at 1 arcsecond, is [or would have been] well resolved with the adaptive optics at Keck, which can resolve down to ~0.05 arcseconds which, I must say, is one of the most awesome things ever). In this case, resolved means spread out over a number of pixels proportional to angular diameter^2, thus the flux PER PIXEL is down by a factor of angular diameter^2. But the angular diameter is proportional to the (real diameter/distance). So the brightness per pixel, which matters, is only proportional to (the distance ratio)^2 = 7^2=49. Still too much for Keck, though, sadly.

Another way to think about this is that if we suddenly made Neptune twice the diameter, it would be 4 times brighter in total, but if you were resolving it, the surface would still be just as bright, there would just be 4 times more surface.

The REAL reason Europa was saturated at Keck though is because astronomers always design these things to look at distant faint galaxies and forget that every once in a while someone will come along and want to look in our neighborhood to check on the whales on Europa.

Mike

]]>I may never fully recover!

]]>Curse, you, sir, for blowing my thesis out of the water!!

Over here in the UK, Ariel is a brand of washing powder, which (I guess) is probably why it is not a very popular girl’s name.

]]>I have a cat named Luna, and I know a woman named Ariel. For what it’s worth…

]]>Area is a function of radius squared, NOT diameter squared

Except CB (47) said:

A = pi*r^2 = pi*(d/2)^2 =

0.25 * pi * d ^ 2= [etc.]

[My bolding]

So, Screamin Chicken, what exactly are you trying to say? That equations should never be rearranged from their simplest form? Or what?

]]>“Screamin Chicken (25), Chris (26): So you’re saying area does not scale as diameter squared? Are you sure you want to argue that? To make it more clear: if I double the diameter of a circle, what happens to the area?”

You obviously didn’t read my statement correctly. What I stated is 100% correct. Area is a function of radius squared, NOT diameter squared, but since you’re taking ratios it doesn’t matter. That ratios bit is the same thing as SCALING (and you know it).

If you had used proper English I wouldn’t have commented at all. You said, “area goes as diameter squared”, which is not clear at all. What does “goes” mean??? Obviously you meant “scales” and not “equals”. You could have just replied saying, “Yes, I meant scale when I said goes” instead of ignoring the rest of my post and saying I’m arguing something that I wasn’t.

@ 45. BJ:

“People, seriously, if you’re going to try and correct someone’s math error, be really, really sure that you’re correct. Radius, diameter, and circumference are all related by constant factors; thus, scaling of quantities with one of these factors is the same as scaling with the other two. ”

Duh! I never said anything different. Phil used the word “goes” which is ambiguous at best. That’s the only thing I had a problem with. And just to be clear I did state that there was nothing wrong with Phil’s math since they do scale.

]]>*Hmmm…*

“Equals” is an interesting and powerful math concept.

]]>