How does the reappearance of Io during an eclipse relate to linear lighting?

[big man]

Hey guys,

I've got a query. When one of Jupiter's satellites reappears from its shadow wouldn't the satellite light up linearly? I mean if you assume it is circular wouldn't the amount it's lit depend on the area of a sector of a circle?
And that would mean that since the area of a circle depends on the central angle it is a linear relationship.

The thing confusing me is that someone said it wasn't a linear relationship.
Why would that be? Is there something I'm not considering here?

This is something I observed a couple of nights ago and I thought I'd check the lightcurve using some freeware. The lightcurve doesn't show a linear relationship, but I don't think I can rely on that 'cause I think the lightcurve is a bit dodgy due to the proximity of Io and Jupiter.

 

[russ_watters]

What you are saying there doesn't make a whole lot of sense to me, but it seems pretty obvious that a circle moving linearly out of a nearly linear shadow is going to brighten in essentially a half-sine wave pattern. Make yourself a demonstration if you are having trouble visualizing it, but as the shadow moves across the moon, you'll get larger and larger, then smaller and smaller slices of it, like slicing a tomato.

 

[big man]

Sorry about the poor explanation.

What I was saying is that the area of the part of the moon that is lit up is increasing linearly. Since the area of a sector is given as $$A=\frac {\theta} {2} r^2$$ and since theta is the only changing variable I would have thought that area would increase linearly. Is this right?

 

[russ_watters]

A sector is a pie-shaped slice. When a moon reappears, it doesn't reappear by increasing sector size, it reappears by slices like a sliced tomato.

 

[big man]

ahh yup I'm an idiot...

Thanks for clearing that for me.

I was looking at a sector when I was meant to be looking at the formula for a segment.
It's all sorted out now...Thanks!