Model of how temperature depends on latitude?

[21joanna12]

Homework Statement

I was asked to 'Use a very simple model to find how the Earth’s temperature should vary with latitude.'

Homework Equations

The Attempt at a Solution

I was thinking about flux and how this should be proportional to the temperature. So I first thought about the area that is exposed to the sun if I consider the Earth's surface to be composed of rings.

If you have an angle of inclination from the equator of $\theta$ (i.e. this is your angle of latitude), then it creates a ring of circumference $\pi(Rcos(\theta))^2$ where R is the radius of the Earth. So the area that would be formed when considering a 'strip' where the angle is $\theta + \delta\theta$ fives you $\pi R\delta\theta(Rcos(\theta))^2$
Since the Earth is tilted at an angle of 23.5 degrees, I thought that when integrating I would have to find:
$\int \pi R(Rcos(\theta))^2 \dot Pcos(\theta-23.5)d\theta$

I apologise for the mess above! This is the first time I am trying to use calculus to form my own model, so I'm not quite sure if I am using it right... and I am not too sure about how to do calculations with flux either, and I don't think I am using this concept correctly.

Any help is much appreciated! And if anyone knows any good resources/books that have caluclations like these, I would also be very grateful!

 

[BvU]

Circumference (Dimension: length!) is $2\pi R$, not $\pi R^2$.

 

[21joanna12]

Circumference (Dimension: length!) is $2\pi R$, not $\pi R^2$.

Apologies! I am clearly half-asleep today :P

So then I get $2\pi R^2 cos(\theta)\delta\theta$ for my strip of area and $2\pi R^2 P\int cos(theta-23.5)cos(\theta)d\theta$. Does this look OK?

 

[BvU]

And your integral goes from where to where ?
And your 23.5 is a constant ?

 

[21joanna12]

And your integral goes from where to where ?
And your 23.5 is a constant ?

The 23.5 is the angle of the Earth's tilt, and i am assuming the sun's radiation comes in horizontally so that the equator line is inclined at an angle of 23.5 degrees to the sun's radiation. With regards to the integral, I am not sure if I should be integrating at all because I an meant to find an equation for the temperature varying with latitude, and I was thinking about strips of area (with these strips being parallel to the latitude lines). But I don't think this is the best way to think about it because then the poles would have a temperature of zero in my model...

 

[BvU]

Re 23.5: So the folks on the capricorn tropic have the sun straight above their heads ?
:) And here's me thinking the poor chaps on the equator had that !
But not constantly...

Re integrating: you're right. After all, you started out mentioning flux !

Re circumference: would you want to multiply by circumference ? divide by it ? Why, precisely ? (And: think back to this integrating business)

 

[21joanna12]

Re 23.5: So the folks on the capricorn tropic have the sun straight above their heads ?
:) And here's me thinking the poor chaps on the equator had that !
But not constantly...

Re integrating: you're right. After all, you started out mentioning flux !

Re circumference: would you want to multiply by circumference ? divide by it ? Why, precisely ? (And: think back to this integrating business)

I realized that the angle of inclination will differ during different times of the year. On the Winter solstice, the equator line is 23.5 degrees above the horizontal rays of radiation from the sun, on the Summer solstice it is 23.5 degrees above and during spring and autumn the equator gets the greatest sun exposure. So my model would have to take this into account...

I think the strip area would not matter for the temperature- what matters is the intensity, so it is per unit area anyway. I think I should be considering the flux density rather than the flux. In this case I think the equation is simply $T=T_{at equator}cos(\theta-\beta)$ where theta is the latitude and beta is the latitude which has the radiation rays from the sun hitting it straight on...

 

[BvU]

So if you want a really simple model you average over the year and get a decent picture (or http://www.sage.wisc.edu/atlas/maps/avganntemp/atl_avganntemp.jpg )

Or else you introduce a time dependent $\beta =$ 23.5 * (days since March 21)/365

Often, flux density and flux are used for the same something/area (not so in magnetics: $B$ and $\Phi$)