Temperature of a Glass Plate Exposed to the Night Sky

Why Does Dew Form so Readily on My Corrector Plate?

We don't get dew on our telescope corrector plates very often here in Arizona.  However there are times when "dewsaster" strikes.  Usually during some attempt to image some deep sky object or to image some short-lived phenomena like a transit of Jupiter's moons. 

I have always wondered why my car windows can get ice on them when the air temperature never goes below 40 Degrees Fahrenheit.  I looked out on many web sites.  Somehow, I just couldn't let go once I started this.  I wanted to know why and how much - analytically.

After working through all of this, and to eliminate suspense, here is what happens if you have a plate of glass elevated above the ground and exposed to the night sky:

There are three types of heat transfer, conduction, convection and radiation. 
  1. Convection heating from the air warms the top and bottom surfaces of the glass plate. 
  2. Radiation from the ground warms or cools the bottom surface of the glass plate.
  3. Conduction allows heat to flow through the glass from the bottom surfaceto the top surface.
  4. Radiation from the top of the glass plate to the sky cools the top surface of the glass plate. 
Radiation transfers MUCH more heat to the night sky than intuition led me to believe.  It is proportional to the difference between the  fourth power of the temperature of the top surface of the glass plate and the fourth power of the night sky temperature.

Lots of semi-misinformation out there on the internet.  Some said the night sky temperature is 3 Degrees Kelvin (which is the background temperature of the universe.)  However, the night sky is full of air that radiates back at us.  So the generally accepted value is approximately -70 degrees Celsius or 203 Kelvin.

I did not find any solution on the internet to the problem that took the thickness of the glass into account.  Now I know why.  It is a tougher problem than I originally envisioned.  The only relevant exposition I found was an analysis of the temperature of an extremely thin leaf exposed to the night sky.  (This has significance for agriculture and is probably the most practical application.)  As it turns out, for thicknesses on the order of 1/4 inch, the contribution is significant but not overwhelming.  Of course the only way to be sure how much it would contribute was to work it all out.

When you try to take the thickness of the glass into account, you are immediately presented with one additional unknown (the temperature of the bottom surface of the glass plate) and it was very hard for me to see how to handle it.  The solution was to observe that the heat flow through the glass had to equal the heat transferred through the top surface of the glass.  This provided an additional equation so now there was a system of two equations and two unknowns which could be solved.  Even so, Mathcad was unable to compute a closed-form solution.  I had to settle for numerical calculations.  This means that so far, I am unable to write a program to do the calculations.  They have to be done using Mathcad.

So, how cold can the top surface of the glass get?  With air temperature of 40 F, ground temperature of 45 F and night sky of -70 C, the top surface of a 1/4 inch glass plate goes down to 25.2 F!  That is 6.8 Degrees F below freezing!  No wonder "dewsaster" strikes!  No wonder car windows ice up!  No wonder roofs get frost on them even though the air temperature stays well above freezing! 

Of course the body of a telescope slows this process down a little but I shudder to think of the calculations that would be required to characterize a complete system.  However, experience in the field makes me think that the telescope does not contribute very much to the situation and that a corrector plate mounted on a telescope would not fare much better than a corrector plate suspended in air with no telescope body.  (This could be tested in the field.) 

A word here about dew shields:
Note that "dew shields" DO work because they do not allow the corrector plate to "see" the whole night sky.  The corrector plate "sees" a little circle of the night sky and it sees the dew shield.  The dew shield also radiates to the night sky but is also warmed by the air.  This delays the cooling of the top surface of the corrector plate and delays formation of dew on the corrector plate. 

Possible improvement:  If the dew shield were constructed of two concentric cylinders with stand-offs to create about a 1/2 inch space between them, then, as the outer cylinder cooled, it seems likely that a small convection current would be set up in the space between the two cylinders since the outer cylinder would be cooled by radiation.  Cool air would then flow out the opening at the bottom.  This would draw relatively warm air in at the top which would convectively warm the inner surfaces.  This would seem to be a better dew shield that might prevent dew from forming on the corrector plate.  This principle would also of course apply to Newtonian reflectors.  An additional cylinder about the telescope tube with stand-offs might be a good idea if dew is a major consideration...

So, if you have Mathcad 12, here is the mathcad file:  dewtell2.mcd.  This can be used to do your own calculations using different temperatures.  (Arrrrghhh  ISP doesn't like .mcd suffix so fails to deliver the file!  Here is a zip file containing dewtell2.mcd that hopefully will get around their whacko filters:  dewtell2.zip )

Also, here is a PDF file that might be useful:  dewtell2.pdf.

Otherwise, here is the content of dewtell.pdf as a set of JPEG files:

Copyright 2005 Howard C. Anderson