How much does temperature affect the time of sunrise? Please forgive me if this is a dumb question, or if my understanding of basic physics is wrong. Please feel free to correct me.
As I understand it, if the Earth didn't have any atmosphere, then the time of sunrise would be the point when the Sun's rays approached your position at a tangent. For example, imagine that the Earth is a perfect sphere, with you standing on top, and the (apparent) motion of the Sun was it travelling clockwise as follows (scales completely wrong, but hopefully the concept is correct)...

Now, assume for simplicity that the Earth's atmosphere is of constant destiny, and starts at a defined height above the planet's surface, then (if I understand correctly), the Sun's rays would be refracted as they entered the atmosphere, meaning that you would see the sun slightly earlier (exaggerated)...

Obviously, this is very simplified, not least because the atmosphere is a gas, and therefore of variable density, presumably being less dense the higher you went. I imagine that the variation in density would mean that the rays appeared to curve, rather than take a sudden turn as shown above.
My question is, how much difference does the air temperature make to the amount of diffraction, which in turn affects the time at which you would see the sunrise? My feeling is that if it were cold across the Earth, then the air would be more dense, resulting in a greater degree of refraction, and so an earlier sunrise. By contrast, a higher temperature would mean lower density, less refraction and a later sunrise.
Anyone able to give me some estimates of how much difference you would expect to see between a warm summer's day and a cold winter's day, assuming normal parameters for "cold" and "warm" for our planet?
 A: The biggest cause of refraction is the change in density of the atmosphere with altitude, not changes caused weather conditions at the surface.
There are formulas to calculate this effect assuming standard values of temperature and pressure at ground level. The apparent change in position of the sun in those conditions is about the same as the sun's visible diameter. 
The time difference this causes depends on the angle at which the sun rises above the horizon, which depends where you are on the earth and what time of the year it is. If the sun rises vertically, the time difference is about 2 minutes, but if it rises at a shallow angle to the horizon it may be much longer.
Changes in air temperature and pressure also have an effect, which is easy to observe (from the known position of the stars, not just by observing the sun) but difficult to predict in a useful way. As a consequence of this, it is not very useful to predict sunrise and sunset times to more accuracy than the nearest minute.
See https://en.wikipedia.org/wiki/Atmospheric_refraction
