The red shift on the sun is obviously not Doppler since the sun is not moving away from us. This shift shows a variation in magnitude that correlates with the number of electrons along the line of sight. It is smallest at the solar center and greatest at the limb where we are looking through the thickest part of the sun's atmosphere. John Kierein and Brooks Sharp showed this correlation as a Compton effect interpretation in the journal "Solar Physics" in March of 1968. Compton himself believed this was the cause of the solar red shift. The electrons on the sun are concentrated in altitude by gravity with the greatest density near the sun's surface (the photosphere) to produce the sun's intrinsic red shift. Similarly, the quasar red shift have an intrinsic Compton effect red shift concentrated at or very near the object's surface.
In addition to this red shift on the sun, which is there all the time and is on the order of 1 part in a million, there has been measured a gamma ray red shift that occurred only during a large solar flare. This solar flare red shift was nearly 1 percent or one part in a hundred! It was measured by the RHESSI satellite. The red shift varied by the element, the heavier element having a bigger red shift. This red shift is also due to the Compton effect and is caused by the gamma rays ionizing the elements and releasing electrons from these element "targets". The heavier elements have greater numbers of electrons to release and consequently have multiple Compton collisions and greater red shifts. Other gamma ray red shifts such as this are also intrinsic Compton effect red shifts.
For the Compton effect to cause the cosmological red shift, intergalactic space must have a density of free electrons and/or positrons. The further light travels through this transparent medium, the greater the red shift - and Hubble's law follows. The existence of electrons and positrons in intergalactic space has been shown by observations of electron-positron annihilation gamma rays coming from above our galactic plane. This is the direction our galaxy is plowing into the intergalactic medium. This is observed from the, appropriately named, Compton Gamma Ray Observatory in orbit above the Earth's atmosphere.
Indeed, while intergalactic space was once thought to be empty, now we know it is filled with clouds of high velocity gas that contain molecular hydrogen. This molecular hydrogen comes from the condensation of hydrogen atoms that are just free electrons and protons. When light hits these free electrons, it produces the Compton effect red shift.
Since Compton effect causes the red shift, the universe is not expanding, but rather is "static". Max Born did an analysis of the background temperature of such a universe and found that it doesn't differ greatly from the observed 3 degree kelvin background.