Stefan-Boltzmann law, is the statement that the total radiant heat power emitted from a surface is proportional to the fourth power of its absolute temperature.
As an equation this is written
E = σT4,
E is energy, σ is a constant and T is temperature. The amount of energy emitted increases with the fourth power of temperature. Such radiant heat emission is the only way a body, such as the earth, can shed heat. So if the earth starts to heat up it will continue to do so until the temperature increases enough to boost radiant emissions and shed the extra heat. But to melt ice takes heat without raising the temperature. So if a body, such as the earth, is taking in more heat than it emits from some source, such as the sun, and it contains ice that is melting, then a certain portion of that extra heat has been absorbed without raising the temperature. And of course even that portion of the heat that does raise the temperature cannot raise it enough to produce a balance, for if the temperature had been raised enough to shed the extra heat, the ice wouldn’t have melted. As long as ice is melting the temperature cannot increase enough for heat balance.
Now these radiant energy emissions also depend upon σ, the constant, which is calculated for an ideal “black body” and can differ with the color of the object, in this case the planet earth. Whatever its value for earth, it is more or less constant for earth. A change is often marked by adding another symbol εac. If something made it darker or lighter, then σ would change but increases or decreases in emissions would, during periods that σ remained constant, still be a function of T. Thus things like volcanic eruptions that spread dark particles or changes in greenhouse gas levels, which change the color, but outside the visible spectrum, can possibly reverse the process. And this would easily be apparent, for ice would stop melting.
But if ice is continuously melting, then this has not happened. Heat will continue to make ice melt and the earth will continue to absorb extra heat without raising the temperature or earth as a whole. This will be true even if we add no more greenhouse gases to the atmosphere. A pot over a small flame will eventually boil. As it is the earth will continue to take in extra heat until the ice is melted. Then the temperature of the body as a whole will go up and will stabilize at some value. In the temperature variations of the past the extra heat was so small that changes in σ, usually a decrease in greenhouse gases caused by the weathering of rocks or expansion of oceans and therefore the CO2 absorbent ocean surface, was enough to reverse the process. Natural processes could reverse the flow because there was so much less extra heat. The change from plus to minus was small.
The rate at which ice melts is a function of the amount of extra heat. This is a lot of melt over a very short time. I have often heard it said that we have to cut our emissions by 30%, 50% or whatever. But the imbalance right now, if we do nothing, is enough to continue to melt ice without raising the temperature enough to shed the extra heat until the ice is gone.
So why has the earth heated at all? The second law of thermodynamics says, roughly, that heat moves from high temperature to low. Were this perfectly efficient the temperature would not rise above 273.15° K, the freezing point, until all the ice is gone. Heat moves from hot to cold, but inefficiently. It is because of this inefficiency that there is an arctic and a tropics and not a ball of uniform temperature. The T is the equation is an average temperature. Heat travels slowly via ocean currents and wind. Thus even though the total average temperature of the earth is not enough to throw off the extra heat, much of it still has temperatures well above 273.15°.
The earth has increased in temperature even though there is still ice because of the inefficiency of the movement of heat from the tropics to the icy regions. But these are only inefficiencies, and if a balance were reached, because of a warming in the tropics, heat would still leak north, make ice melt, and lower the temperature again. Then there would again be an imbalance. It will not stop at 1.5 ° or 2° unless the ice is gone or there is no heat transfer to icy climes.
In winter ice can, of course, be colder than the freezing point. Extra heat in the arctic in winter can raise the temperature of ice from, for example 243° K on average to (I’m eyeballing here) 247°, which is what has happened. Below the freezing point extra heat does raise the temperature in ice. This flattening of the temperature gradient north to south can affect the inefficiencies of heat transfer, in this case by distorting what is called the polar vortex. This circular wind which, so to speak, keeps the cold in the arctic, has been letting it out, and these winds meander south, pouring cold air south and warm air north, reducing the inefficiencies of heat transfer. In this way increased heat overall can produce local unusual cold snaps.
In the summer the arctic, because of ice melting, remains at roughly 273.15° K, the freezing point of water, while the rest of the hemisphere is heating up. The Hadley, Ferrel, and arctic atmospheric cells keep winds from blowing from the tropics to the arctic and so limit that amount of hot air that moves north. Not convection but conduction, a slower process, has to move heat between cells. As the tropics warm, because of these inefficiencies, the Hadley cell is expanding. Also, large powerful hurricanes, whatever else they do, move a lot of warm air north. An increased temperature gradient north to south breaks down the inefficiencies.
But that is neither here nor there. As long as the second law of thermodynamics holds, heat will move to the icy regions from warmer climes, the ice will melt, and the earth will, to that extent, absorb heat without raising temperature. And therefore, to that extent, according to the Stefan-Boltzman law, earth will not be able to throw off the excess heat coming in until the ice is gone.