Possibility of a venus-like planet with a magma ocean?

[Armchair Rocket Scientist]

So, something I've been wondering about for a while: is it possible for the greenhouse effect in the atmosphere of a venus-like planet to raise its surface temperature so high that the entire surface becomes molten?

First, according to this link, nearly all igneous rocks that would be found on a planetary surface should be molten at a temperature of 1200 *C, or 1500 K. I am assuming that this is still approximately accurate under a pressure of several hundred atmospheres.

We can assume that a planet with a Venus-like atmosphere will have a nearly constant surface temperature across its entire surface, even if it is tidally locked to its star, because we know Venus's surface temperature varies by only a few degrees, and its rotation is so slow that it's solar day is over a hundred days long. We will also assume a near-circular orbit, giving a constant planetary temperature, and assume that the internal heat generated by the planet is negligible.

Therefore, the planet should radiate the same amount of energy as it receives from its star, and we may use the equations here.

P_in = Insolation * (1 - albedo) * pi * planet radius², and P_out = emissivity * the Stefan-Boltzmann constant * 4 * pi * planet radius² * T⁴. Setting these equal to one another:

Insolation * (1 - albedo) = 4* emissivity * the Stefan-Boltzmann constant * T⁴

The original page sets emissivity equal to 1 for the black-body temperature, but we won't do that, so:

T = ((1-albedo)*insolation/(4*Stefan-Boltzmann constant * emissivity))^1/4

Note that the temperature is measured at the altitude where the atmosphere becomes effectively transparent to thermal radiation. This should be a bit lower than the actual surface temperature, but we will set them equal for convenience.

We will set T = 1500 K, albedo = 0.90 (the same as Venus), and put the other constants together:

(1500 K)⁴ = insolation * 0.1 / (4 * 5.67*10^-8 W/m²*K⁴ * D² * emissivity)

insolation / emissivity = 1.48*10⁷ W/m²

To make life easier, we will change insolation to units of Earths (1366 W/m²)

This gives us emissivity = insolation / 10,800 earths.

For a planet receiving the insolation of Venus, the required emissivity is therefore 1.77*10^-4, which is about 16x lower than it would be for Venus (half the surface temperature, everything else is the same, and this is proportion to T⁴). With Mercury's insolation, this rises to 6.18*10^-4. If the planet has equal emissivity to Venus, it must be roughly 4 times closer to the sun.

Now, I still have some questions:

1. How do I determine at what temperatures and pressures CO2 will be stable? Note that unless photodissociation and ionization occurs, a rocky planet of Earth's mass or larger should have no problem retaining CO2, even at high temperatures.

2. How can I approximate a planet's emissivity from the amount of CO2 (mass per unit of surface area) in its atmosphere.

3. At higher insolations, will the sulfur clouds seen on Venus disappear, raising the planet's albedo?