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I have a planet around 11242 km in diameter. It is very similar to Mars, with most surface water trapped at the poles. For this, I want the polar temperature below 0 C, but the equatorial temperature to be consistently above 50 C; this way, I can have the water stay at the poles and not evaporate or melt. Assuming that the star's radius is 1.21 solar radii and 5992 K, how can I figure out the temperature of a region at the poles of the planet, as opposed to the temperature of the equator, so I can accurately determine the distance at which my planet needs to orbit? A simple equation would help.

I have thought of using an average surface temperature calculator such as this one to determine the temperature of a small asteroid orbiting at the distance at which the poles are from the sun and another asteroid orbiting at the distance of the equator and using those as my temperature values, but I don't know how to compensate for heat transport.

Note: the planet only has one atmospheric cell.

UPDATE:

On Medium, at this article, it states that the relationship between latitude and temperature is roughly sinusoidal, so I suppose the question now is: how do I find the extremes of the graph?

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The nature of this question is less about orbital position but rather planetary composition, and, as far as I am aware, with the conditions provided, this is impossible, with the exception of one extremely specific end condition that probably provide some satisfaction, and one questionable solution that might work without considerably altering your world. The main stumbling block is that your planet does not have differentiated atmospheric cells. Proximity alone is insufficient, as without enough greenhouse shielding, your planet would behave like Mercury, being at one extreme during daytime, but extremely cold at night.

This necessitates a large atmosphere with a high concentration of greenhouse gases, to keep the night-side of the planet cool. Similarly, this insulation, with only one atmospheric cell, would inevitably leak into the polar zones and make temperatures there far too warm.

So here are the solutions:

1) The planet is tidally locked to the parent star in a 1:1 resonance with limited concentrations of greenhouse gases. Depending on the axial tilt, either one or both polar regions would remain cold enough, and the equator warm enough, but the glaring issue here is that the entire dark side of the planet would be a freezing wasteland.

2) The planet has earthlike atmospheric characteristics, however the majority of the surface near the equator is pitch black, for whatever geological reason (could be igneous volcanic rock or any number of factors), and the polar zones are covered in white, however this white region must go beyond the areas covered by snow ice. This would work by, in essence, creating a very crude system of atmospheric cells, with the white, reflective regions being cooled, and the absorbent, black regions being warmed.

I understand your question was about the orbital mechanics of this planet, but I hope this geological argument works to aid your design.

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