I can't answer this in full (a lack of any orbit itself being an absurd idea), but a lack of rotation would mean a tidally locked planet. If you're looking for a practical understanding of climatology for the sake of building a fictional world, perhaps my answer to this question will prove useful to you.

"DOES NOT rotate, DOES NOT orbit" - That's effectively the same thing as rotating once per orbit. That is, your situation is the same for a tidally locked Earth-turned-billiard-ball.

@Palarran, of course it's absurd, but I'm not looking for the entire book in this answer, just the first chapter. Your answer is good, but it doesn't address the detail I eventually want to obtain, nor does it address the beginning. Like many answers to climate questions on this site, it makes substantial assumptions. I'm looking to avoid those.

@Samuel, It's only the same thing as a tidally-locked planet if the orbit is perfectly circular. Then, yes, it would be. It's not Earth-turned-billiard-ball due to the Earth's elliptical orbit, which leads to a dynamic process. This is the first in a series of qustions that will help me (and others) really understand the process of building a climate, and dealing with tidally-locked worlds is one of the steps along that path.

@kingledion, I want to deal with the effects of the surface on climate in a later question. Right now, you should not assume any reflected heat from the surface, or anything else. The gravity well is climatologically inert. I'm hoping that I've reduced the question to one of basic gas dynamics.

The rotation of the planet is crucial. There is exactly zero similarity between the climates on a tidally locked planet and on one with a resonable rate of rotation. If the planet does in fact have reasonable days and nights, then you can look at Earth's southern ocean for a hint.

"Earth's elliptical orbit": You can safely assume that Earth's orbit is circular for a first approximation; the excentricity is less than 2%.

@AlexP the rotation of the planet is crucial LATER. You can't understand calculus without first learning how to add.

@JBH If you think the answer for a perfectly circular orbit and a orbit with an eccentricity of ~0.0167, then you better fix the Sun's output too, it's not perfectly constant after all.

SPHERICAL HORSE! (you all have no idea how hard I'm laughing right now.) This is a first-order approximation to establish a fundamental understanding of pressure and turbulence on a sphere. I'm judging a static condition. @Samuel, you're inviting variables where none should exist. We'll get to reality later (like, a half-dozen questions later). I did fix the solar luminosity to 1.0.

Exactly HOW do you get away with the superfluous joke, and not get asked to edit it out? Is there a discriminatory policy in effect here? A science joke for your Saturday morning

@kingledion, while I look forward to your answer, please bear in mind that I'm going to ask questions indroducing water, soild, etc. later. One step at a time unless you absolutely can't stand it, please.

It got posted before I gave the reference. Sorry, slip of the enter key.

@JustinThyme, HAH! I like that version better! I'm quoting from a 30-year-old memory my Physics 101 instructor told the class.

@JBH The easiest stepping stone would be to use a smooth Earth and diverge from there. Stopping the rotation, something that exists on all planets we can observe climate on, complicates the issue rather than simplifies it.

One more PS, I think you should add the hard-science tag here, since I think that is what you want.

"The surface DOES NOT contribute to climatological effects": what does this even mean? Gases are very very good at transferring heat through convection and relatively poor at absorbing the energy of photons. The consequence is that the temperature of the surface is very very important to the behavior of the gaseous atmosphere. What happens to a photon when it hits then surface of the planet? Either it gets reflected, or it gets absorbed. If it gets reflected, this doubles its chances of warming the gas; if it is absorbed, it warms the surface, which then immediately goes to warm then gas...

@kingledion, I had no intention of adding the hard-science tag. Honestly, I hate that tag.

@JBH Then you are going to get answers like the one JustinThyme just put in :)

@AlexP, it's very important LATER. I understand what you're saying. I'd even understand a more complicated answer, but I'm looking for simple. Given a long period of time, pressure and turbulence would be established and would stabilize no matter how poorly the atmosphere absorbs energy.

Please understand, my point is that really, there is no such thing as superfluous data or information as long as it shapes the question. What I object to is the process of labeling information as superfluous before any thought is put into why it is pertinent. Even when people are hit over the head with the background information, they still tend to shape the question into one they want to answer and ignore the rest.

@kingledion You mean an answer that actually addresses the OP's question, without reading any more into it? The OP wanted KISS, I gave KISS.

@JustinThyme, that wasn't KISS, it was sarcastic. The atmosphere is heated from one direction. Gravity holds the atmosphere onto the sphere. Theoretically (ignoring such inconveniences as the solar wind ripping the atmosphere away due to the lack of a magnetosphere becuase that's what you do when you use spherical horses) a process of convection is set up with turbulent zones surrounding a circular area of heated high pressure. No answer yet even points this out. Everybody's overthinking this something awful. Climate is a series of layered effects. This should be where it starts.

@JustinThyme, also, you're not an engineer (if you are, I apologize, but you don't act like one). If engineers tried to accept all the "pertinent" information at the same time they wouldn't get anything done on time or correctly. Most complex systems cannot be studied or understood by a single person as they require many disciplines. Trying to listen to everyone at once is a good way to get lost in the morass of data, forcing you to redesign and redesign as you must accomodate details that were inefficiently considered all at once.

"they wouldn't get anything done on time or correctly" -- As they say, you can only have two of on time, under budget, and correct.

It was done in the same vein as your spherical horse. If you completely over-simplify, you get an over-simplified answer, One version of the joke goes 'Easy answer, All you have to do is ASSUME a spherical horse', and that is exactly what you have done. Engineering gets away with 'simplifying' because they always add a safety margin of 100%, after assuming a risk factor for every variable of 100%. Engineers OVER-build, to account for everything that is fed into error. You have over-simplified too much, and then assumed an atmosphere under such conditions.

Mars had an atmosphere, and then lost it. Now, there is nothing to REPLENISH it. Gravity alone is not sufficient for an atmosphere. You need the conditions to KEEP it. Without some mechanism for heat transfer between the hot and the cold side, I am afraid the answer is trivial. And I gave the trivial answer. Atmosphere alone will not do it.

I totally agree with your attempt to simplify down to first principles. I am just suggesting that you have simplified too much. I gave you the starting conditions for your over-simplification, now you have to add variables that alter from what I gave you. I suggest you start with some form of heat transfer, such as rotation or water. Atmosphere is NOT the first variable that should be considered.

BTW, and FYI, some engineers and architects are now refining their inputs and calculations, and are proposing that 30 story buildings can be safely put on a footprint no wider than standard residential lots. See this.

Oh, and apology accepted.

Ignoring the surface is proving a bit...odd. The sun's rays only somewhat heat up the atmosphere. The atmosphere actually warms more due to the surface (high frequency wavelengths pass through the atmosphere, warms the surface, and is then radiated into the atmosphere as infrared and other low frequency wavelengths that the atmosphere absorbs more readily). The rate the surface warms has alot to do with it's albedo (whats reflected vs what is absorbed). doubtful you can answer 'It would be cool if the answer could accomodate variations in solar luminosity' without that info