In my setting, a McKendree cylinder (a rotating habitat so large it holds its own atmosphere) is orbiting the sun, with the top of the cylinder pointing at the sun.
How do we make the sun appear to rise and fall from the livable areas of the cylinder?
Would be ideal if the rise and fall was same regardless of where you are standing.
A McKendree cylinder is simply a large enclosed O'Neill cylinder, spinning for gravity. Because of the speed it spins, any light entering from 'windows' along the length of the cylinder would create a fast strobing light effect.
Don't use long slit windows as in the proposal by O'Neill, instead:
An example, with a window at the far end, and a ball mirror assembly that can move along the axis of the McKendree cylinder:
The mirror assembly essentially reflects the sun to all points within the cylinder. The movement from one end to the other would simulate the day cycle, and close the window (or re-orient the mirrors) to simulate night.
This would allow you to control:
NB: in the example above, the 'mirror assembly' can also be a ball of diffusing crystal - shaped and designed to refract and reflect light in all directions to simulate a sun.
This is difficult to achieve as you've stated it. I'll begin with how the system works by default.
Because the habitat is rotating to make artificial gravity, it will be acting as a gyroscope which means it will not be spinning around its axis in other planes. If at one point in its orbit it points one end towards the sun, then a quarter or an orbit later it will have the sun directly at its side. Quarter of a turn again, and the sun faces the opposite end of the cylinder.
In the original position, which I'll call 'summer', the sun will always be in the sky just above the horizon. As we pass into 'fall', the sun will start making bigger and bigger circles around the midpoint. Soon it will start passing behind other parts of the cylinder, and you'll get something like a day-night cycle going.
You didn't specify the number of arms, but let's say it's three for simplicity. As the sun shines through the gap between two arms, the opposite arm will be directly illuminated - have a bright day. As the day goes on, the sun passes behind an arm and you will have night. Then it's in an opening again, but now illuminating a neighbor arm - giving you reflected sunlight, a dim day. Then it's behind you and you get night again, after which the cycle repeats in reverse.
The sun path will move across the cylinder during fall, and will go into the bottom of the cylinder in winter. Spring is the same deal in reverse - including the sun spiraling in the other direction now, switching clockwise and counterclockwise.
So by default you would have a sunrise and sunset cycle, albeit a bit of a complex one, and not around the solstices. Having the same sunrise and sunset all the time is very difficult - how would you have the sun rise at the same time for two people whose definition of 'up' is at right angles to each other, or pointing in opposite directions? You could do something complex with orbits and a wobbling cylinder, but then you'd have to give up your rotation which I assume is more important. The only solutions left would have to be either a setup of mirrors or altogether artificial lightning, but than you don't really have a sunrise anymore. I'm going to have to give a pass on the simultaneous part.
Just to clarify (because the original question is kinda vague)- when you say the McKendree cylinder is "open to space", you don't literally mean its open to hard vacuum. You mean it has three massive transparent areas, like a classic O'Neill cylinder, which allow sunlight in- right?
Assuming that is true:
I'm pretty sure the orbit is unstable. If I'm reading your question right, you want the same end constantly pointing towards the sun- like a pencil orbiting, the pointy side always aimed at the sun, not unlike the way a tidally locked planet always has the same side facing the sun. The problem is a McKendree cylinder isn't a planet. Planets are spheres; a McKendree cylinder is, as the name suggests, a cylinder. If it orbited end-on to the sun, I'm pretty sure it would eventually start to wobble, to become unstable, and eventually turn to become side on to the sun. This is because of the shape-the two ends would be orbiting at different speeds. (Now, this wouldn't happen quickly assuming the cylinder is abound earth-distance from the sun, it would likely take hundreds or thousands of years- but it would happen.)
But before that happened...
Assuming that the transparent parts are in the sides of the cylinder, and the cylinder is end-on to the sun, no natural light would enter the cylinder without the use of mirrors to reflect light. In that case, a sun-set/sun-rise setup could be achieved by having the mirrors on a timer, rotating their facing to allow sunlight in at specific times only.
On the other hand, assuming that its the end of the cylinder that is transparent (not the sides), you could have a large, slowly rotating object between the cylinder and the sun like a massive thin sheet of metal (or many small thin sheets of metal). Properly created and positioned, they could be designed to allow a day/night system, their mass shielding the cylinder from the sun for the 'night', and side-on to allow sunlight through during the 'day'.
