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Recent nights where I try to make my first observations with the entry levell telescope Seben with 114 mm mirror it seems that for lenses less than 6 mm details go lost due to warm air refraction (as observed with the Moon contour).

Everywhere they say, it shouldn't be too warm: is it possible to recommend a rough temperature range where you could go for say 4 mm and still see details?

(Side note: the overall visibility index was about 77%).

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The air is only part of your problem here. There are several factors involved.

it is absolutely true that the air can cause a loss of detail in the image you see. But even if there's no atmosphere between you and the moon or anything else you're looking at, there is only so much detail you can see. Technically speaking, you can magnify as much as you want ad infinitim. But there is only so much USABLE magnification you can get out of a telescope. This is due to the diffraction of light.

When the light from the moon or any other object enters your telescope tube, it begins diffraction. If you want to better understand the physics behind it, there's a fairly easy-to-undersatand Khan Academy video about it. Essentially, what is happening is interference patterns form among the light waves and the result is a blurring of detail. This blurring is less noticable at lower magnifications, and gets worse the more you magnify. While the specifics are a little different from the result of blowing up a picture, the effect is similar. I put together this example of the effect. In my example, I started with a picture I took of the moon in May or June and each image in the series is doubled in size then cropped. I go from a full image of a waxing gibbous moon to zoom in on the crater Herschel. As the magnification increases, you start to see the image getting fuzzier and fuzzier. In the case of a picture, this is because there is only so much data to work with, depending on the geometry of the image sensor in the camera. Technically, it is also the case with the light that there is a fininte amount of information/data to work with, but the mechanics are a bit different. In the end, the result is the image blurs past a certain point and you can't get a clearer image out of it.

(Technically speaking, using speckle interferometry, a camera CAN increase detail when multiple images are captured and analyzed together, but there's still a finite limit, and this doesn't apply at all to visual observing.)

Your telescope has an aperture of 114mm, as you indicated. I did a quick search for Seben 114mm telescopes and the only ones I found showed a focal length of 1,000mm. As it turns out, it appears this is not a true Newtonian telescope, but a Bird-Jones Newtonian (or Jones-Bird, depending on who you ask), which is actually a type of catadioptric design and has some inherent issues, which I'll circle back around to.

They eyepieces I found listed that come with this telescope include an SR4mm and a H6mm. The SR4mm is a symmetrical Ramsden design eyepiece with a 4 mm focal length. The H6mm is a Huygens design with a 6mm focal length. The Huygens eyepiece design was invented in about 1662 by Christian Huygens, and was an improvement upon the original basic single-lens eyepiece design of the first Lippershey and Galilean telescopes, and Kepler's minor improvements in 1611. Huygens added a second element and produced an eyepiece that provides better eye relief (the distance from the eyepiece to your eye, and less chromatic aberration (the separation of colors of light caused by refraction. These eyepieces, however, have fairly narrow apparent fields of view and perform poorly in telescopes with focal ratios under f/10. The next major upgrade in eyepiece design was Jesse Ramsden's invention in 1782. Ramsden's eyepiece provided better apparent fields of view, but decreased eye relief and increased chromatic aberration. When used in longer focal lengths, this is less of a problem,b ut still a problem. The Ramsden eyepiece is also susceptible to "ghosting" which is caused by reflections within the eyepiece. These eyepieces aer fairly inexpensive to manufacture, however, which makes them common in lower-cost telescopes such as yours.

laying aside, for the moment, the inherent issues with these eyepieces, the sizes you specified are 4 mm and 6 mm. Based on the design types, I am going to assume an apparent field of view (AFOV) of about 40° for both. WhenI do the calculations, I find that the 6 mm gives you a magnification of about 167X with a TFOV (true field of view) of about 14 arcminutes. The 4 mm gives you 250X magnification with a TFOV around 10 arcmin.

As I mentioned above, however, aperture limits usable magnification. In particular, it limits the angular size of details you can resolve. A 114 mm aperture can provide details no larger than 0.93 arcseconds in size. At the distance to the moon (about 384,400 km) this means the smallest details you can see are about 1.73 km in size. But this is the theoretical limitation.

In the mid 1800's, William R. Dawes produced a method of calculating how far apart two stars must be to be able to determine that they are, in fact, two separate stars. His formula, which provides a measure in arcseconds we call the Dawes' Limit, is one of the most commonly referenced ways to describe how small a detail you can see with a given telescope aperture. For your telescope, that would be about 1.02 arcseconds. By this measure, you could not see details any smaller than about 1.9 km in size on the moon. A third measure, known as the Raleigh Criterion, which is also measured in arcseconds, suggests the smallest details visible in a 114 mm telescope would be about 1.21 arcseconds in size or 2.25 km at the distance of the moon.

Based on these formulae, then, if there was a particular crater on the moon that was about 2km in size, you could probably just make it out. You could magnify this as much as you like, but you won't see more detail, just a blurrier and blurrier hint of a crater. Eyepiece size/design won't change that, nor will your focus ability.

Depending on who you ask, for a given inch of aperture, you can only get about 25 to 60x of usable magnification - meaning that anything over that level is going to be so blurry as to not be worth viewing. On a typical night, that would be toward the lower side, and on a night with very good air and sky conditions, toward the higher end. Based on this, I'd say on a typical night you shouldn't expect more than about 135X magnification, and on a really nice night, maybe 225X. Technically, the 4mm gets you much higher than that, and with the Barlow lens, double that (to around 500X) but there's absoloutely nothing you can do to improve the sharpness of the image - the limits of physics dictate that the telescope simply cannot do it. Anything over 135X on an average night is likely to be fuzzy and get fuzzier still as you try to increase magnification.

But wait, there's more!

As I mentioned, your telescope isn't a true Newtonian. A true Newtonian uses a mirror with a curvature defined by the formula of a parabola. These kinds of mirrors are a bit harder to manufacture, so many manufacturers of lower-priced telescopes cheat a little and use a mirror with a spherical curvature. The spherical curvature is a constant-radius curve, and this makes it MUCH easier to manufacture. However, it introduces a problem known as spherical aberration Luckilly, spherical aberration and most other aberrations are decreased by increasing focal length. The Bird-Jones design uses a shorter-focal length spherical mirror and adds a correcting lens (almost identical to a Barlow lens) in the path of light to correct the errors. If you look into your focuser without an eyepiece, you'll probably notice a lens between your eye and the secondary mirror - this is the Bird-Jones corrector. It adds more glass to the optical path. The more glass, the more chance for distortion and further aberration. Additionally, this is not likely to be a very high quality optical component, making it more likely to produce a poorer image. But worse, it makes the process of collimation very difficult indeed.

I'm guessing, though I may be wrong, you have little or no understanding of collimation. Simply put, collimation is the proper alignment of the optical components in the telescope. If the mirrors and lenses are not properly aligned, it introduces further errors in the path of light, which manifest themselves in image aberration and distortion. When collimation is off, the image quality degrades and sharpness is lost. Unfortunately, this built-in lens makes collimation more difficult, which means it's likely that your collimation is off somewhat, which will FURTHER degrade your image quality at higher magnifications.

So, what can you do?

Honeslty, not much. While warmer air will cause issues inside the tube of the telescope, this can be reduced through acclimatization - letting the telescope come to ambient temperature. If you can collimate your telescpe accurately, that will improve your image. And replacing your eyepieces with better designs (even lower-price Plossl eyepieces are significantly better) will help. Even then, you're not going to get sharp detail at higher magnification. It is simply not possible. To get sharper detail, you need a larger aperture and better air conditions.

My personal recommendation for a beginner telescope is an 8" Dobsonian. Though somewhat more expensive, the 8" aperture provides a lot more detail. And a Dobsonian with a true Newtonian OTA is far easier to collimate and get a clearer image out of.

For now, keep using your 114, but accept its limitations. And start looking to your next scope.

I would also recommend finding a local astronomy club you can join and learn more about how scopes work, how to maintain them, and ways to better enjoy your observing sessions.

Good luck and clear skies!

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The temperature is not important, at least not in the way that you are asking. The temperature of the telescope, the air around it, and the miles (kilometers) of air that you are looking through are what makes a difference.

Obviously you cannot control the miles (kilometers) of air, so the only control you have is the temperature of the scope compared to the local temperature. You want that to be as equal as possible, so let the scope acclimate to the air temperature (usually meaning to let the scope cool down). Of course, avoid looking over terrain that had uneven temperatures such as hot rooftops, roads, concrete, and do forth.

It is likely that the 4 mm eyepiece will never give a good view (depending on the quality) or may give an acceptable view on rare occasions,

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What really matters is that the telescope and the ambient air are at the same temperature. A simple way to accomplish this is to take the scope outside one hour before you begin observing.

warm air refraction (as observed with the Moon contour)

This is called seeing, and it's within the instrument, ambiental and environmental.

Environmental seeing is simply called seeing. It's basically air turbulence through the whole atmosphere. It varies in time; when it's really bad, the image is blurry and there's nothing you can do about it.

https://en.wikipedia.org/wiki/Astronomical_seeing

Within the instrument it's just air convection in a hot telescope used outside in the cool night air.

https://garyseronik.com/beat-the-heat-conquering-newtonian-reflector-thermals-part-1/

https://garyseronik.com/beat-the-heat-conquering-newtonian-reflector-thermals-part-2/

Ambiental seeing is convection from a hot rooftop or concrete slab. Don't observe stars that are just above a house or tall building - convection tends to blur the image.

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