The function of the primary mirror is to bring the light that is received into an exact focus. To do this for a light source that is positioned at infinity, the mirror must have a paraboloidal shape. The size (mirrors are classified by their diameter) of the mirror determines two fundamental properties. First, it determines the amount of light that is received. Second, it determines the maximum possible angular resolution.
The faintest possible star that can be seen is directly related to the mirror diameter. An approximation for this is given in the table below. (Note that these values vary from observer to observer, and from place to place)
The above table is only valid for pinpoint light sources (see side note). For extended objects the properties of the eye/brain combination become more complex, but the same principle as above is valid, i.e. larger diameter = better detectability. See "Visual Astronomy; Optimum Detection Magnification" by Mel Bartels.
For photography the situation is a little bit different, since the film is able to integrate the amount of light it receives over a certain time. The table below gives an indication of how the exposure time and mirror diameter influence the detectable magnitude. Apart from the fact that longer exposures are more difficult, there is an upper limit as to what the film can handle.
CCD's also integrate the amount of light, but at low light levels their detection curve is linear. They suffer from thermal noise which increases with temperature. Unfortunately I don't have any data on the limiting magnitude.
As written above, the best attainable angular resolution of a scope depends on the diameter of it's mirror. The formula is given below, it yields the resolution in radians. To convert this to arc seconds, multiply with 206265. The wavelength to which the dark adapted eye is most sensitive is 0.56 E-06 meters (0.56 micro meter).
The use of this formula is to give an indication of what to expect. To seperate double stars the resolution is actually a little bit better.
The magnification of a scope gives the relation between mirror diameter and exit pupil as shown in the formula below:
The exit pupil is the diameter of the bundle of light coming out of the scope. Due to the characteristics of the eye, the usable pupil diameter is from 6 mm to 0.5 mm. (but different people have different beliefs (or eye's :-)). This means that the ratio between lowest magnification and highest magnification is 12. The usable magnification range is thus also determined by the primary mirror diameter.
The highest usable magnification for a (near to) perfect mirror is about twice its diameter in mm (or 50 times its diameter in inches). However, several observers report that using higher magnifications can be useful (under special conditions) to extract the last possible detail from an image. But one does need considerable observing experience.
In order for the mirror not to sag under it's own weight (which would deform its surface, and we're talking about much less than one wavelength !) the mirror needs a minimum thickness. This thickness depends of course of it's diameter, but also on the mirror support, called the cell. The older/simpler cell system using three support points needs a thicker mirror than the supports that utilize 9, 18 or 27 points. A mirror is said to have "full thickness" if its sagging on a three point cell will not seriously degrade its performance. The relation between diameter and "full thickness" is given below.
The weight of the mirror depends on it's volume. As can be seen from the above relation, the volume will increase dramatically for larger mirrors. This is shown in the table below.
Above 250 mm (10") the weight will severely impact the mechanical design of a scope. Is has become common to use thinner mirrors (also called "thin mirror") with a more elaborate cell. Scopes with a mirror of 500 mm (20") with a thickness of 50 mm (2") have successfully been build. Unfortunately it is rather complicated to calculate the exact position of the support points so that approximations supported by experience are used. (See "the cell" for more information on cell design)
When you want to grind your first mirror, there seems to be a consensus among ATM'ers that the best diameter to start with is 150 to 250 mm (6 to 10"). Smaller mirrors become actually harder to make, as do the bigger mirrors (but for different reasons). Probably the biggest mirror you can make on your own is about 400 to 500 mm (16 to 20"). Above this you need a machine, or assistance.
The principal method for mirror making is to grind two round glass plates over each other with a mixture of water and abrasive in between. The process is divided into 4 phases :
Rough grinding : During the first phase one grinds with a coarse abrasive until the required sagitta is reached. (See below for it's formula: "Spherical Curvature")Fine grinding : During this phase, one uses smaller and smaller abrasives to smooth the mirror's surface. Each abrasive has to eliminate the traces (pits and grooves) of it's predecessor.
Polishing : The result from this phase should be a spherical mirror with a surface that has no pits or grooves left. The surface should be smooth to within 1/8 wavelength.
Figuring : This process should change the spherical shape of the mirror into the parabolic shape without loosing the smoothness obtained during polishing.
After grinding and polishing one ends up (or hopes to) with a sphere. The formula for it's curvature is given below.
In this formula the distance is calculated between the mirror surface and an imaginary plane that is tangent with the sphere at the optical axis. The maximum value for 's' is also called the 'sagitta', this is the depth of the center of the mirror.
Note: The focal length is halve the radius of curvature.
The curve of the main mirror for a newton telescope should be a parabola along all diagonals. This shape is called paraboloidal, but often referred to as parabolic. The formula that describes this curvature is given below.
In this formula the distance is calculated between the mirror surface and an imaginary plane that is tangent with the parabolic at the optical axis.
The figure of the mirror also depends on mechanical stress. A good cell does not put any mechanical stress on the mirror. The temperature adjustment is then the only remaining factor. When the mirror is brought from a warm room into the cold of night, the mirror will cool down. This cooling is not homogeneous, so while it is cooling it's figure will be deformed and the mirror cannot perform optimal. Sometimes a fan is used to speed the temperature adjustment.
During the night, the outdoor temperature is (often) not stable but gently falling. This is the cause for a permanent adjustment of the mirror's temperature, and thus for a permanent deformation. Some people compensate for this by using a slightly "under corrected" figure instead of the ideal paraboloid. The benefits of this are not clear enough to make this a general rule.
The requirement on the stability of the mount is also dependent on the primary. A heavier (i.e. larger) mirror needs a very stable mount. Also the higher magnifications that go with the larger diameter more easily show any deficiencies in the mount design.