A telescope does not usually enlarge an object by making a larger linear image at your eye. Instead, it increases the
angular size under which the object is seen. That is why the key quantity for a telescope is the
angular magnification, not the ordinary linear magnification used for a single image on a screen.
In an afocal telescope, the objective lens first forms an image of a distant object near its focal plane.
The eyepiece then acts like a magnifier for that image and sends light out again as a parallel bundle.
Because the outgoing rays subtend a larger angle at the eye than the incoming naked-eye view, the object appears magnified.
Using the signed focal-length convention adopted in this calculator, the angular magnification is
\[
m_{\mathrm{ang}}=-\frac{f_o}{f_e},
\]
where \(f_o\) is the signed focal length of the objective and \(f_e\) is the signed focal length of the eyepiece.
A converging lens has positive focal length and a diverging lens has negative focal length.
For an astronomical (Keplerian) telescope, both the objective and the eyepiece are converging, so \(f_o>0\) and \(f_e>0\).
That makes the signed magnification negative:
\[
m_{\mathrm{ang}}<0.
\]
The negative sign indicates that the final image is inverted. In many astronomical observations that does not matter much,
because one is mainly interested in angular detail rather than in upright orientation.
The magnitude of the magnification is
\[
|m_{\mathrm{ang}}|=\frac{f_o}{f_e}.
\]
For example, if \(f_o=1000\ \text{mm}\) and \(f_e=20\ \text{mm}\), then
\[
|m_{\mathrm{ang}}|=\frac{1000}{20}=50,
\]
so the telescope gives a \(50\times\) view.
For a Galilean telescope, the objective is converging but the eyepiece is diverging, so \(f_o>0\) and \(f_e<0\).
In that case the same signed formula gives a positive angular magnification:
\[
m_{\mathrm{ang}}>0.
\]
That means the final image is upright. Galilean systems are therefore attractive for opera glasses and compact low-power telescopes,
although they usually have a smaller field of view than astronomical designs.
Another important quantity is the tube length for an afocal arrangement. With signed focal lengths it is
\[
L=f_o+f_e.
\]
This automatically gives the familiar results: \(L=f_o+f_e\) for an astronomical telescope with both lenses converging,
and \(L=f_o-|f_e|\) for a Galilean telescope with a diverging eyepiece.
Telescopes are also characterized by their exit pupil. The exit pupil is the image of the objective formed by the eyepiece.
It tells you how wide the outgoing beam is and where your eye should be placed to receive the full light bundle.
The exit pupil diameter is
\[
D_{\mathrm{exit}}=\frac{D_o}{|m_{\mathrm{ang}}|},
\]
where \(D_o\) is the objective diameter. A larger magnification makes the exit pupil smaller, which is why very high magnification can produce a dimmer-looking view.
The position of the exit pupil relative to the eyepiece can be found by treating the eyepiece as forming an image of the objective.
If the objective is at distance \(L\) from the eyepiece, then
\[
\frac{1}{f_e}=\frac{1}{L}+\frac{1}{d_{\mathrm{exit}}}.
\]
In an astronomical telescope, \(d_{\mathrm{exit}}\) is positive, so the exit pupil is real and lies outside the eyepiece.
In a Galilean telescope, \(d_{\mathrm{exit}}\) is negative, so the exit pupil is virtual and lies inside the tube.
Finally, angular magnification lets us compare the naked-eye angular size \(\alpha\) of an object with the apparent telescopic angular size \(\beta\):
\[
\beta=|m_{\mathrm{ang}}|\,\alpha.
\]
This is why the Moon, which appears about half a degree wide to the unaided eye, can appear about \(25^\circ\) wide in a \(50\times\) telescope.
At more advanced levels, telescope performance also depends on field of view, aberrations, eye relief, diffraction, and pupil matching with the human eye.
But the central backbone remains simple: the objective sets the image scale, the eyepiece sets the angular expansion, and the ratio of focal lengths determines the angular magnification.
