Spread the love

Astronomical Telescope Class 12 | Astronomical Telescope

Astronomical Telescope Class 12 | Astronomical Telescope :- An optical instrument designed for astronomical observation that uses lenses to collect light from celestial objects, resulting in detailed, magnified and clear images of stars, planets, galaxies and other astronomical phenomena, is called an astronomical refracting type telescope.

Astronomical Telescope Class 12 – Construction

In this, an achromatic convex lens (O) of longer focal length and larger aperture is fitted at one end of a cylindrical metal tube, which is called the objective lens, which is towards the object. At the other end of this tube, another small tube is attached, on the outer end of which an achromatic convex lens (E) is fitted, whose focal length and aperture are less than that of the objective lens. This lens is close to the eye, hence it is called eye piece. The distance between these two lenses can be increased or decreased by sliding the smaller tube inside the larger tube through the rack and pinion arrangement.

Astronomical Telescope Class 12 – Ray Diagram

Astronomical Telescope Class 12 – Working

The incident light rays from the object AB situated at infinity are incident on the objective lens (O) at an angle α with the principal axis. The objective lens (O) forms a real and inverted image A’B’ of the object AB. This image acts as an object for the eye lens (E).

The distance between the two lenses is adjusted using the rack and pinion arrangement in such a way that the image A’B’ is formed between the focus (F_E) and the optical center (O₂) of the eye lens. Now the eye lens acts like a simple microscope and forms a virtual, highly magnified image A”B” of the intermediate image A’B’. This final image A”B” is formed inverted with respect to the original object AB.

Astronomical Telescope Class 12 – Magnifying Power

The magnifying power of an astronomical telescope is defined by the ratio of the angle subtended at the eye (β) by the final image A”B” to the angle subtended at the eye (α) by the object AB located at infinity.

[You might be thinking that the α angle is being formed on the objective lens (O) and not on the eye. But if you look carefully, the light rays which are incident on the objective lens (O) at an angle α, will also be incident on the eye at an angle α.]

m = angle subtended by the final image A”B” at the eye (β)/ angle subtended by the object AB at the eye (α)

$\displaystyle m=\frac{\beta }{\alpha }$

The angles β and α are very small, hence $\displaystyle \beta \approx \text{tan}\beta ,\text{ }\alpha \approx \text{tan}\alpha$

$\displaystyle \Rightarrow m=\frac{\text{tan}\beta }{\text{tan}\alpha }$ …..(1)

In the triangle A’B’O₂,

$\displaystyle \text{tan}\beta =\frac{A'B'}{{{O}_{2}}B'}$ …..(2)

Similarly in triangle A’B’O₁,

$\displaystyle \text{tan}\alpha =\frac{{A}'{B}'}{{{O}_{1}}{B}'}$ …..(3)

Putting the values of equations (2) and (3) in equation (1),

$\displaystyle m=\frac{{}^{{A}'{B}'}\!\!\diagup\!\!{}_{{{O}_{2}}{B}'}\;}{{}^{{A}'{B}'}\!\!\diagup\!\!{}_{{{O}_{1}}{B}'}\;}$

$\displaystyle \Rightarrow m=\frac{{{O}_{1}}{B}'}{{{O}_{2}}{B}'}$ …..(4)

Using sign convention,

O₁B’ = + v_o= + f_o

O₂B’ = – u_E

Putting all these values in equation (4),

$\displaystyle m=\frac{+{{f}_{0}}}{-{{u}_{E}}}$

$\displaystyle m=-\frac{{{f}_{0}}}{{{u}_{E}}}$ …..(5)

Equation (5) is a general formula for the magnifying power of an astronomical telescope. For different values of u_E, different values of magnifying power will be obtained. Let us now find the value of magnifying power in some special cases.

Special Cases

(1) When the final image is formed at the least distance of distinct vision (D)

For the eye lens, when the final image is formed at D,

u = -u_E

v = v_E = -D

f = +f_E

Using lens formula on the eye lens,

$\displaystyle \frac{\text{1}}{f}=\frac{1}{v}-\frac{1}{u}$

$\displaystyle \Rightarrow \frac{\text{1}}{\left( +{{f}_{E}} \right)}=\frac{1}{\left( -D \right)}-\frac{1}{\left( -{{u}_{E}} \right)}$

$\displaystyle \Rightarrow \frac{1}{{{u}_{E}}}=\frac{1}{{{f}_{E}}}+\frac{1}{D}$

Putting this value in equation (5),

$\displaystyle m=-\frac{{{f}_{0}}}{{{u}_{E}}}=-{{f}_{0}}\left( \frac{1}{{{u}_{E}}} \right)=-{{f}_{0}}\left( \frac{1}{{{f}_{E}}}+\frac{1}{D} \right)$

$\displaystyle \Rightarrow m=-\frac{{{f}_{0}}}{{{f}_{E}}}\left( 1+\frac{{{f}_{E}}}{D} \right)$ …..(6)

Here the negative sign indicates that the final image (A”B”) is inverted with respect to the object (AB).

Length of the tube (The distance between the optical center O₁ of the objective lens and the optical center O₂ of the eye lens is called the length of the tube) :-

$\displaystyle L={{f}_{o}}+\left| {{u}_{E}} \right|$ …..(7)

While determining the length of the tube in equation (7), the value of u_E is to be taken positive.

(2) When the final image is formed at infinity (∞)

For the final image to be formed at infinity, the position of the image A’B’ in front of the eye lens (E) should be at its focus point (F_E). To do this, the eyepiece is moved back a little using the rack and pinion arrangement, so that the intermediate image A’B’ comes at the focus (F_E) of the eyepiece. Putting u_E = f_E in equation (5), we get,

$\displaystyle m=-\frac{{{f}_{0}}}{{{f}_{E}}}$ …..(8)

Here the value of u_E is not taken negative because while deriving equation (5) we have already kept the negative sign for u_E.

Length of the tube :-

$\displaystyle L={{f}_{O}}+{{f}_{E}}$ …..(9)

Note :-

(1). In equations (5), (6) and (8) the negative sign indicates that the final image A”B” is inverted with respect to the object AB.

(2). The aperture of the objective lens should be large and the aperture of the eye lens should be small.

(3). From equations (6) and (8), the focal length (f_E) of the objective lens should be less to obtain greater magnification.

Astronomical Telescope Class 12 | Astronomical Telescope

Astronomical Telescope Class 12 – Construction

Astronomical Telescope Class 12 – Ray Diagram

Astronomical Telescope Class 12 – Working

Astronomical Telescope Class 12 – Magnifying Power

Like this: