(a) How many astronomical units (A.U.) make 1 parsec?

(b) Consider a sunlike star at a distance of 2 parsecs. When it is seen through a telescope with 100 magnification, what should be the angular size of the star? Sun appears to be (1/2)° fromthe earth. Due to atmospheric fluctuations, eye can’t resolve objects smaller than 1 arc minute.

(c) Mars has approximately half of the earth’s diameter. When it is closest to the earth it is at about 1/2 A.U. from the earth. Calculate what size it will appear when seen through the same telescope.

(Comment : This is to illustrate why a telescope can magnify planets but not stars.)

(a) Parsec is defined as the distance at which 1 A.U. long arc subtends an angle of 1s. The angle subtended by an arc of length L and radius r is given as:

Here L = 1 A.U., θ = 1s and r = 1 parsec. Therefore

(b) Given:

Sun’s diameter as seen from earth =

Star’s distance from earth = 2 parsecs

Magnification of the telescope = 100

Minimum resolution eye due to atmosphere = 1’

The size of the star as seen from the earth can be considered by placing the sun at the same distance from earth as the star since it is sun like. Thus if sun appears to be in diameter from earth at 1 A.U. then for 2 parsecs or as calculated in (a) from 2 x 2.06 x 10⁵ A.U. its angular diameter will be,

When seen from the telescope, the star will be 100 times magnified and therefore,

Converting to arc minutes the angular diameter will be 7.25 x 10^-5 arc minutes. Since due to atmospheric fluctuations, the eye cannot resolve below 1 arc minute, the sun like star appears to be 1 arc minute in angular diameter.

(c) Given:

Ratio between mar’s diameter and earth’s diameter =

Let D_e, D_ma and D_s be the diameters of earth, mars and the sun. Then we have,

Also, we know that

Therefore, we have,

To find the size of mars as seen from earth, we first place mars at distance equal to the earth-sun distance form earth i.e. 1 A.U. At 1 A.U. the sun appears to be in diameter, therefore for mars,

So, at mars’ original distance of A.U. mars will have a diameter of

With the telescope used in the previous problem,

Therefore, after magnification mars appears to be 30 arc minutes which is more than 1 arc minute and hence is not influenced by atmospheric fluctuations and is magnified. Thus, the distance of stars are so large in comparison to planets that stars cannot be magnified on earth while planets can be. To study stars, the stars are made brighter by allowing more light to enter the telescopes by using different designs.