Question

Question 2.19 The principle of ‘parallax’ in section 2.3.1 is used in the determination of distances of very distant stars. The baseline AB is the line joining the Earth’s two locations six months apart in its orbit around the Sun. That is, the baseline is about the diameter of the Earth’s orbit ≈ 3 × 1011m. However, even the nearest stars are so distant that with such a long baseline, they show parallax only of the order of 1” (second) of arc or so. A parsec is a convenient unit of length on the astronomical scale. It is the distance of an object that will show a parallax of 1” (second) of arc from opposite ends of a baseline equal to the distance from the Earth to the Sun. How much is a parsec in terms of meters?

Chapter Units And Measurements Page 36

Answer 1

Q.1 Diametre of earth's orbit = 3 × 10¹¹ m
Radius = diametre/2
= (3 x 10¹¹)/2
= 1.5 × 10¹¹ m

∅ = 1"
We know,
3600" = 1°
And 180° = π rad
1° = π/180 rad
1" = π/(180× 3600 ) rad = 4.8457× 10^-6 rad
∅ = r/d
Where d is the distance between sun and earth .
4.8457 × 10^-6 = 1.5 × 10¹¹ /d
d = 1.5 × 10¹¹/4.8457× 10^-6
= 3.09552 × 10^16 m{ 1 parsec }

Answer 2

Diameter of earth's orbit = 3*10^11 m 

so radius = 1.5 * 10^11m 
let the distance parallax angle is 1'' =  4.8 *10^-6 
and distance of star from earth be D
now ,
we know that Ф = r/D
so 
D= r/Ф
hence 
D = 1.5*10^11/4.8 * 10^-6 
=> D ≈ 3.07 ×10^16
so 1 parsec = 3.07 ×10^16