Question

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?

Answer 1

Diameter of Earth’s orbit = 3 × 10¹¹ m

Radius of Earth’s orbit, r = 1.5 × 10¹¹ m

Let the distance parallax angle be 1”= 4.847 × 10^–6 rad.

Let the distance of the star be D.

Parsec is defined as the distance at which the average radius of the Earth’s orbit subtends an angle of 1.

:. We have `theta = r/D`

`D=r/theta = (1.5xx10^(11))/(4.847xx10^(-6))`

`=0.309 xx 10^(-6) ~~ 3.09xx10^(16)m`

Hence, 1 parsec ≈ 3.09 × 10¹⁶ m.

Answer 2

From parallax method we can say

θ=b/D,where b=baseline ,D = distance of distant object or star

Since, θ=1″ (s) and b=3 x 10¹¹ m

D=b/20=3 x 10¹¹/2 x 4.85 x 10^-6 m  or D=3 x 10¹¹/9.7 x 10^-6 m =30 x 10¹⁶/9.7 m

= 3.09 x 10¹⁶ m = 3 x 10¹⁶ m.