Question

How will you ‘weigh the sun’, that is estimate its mass? The mean orbital radius of the earth around the sun is 1.5 × 10⁸ km.

Answer 1

Orbital radius of the Earth around the Sun, r = 1.5 × 10¹¹ m

Time taken by the Earth to complete one revolution around the Sun,

T = 1 year = 365.25 days

= 365.25 × 24 × 60 × 60 s

Universal gravitational constant, G = 6.67 × 10^–11 Nm² kg^–2

Thus, mass of the Sun can be calculated using the relation,

`M = (4pi^2r^3)/(GT^2)`

= `(4xx(3.14)^2 xx (1.5xx10^11)^3)/(6.67xx10^(11)xx(365.25xx24xx60xx60))^2`

=`(133.24xx10)/(6.64xx10^(4)) = 2.0xx10^(30)` kg

Hence, the mass of the Sun is 2 × 10³⁰ kg.

Answer 2

The mean orbital radius of the Earth around the Sun
R = 1.5 x 10⁸ km = 1.5 x 10¹¹ m

Time period, T = 365.25 x 24 x 60 x 60 s

Let the mass of the Sun be M and that of Earth be m.

According to law of gravitation

`F= G(Mm)/RA^2`  ...(i)

Centripetal force

`F = (mv^2)/R = m.R.omega^2` ...(ii)

`(GMm)/R^2 = m.R.omega^2`

=`(mR.4pi^2)/T^2`

`:. M = (4pi^2R^3)/(G.T^2)`              [∵ `omega = (2pi)/T`]

= `(4xx(3.14)^2xx(1.5xx10^11)^3)/(6.67xx10^(-11)xx(365.25 xx 24 xx 60xx60)^2)`

= `2.009 xx 10^(30) kg = 2.0 xx 10^(30) kg`