Io, one of the satellites of Jupiter, has an orbital period of 1.769 days and the radius of the orbit is 4.22 × 10⁸ m. Show that the mass of Jupiter is about one-thousandth that of the sun.

Question

Orbital period of I₀ , T_I0 = 1.769 days

= 1.769 × 24 × 60 × 60 s

Answer 1

Orbital period of I₀ , T_I0 = 1.769 days

= 1.769 × 24 × 60 × 60 s

Orbital radius of I₀ , R_I₀ = 4.22 × 10⁸ m

Satellite I₀ is revolving around the Jupiter

Mass of the Jupiter is given by,

M_J = 4π²R_I0³ / GT_I0² ...(i)

where,

M_J = Mass of Jupiter,

G = Universal gravitational constant,

Orbital period of the earth,

T_e= 365.25 days

= 365.25 × 24 × 60 × 60 s

Orbital radius of the Earth,

R_e= 1 AU = 1.496 × 10¹¹m

Mass of sun is given as,

M_s = 4π²R_e³ / GT_e² ...(ii)

Therefore,

M_s / M_J =

=

Substituting the values,

=

= 1045.04

∴ M_s / M_J ~ 1000

M_s~ 1000 × M_J

Hence, it can be inferred that the mass of Jupiter is about one-thousandth that of the Sun.

Answer 2

Escape velocity is given by,

straight v subscript straight e equals square root of 2 gr end root space equals space square root of fraction numerator 2 GM over denominator straight r end fraction end root

space space space equals square root of fraction numerator 2 GM over denominator straight R plus straight h end fraction end root

where,

g is the acceleration due to gravity at the point from where the body is projected,

r is the distance from the centre of the earth from where the body is projected, and

h is the height from the surface of the earth.

(a) Escape velocity, v_e does not depend on the mass of the body.

(b) Escape velocity, v_e depends on upon g and at different locations g is different (g_λ = g - Rω²cos² λ ), therefore, v_e slightly depends on upon the angle of latitude ( ∵ Rω²cos ² λ ≤ 3.36 cm/s²).

(c) Escape velocity, v_e does not depend on upon the direction of projection.

(d) Escape velocity, v_e depends on upon height from the surface of the earth.

Answer 3

The escape velocity is independent of the mass of the body and the direction of projection. It depends upon the gravitational potential at the point from where the body is launched. Gravitational potential depends slightly on the latitude and height of the point, therefore, the escape velocity depends on these factors.

Answer 4

Mass of our galaxy Milky Way, M = 2.5 × 10¹¹ solar mass

Solar mass = Mass of Sun = 2.0 × 10³⁶ kg

Mass of our galaxy, M = 2.5 × 10¹¹ × 2 × 10³⁶

= 5 × 10⁴¹ kg

Diameter of Milky Way, d = 10⁵ ly

Radius of Milky Way, r = 5 × 10⁴ ly

1 ly = 9.46 × 10¹⁵ m

∴ r = 5 × 10⁴ × 9.46 × 10¹⁵
= 4.73 ×10²⁰ m

A star revolves around the galactic centre of the Milky Way.

Time period is given by the relation,

T =

=

= 1.12 × 10¹⁶ s

1 year = 365 × 324 × 60 × 60 s

1s =

years

Therefore,

1.12 × 10¹⁶ s =

= 3.55 × 10⁸ years.

Answer 5

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

T_e = 1 year

Orbital radius of the Earth in its orbit, R_e= 1 AU

Time taken by the planet to complete one revolution around the Sun, T_P = ½T_e

= ½ year

Orbital radius of the planet = R_p
From Kepler’s third law of planetary motion,

Hence, the orbital radius of the planet will be 0.63 times smaller than that of the Earth.

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Io, one of the satellites of Jupiter, has an orbital period of 1.769 days and the radius of the orbit is 4.22 × 10⁸ m. Show that the mass of Jupiter is about one-thousandth that of the sun.

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Gravitation

Io, one of the satellites of Jupiter, has an orbital period of 1.769 days and the radius of the orbit is 4.22 × 108 m. Show that the mass of Jupiter is about one-thousandth that of the sun.

Io, one of the satellites of Jupiter, has an orbital period of 1.769 days and the radius of the orbit is 4.22 × 10⁸ m. Show that the mass of Jupiter is about one-thousandth that of the sun.