Explanation Given information: The eccentricity ( ε ) of Hyakutake comet is 0.999887 . The minimum distance ( r 0 ) between the comet and the sun is 0.230 R E . Calculation: Write the equation of velocity of the circular orbit of earth ( v 0 ) . v 0 = G M R E (1) Here, G is the gravitational constant, M is the mass of distance between earth to sun and R E is mean distance from the sun to the earth. The periodic time of earth ( τ 0 ) is 1 year. Find the equation of the periodic time of earth ( τ 0 ) . τ 0 = 2 π R E v 0 (2) Substitute Equation (1) in Equation (2). τ 0 = 2 π R E G M R E τ 0 = 2 π R E 3 2 G M G M = 2 π R E 3 2 τ 0 (3) In polar axis θ = 0 . Therefore, cos θ = 1 . Write the equation of trajectories of earth orbits. 1 r 0 = G M h 2 ( 1 + ε ) (4) 1 r 1 = G M h 2 ( 1 − ε ) (5) Divide Equation (4) by (5). 1 r 0 1 r 1 = G M h 2 ( 1 + ε ) G M h 2 ( 1 − ε ) r 1 r 0 = 1 + ε 1 − ε r 1 = 1 + ε 1 − ε r 0 Write the equation of distance of major axis of earth ( a ). a = 1 2 ( r 0 + r 1 ) Substitute 1 + ε 1 − ε r 0 for r 1 . a = 1 2 ( r 0 + 1 + ε 1 − ε r 0 ) = r 0 2 ( 1 − ε + 1 + ε 1 − ε ) = r 0 2 ( 2 1 − ε ) = r 0 1 − ε (6) Write the equation of distance of minor axis of earth ( b ). b = r 0 r 1 Substitute 1 + ε 1 − ε r 0 for r 1 . b = r 0 ( 1 + ε 1 − ε r 0 ) = r 0 1 + ε 1 − ε (7) Find the angular momentum per unit mass h using the equation

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Chapter 12.3, Problem 12.109P

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Find the periodic time of comet.

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Find the periodic time of comet.

Solution Summary: The author explains the periodic time of the Hyakutake comet. The eccentricity is 0.999887 and the minimum distance between earth and sun is 0.230R_E.

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