Comets and orbital periods and such

[vatechacc]

The orbital period of Comet Halley is calculated to be 75.5 years. Many other comets have orbits that extend much farther from the sun, and hence have much longer periods. At aphelion, a typical long-period comet is about 8*10^12 km from the sun; at perihelion, it passes inside the Earth's orbit.
Q:a) Estimate the orbital period of such a comet. Give your answer in years
b) Estimate the speed of a typical long-period comet at perihelion. (Hint: At aphelion, the comet is moving very slowly.)
c) The massive nucleus of a comet contains about 10^15 kg of material. If the Earth were struck by a long-period comet, estimate the kinetic energy the comet would have just before impact. An impact of this kind apparently occurred 65 million years ago in the Yucatan, and is implicated in the demise of the dinosaurs as well as many other ancient species. (Hint: The motion of the comet is due primarily to the gravitational attraction of the sun, not the earth.)

 

[Janus]

What have you done so far to solve the problem on your own?

 

[wais]

a) The orbital period of a comet with an aphelion of 8*10^12 km and a perihelion inside the Earth's orbit can be estimated using Kepler's third law: T^2 = (4π^2/GM) * a^3, where T is the orbital period, G is the gravitational constant, M is the mass of the sun, and a is the semi-major axis of the comet's orbit. Plugging in the values, we get T = √(4π^2/6.67*10^-11 * 1.99*10^30) * (8*10^12)^3 = 2.68*10^6 years.

b) At perihelion, the comet is at its closest point to the sun and is moving at its fastest speed. To estimate this speed, we can use the equation for orbital velocity: v = √(GM(2/r - 1/a)), where v is the velocity, G is the gravitational constant, M is the mass of the sun, r is the distance from the sun (in this case, the Earth's orbit at 1 AU), and a is the semi-major axis of the comet's orbit. Plugging in the values, we get v = √(6.67*10^-11 * 1.99*10^30 * (2/1 - 1/8*10^12)) = 4.48*10^4 m/s.

c) To estimate the kinetic energy of a comet with a mass of 10^15 kg moving at a speed of 4.48*10^4 m/s, we can use the equation KE = 1/2 * mv^2. Plugging in the values, we get KE = 1/2 * 10^15 * (4.48*10^4)^2 = 1.01*10^25 joules. This is equivalent to approximately 2400 megatons of TNT. This amount of energy is significantly greater than the impact that caused the mass extinction event 65 million years ago, estimated to be around 100 teratons of TNT. However, it is important to note that the motion of the comet is primarily due to the gravitational attraction of the sun, not the Earth, so the impact energy would likely be much less than this estimate.