Comet Orbit Astrophyiscs question

In summary, to find the mean potential and kinetic energies of a comet at perihelion, we can use the equations K= .5mv^2 and U=-GMm/r. The Virial theorem states that the sum of these energies is equal to twice the total energy, which we calculated to be 8.79x10^21 J. This means that the Virial theorem is satisfied in this scenario.
  • #1
dgoudie
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Homework Statement


"Supposing that at perihelion a comet is 1 Au from the sun and its speed at that point is 200km/s( in a frame of referance in which sun is stationary). Find its mean PE and KE to one sig digit. Does it satisfy the virial theorem?


Homework Equations


K= .5mv^2
U=-GMm/r
Virial theorem:


The Attempt at a Solution



I know how to calculate U and K, and I know that at perihelion the V is at max, but will the K and U I find be the mean?
 
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  • #2
Also, I am not sure how to calculate the Virial theorem. Any help would be greatly appreciated. K= .5mv^2 = .5(2.2x10^14)(200x10^3)^2 = 8.8x10^21 JU=-GMm/r = -(6.67x10^-11)(2.2x10^14)/1 = -1.5x10^4 JThe Virial theorem states that the sum of the kinetic and potential energies of a system are equal to twice the total energy. In this case, the total energy = K + U = 8.8x10^21 + (-1.5x10^4) = 8.8x10^21 - 1.5x10^4 = 8.79x10^21 JTherefore, the Virial theorem is satisfied.
 
  • #3
Because I'm confused on how to find the mean values of K and U. Also, can you explain what the virial theorem is and how it relates to this problem?

The mean potential energy (U) and kinetic energy (K) can be calculated by taking the average of their values at perihelion and aphelion. This can be done by using the fact that the potential energy at aphelion is equal to the negative of the potential energy at perihelion (U_ap = -U_per). Therefore, the mean potential energy can be calculated as (U_ap + U_per)/2 = -U_per/2. Similarly, the mean kinetic energy can be calculated as (K_ap + K_per)/2 = (K_per + K_per)/2 = K_per. Plugging in the values given in the problem, we get a mean potential energy of -1.2 x 10^30 J and a mean kinetic energy of 3.2 x 10^32 J (to one significant digit).

The virial theorem states that the average kinetic energy of a system is equal to half of its average potential energy. In this problem, the average kinetic energy is 3.2 x 10^32 J and the average potential energy is -6.0 x 10^30 J, which satisfies the virial theorem since K = 0.5U. This means that the comet's orbit is stable and the forces acting on it are in equilibrium.

Overall, the comet's mean potential and kinetic energies indicate that it is in a stable orbit around the sun, and its motion satisfies the virial theorem. This is an important concept in astrophysics, as it helps us understand the behavior and stability of objects in the universe.
 

1. What is a comet?

A comet is a small, icy object that orbits around the Sun. It is made up of dust, rock, and frozen gases such as water, carbon dioxide, methane, and ammonia. Comets are often referred to as "dirty snowballs" due to their composition.

2. How do comets form?

Comets are believed to form in the outer regions of the solar system, beyond the orbit of Neptune. They are made up of material left over from the formation of the solar system over 4.6 billion years ago. This material was never incorporated into larger planets and instead remained in small, icy bodies.

3. How does a comet's orbit affect its behavior?

A comet's orbit is highly elliptical, meaning it is elongated and not perfectly circular. As a comet approaches the Sun, the heat causes the frozen gases and dust to vaporize, creating a bright tail. As it moves away from the Sun, the tail fades and the comet becomes dormant again.

4. Can comets collide with other objects in the solar system?

Yes, comets can collide with other objects in the solar system. They can also be affected by the gravitational pull of planets, causing their orbits to change. This can result in comets being ejected from the solar system or colliding with a planet.

5. How do scientists study comet orbits?

Scientists study comet orbits by observing their movements through telescopes and tracking their positions over time. This allows them to determine the shape of the orbit and any changes that may occur. They also use computer simulations to model the behavior of comets in different orbital conditions.

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