As a comet swings around the Sun, ice on the comet's surface...

As a comet swings around the Sun, ice on the comet's surface vaporizes, releasing trapped dust particles and ions. The ions, because they are electrically charged, are forced by the electrically charged solar wind into a straight ion tail that points radially away from the Sun (Fig. $33-39$ ). The (electrically neutral) dust particles are pushed radially outward from the Sun by the radiation force on them from sunlight. Assume that the dust particles are spherical, have density $3.5 \times 10^{3} \mathrm{~kg} / \mathrm{m}^{3},$ and are totally absorbing. (a) What radius must a particle have in order to follow a straight path, like path 2 in the figure? (b) If its radius is larger, does its path curve away from the Sun (like path 1 ) or toward the Sun (like path 3 )?

Key Concepts

Gravitational Force

Gravitational force is the attractive force exerted by massive bodies like the Sun on other masses, including dust particles. For a particle near the Sun, this force acts to pull the particle inward, following an inverse-square law with respect to the distance from the Sun. The balance between gravitational attraction and outward forces (like radiation pressure) influences the trajectory of the particle.

Beta Parameter (Radiation-to-Gravity Ratio)

The beta parameter is the ratio of the force due to radiation pressure to the gravitational force acting on a particle. In cometary physics, this parameter determines the net effect on the dust particle's motion. If beta equals unity, radiation pressure exactly counteracts gravity, allowing the dust particle to travel on a straight, radial path away from the Sun. Variations in this ratio, which depend critically on particle size, alter the curvature of the dust particle’s trajectory.

Radiation Pressure

Radiation pressure is the force exerted by light (photons) when it interacts with matter. In the context of comet tails, it acts on dust particles by transferring momentum from the absorbed sunlight, pushing the particles radially outward from the Sun. This force depends on the luminosity of the Sun, the distance from it, and the optical properties (such as absorption or scattering) of the dust particles.

Dust Particle Dynamics

Dust particle dynamics in comet tails involves the interplay of forces such as radiation pressure and gravity. The size, density, and optical properties of the spherical dust particle determine how these forces act. For instance, a particular radius may result in a balance between the two forces, causing the dust particle to move in a straight line. If the particle is larger (and thus has a smaller beta parameter), gravity will dominate more strongly, causing the particle's path to curve differently with respect to a strictly radial trajectory.

Transcript

0:00 Hi there.

00:01 So for this problem, we are told that as a common swings around the sun, eyes on the common surface vaporizes, releasing trap dust particles and ions.

00:10 Now, the ions, because they are electrically charged, are forced by the electrically charged solar wind into a straight ion tailed that points radically away from the sun.

00:23 So the electrically neutral dust particles are pushed radically upward from the sun.

00:30 Sun by the radiation force and then from the sunlight.

00:34 Now we need to assume that the dust particles are spherical and have a density that is equal to 3 .5 times 10 to the 3 kilograms per cubic meter and are totally absorbing.

00:54 So for part a of this problem, we are asked what radius must a particle have in order to follow a straight? path light path to in the figure now let me show the figure in here so this is a situation that we have we have the path to so in order to solve this we shall assume that the sun is far enough from the particle to add as an isotropic point source of light so the forces that add on the dust particle are deradically upward force and the relatively in one force towards the sun, that is the gravitational force.

01:42 So we know that the radically upward radiation force, the radiation force is defined as the intensity of light times the area divided by the speed of light.

01:56 That we can also write as the pressure by the sun divided by four times pi times the radius square and this, well, the power of the sun, of course, and this times pi times the radius squared divided by the speed of light.

02:20 So we can also write this as the power times the radius squared divided by four times the radius square and this times the speed of light.

02:32 Now, capital r in this case is the radius of the particle, and the area is the contraceptional area of that particle.

02:41 On the other hand, the gravitational force on the particle is given by newton's law of gravitation that we know is going to be equal to the gravitational constant times the mass of the sum times the mass of the dust particle, and this divided by the separation distance are to the square.

03:04 So in here we substitute the mass of the dust particle in terms of the density, because we know that that is, the mass can be written in terms of density as the density times the volume.

03:19 So we have this.

03:20 Now we have the density of the dust particle, and this times the volume of the dust particle that is considered as a sphere is this value right here.

03:36 And now we divide this by the separation distance.

03:43 So we can simplify this expression as 4 times pi times the gravitational constant, the mass of the sun, and that times the density of the particle of dots and the radius of the particle to the cube and times three times the separation distance between these two bodies.

04:07 Now, when the two forces balance, we set this to expression equal to each shorter.

04:16 So we will find that this is the power of the sun times the radius squared, divided by four times the separation distance, art to the square times the speed of light...

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