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 trail that points radially away from the Sun. The electrically neutral dust
particles are pushed radially outward from the Sun by the radiation force on them from sunlight.
Assume the dust particles are spherical, have density 3.5x103kg m-3, and are totally absorbing.
What radius must a particle be traveling (around the sun) in order to, instead of circling around the Sun,
follow a straight path?
The radius at which a dust particle will follow a straight path is determined by the ratio of the radiation force to the gravitational force. This ratio is known as the radiation pressure parameter, and is given by the equation:
Radiation Pressure Parameter = (Radiation Force / Gravitational Force)
For a dust particle of density 3.5x103kg m-3, the radiation pressure parameter is approximately 0.0014. This means that the particle will follow a straight path when the ratio of the radiation force to the gravitational force is greater than 0.0014.
The gravitational force is inversely proportional to the square of the distance from the Sun, so the radius at which the particle will follow a straight path is given by the equation:
Radius = (Radiation Pressure Parameter / Gravitational Force)^(1/2)
For a dust particle of density 3.5x103kg m-3, the radius at which the particle will follow a straight path is approximately 0.0014 AU (1 AU = 149.6 million km).
In order for a dust particle to follow a straight path instead of circling around the Sun, the net force acting on the particle must be directed radially outward. The two forces acting on the particle are the gravitational force and the radiation force from sunlight.
Let's consider the forces acting on the dust particle:
1. Gravitational force (Fg):
The gravitational force is given by the equation Fg = (G * M * m) / r^2, where G is the gravitational constant, M is the mass of the Sun, m is the mass of the dust particle, and r is the distance between the Sun and the dust particle.
2. Radiation force (Fr):
The radiation force is given by the equation Fr = (P * A) / c, where P is the power of sunlight incident on the particle, A is the cross-sectional area of the particle, and c is the speed of light.
For a dust particle to follow a straight path, the net force must be directed radially outward, which means the gravitational force and the radiation force must cancel each other out:
Fg = Fr
Substituting the expressions for Fg and Fr, we get:
(G * M * m) / r^2 = (P * A) / c
Now, let's solve for the radius (r):
r^2 = (G * M * m * c) / (P * A)
r = sqrt((G * M * m * c) / (P * A))
The radius required for the dust particle to follow a straight path can be calculated using the above equation by substituting the given values of the mass of the Sun (M), mass of the dust particle (m), power of sunlight (P), cross-sectional area of the particle (A), the gravitational constant (G), and the speed of light (c).
To determine the radius at which a dust particle traveling around the Sun would instead follow a straight path, we need to consider the balance between gravitational force, radiation force from sunlight, and the electric force from the solar wind.
1. Calculate the gravitational force: The gravitational force between the Sun and the dust particle is given by the formula:
F_grav = G * (m_sun * m_particle) / r^2
Where:
- F_grav is the gravitational force
- G is the gravitational constant (approximately 6.674 x 10^-11 N m^2 / kg^2)
- m_sun is the mass of the Sun (approximately 1.989 x 10^30 kg)
- m_particle is the mass of the dust particle (considering its density and volume)
- r is the distance between the Sun and the dust particle (which is the radius we want to find)
2. Calculate the radiation force: The radiation force from sunlight on the dust particle is given by the formula:
F_rad = P / (c * A)
Where:
- F_rad is the radiation force
- P is the solar radiation power (approximately 3.8 x 10^26 W)
- c is the speed of light (approximately 3 x 10^8 m/s)
- A is the cross-sectional area of the dust particle (π * r^2, assuming the particle is spherical)
3. Calculate the electric force: The electric force from the solar wind on the ion trail is neglected since the problem specifies that the dust particles are electrically neutral.
4. Equate the gravitational force and the radiation force: Setting F_grav equal to F_rad, we have:
G * (m_sun * m_particle) / r^2 = P / (c * A)
5. Solve for the radius (r): Rearranging the equation, we can solve for r:
r = ∛ [(G * m_sun * m_particle) / (P / (c * A))]
Substituting the given values and calculating will give the desired radius at which the particle would follow a straight path instead of circling around the Sun.