Finding the Radius of the Orbit

Oct 13, 2025 by miodeya.com 83 views

The radius of the planet Mercury is $$2.43 \times 10^6 , \text{m}$$ and its mass is $$3.2 \times 10^{23} , \text{kg}$$. Find the period of a satellite orbiting Mercury $ $265,000 \, \text{m}$ [/tex] above its surface.

Understanding the Problem

To find the period of a satellite orbiting Mercury, we need to use the concept of orbital mechanics. The period of a satellite in a circular orbit is given by the equation:

T = 2πr^(3/2) / (G^(1/2) * M^(1/2))

where:

T is the period of the satellite
r is the radius of the orbit
G is the gravitational constant
M is the mass of the planet

Given Values

We are given the following values:

r = 2.43 × 10^6 m (radius of Mercury)
M = 3.2 × 10^23 kg (mass of Mercury)
h = 265,000 m (height of the satellite above the surface of Mercury)

Finding the Radius of the Orbit

The radius of the orbit is the sum of the radius of Mercury and the height of the satellite above the surface:

r = 2.43 × 10^6 m + 265,000 m = 2.695 × 10^6 m

Finding the Period of the Satellite

Now we can plug in the values into the equation for the period:

T = 2π(2.695 × 10^6 m)^(3/2) / (6.674 × 10^(-11) N m^2 kg^(-2)(1/2) * (3.2 × 10^23 kg)^(1/2))

Simplifying the Equation

To simplify the equation, we can use the following values:

G = 6.674 × 10^(-11) N m^2 kg^(-2)
M = 3.2 × 10^23 kg

T = 2π(2.695 × 10^6 m)^(3/2) / (6.674 × 10^(-11) N m^2 kg^(-2)(1/2) * (3.2 × 10^23 kg)^(1/2))

Solving for T

Now we can solve for T:

T ≈ 2.35 × 10^4 s

Converting to Hours

To convert the period to hours, we can divide by 3600 (the number of seconds in an hour):

T ≈ 2.35 × 10^4 s / 3600 s/h ≈ 6.53 h

Conclusion

The period of a satellite orbiting Mercury 265,000 m above its surface is approximately 6.53 hours.

Orbital Mechanics

Orbital mechanics is the study of the motion of objects in space. It involves the study of the gravitational forces that act on objects in space, as well as the motion of those objects in response to those forces. Orbital mechanics is used in a variety of fields, including astronomy, space exploration, and satellite technology.

Gravitational Forces

Gravitational forces are the forces that act between objects with mass. They are responsible for the motion of objects in space, and are the primary force that governs the motion of satellites in orbit around a planet.

Orbital Period

The orbital period of a satellite is the time it takes for the satellite to complete one orbit around a planet. It is determined by the mass of the planet, the radius of the orbit, and the gravitational constant.

Orbital Radius

The orbital radius of a satellite is the distance between the satellite and the center of the planet. It is determined by the mass of the planet, the velocity of the satellite, and the gravitational constant.

Orbital Velocity

The orbital velocity of a satellite is the velocity at which the satellite moves in its orbit around a planet. It is determined by the mass of the planet, the radius of the orbit, and the gravitational constant.

Orbital Energy

The orbital energy of a satellite is the energy required to move the satellite from one point in its orbit to another. It is determined by the mass of the planet, the radius of the orbit, and the gravitational constant.

Orbital Eccentricity

The orbital eccentricity of a satellite is a measure of how elliptical its orbit is. It is determined by the mass of the planet, the radius of the orbit, and the gravitational constant.

Orbital Inclination

The orbital inclination of a satellite is the angle between the plane of its orbit and the equatorial plane of the planet. It is determined by the mass of the planet, the radius of the orbit, and the gravitational constant.

Orbital Longitude

The orbital longitude of a satellite is the angle between the point on the planet where the satellite crosses the equator and the point where the satellite is at its highest or lowest point in its orbit. It is determined by the mass of the planet, the radius of the orbit, and the gravitational constant.

Orbital Latitude

The orbital latitude of a satellite is the angle between the point on the planet where the satellite crosses the equator and the point where the satellite is at its highest or lowest point in its orbit. It is determined by the mass of the planet, the radius of the orbit, and the gravitational constant.

Orbital Plane

The orbital plane of a satellite is the plane in which the satellite moves in its orbit around a planet. It is determined by the mass of the planet, the radius of the orbit, and the gravitational constant.

Orbital Node

The orbital node of a satellite is the point where the satellite crosses the equator of the planet. It is determined by the mass of the planet, the radius of the orbit, and the gravitational constant.

Orbital Periapsis

The orbital periapsis of a satellite is the point in its orbit where it is closest to the planet. It is determined by the mass of the planet, the radius of the orbit, and the gravitational constant.

Orbital Apoapsis

The orbital apoapsis of a satellite is the point in its orbit where it is farthest from the planet. It is determined by the mass of the planet, the radius of the orbit, and the gravitational constant.

Orbital Semi-major Axis

The orbital semi-major axis of a satellite is the average distance between the satellite and the center of the planet. It is determined by the mass of the planet, the radius of the orbit, and the gravitational constant.

Orbital Semi-minor Axis

The orbital semi-minor axis of a satellite is the average distance between the satellite and the center of the planet, perpendicular to the plane of the orbit. It is determined by the mass of the planet, the radius of the orbit, and the gravitational constant.

Orbital Eccentricity Vector

The orbital eccentricity vector of a satellite is a vector that points from the center of the planet to the periapsis of the satellite. It is determined by the mass of the planet, the radius of the orbit, and the gravitational constant.

Orbital Inclination Vector

The orbital inclination vector of a satellite is a vector that points from the center of the planet to the node of the satellite. It is determined by the mass of the planet, the radius of the orbit, and the gravitational constant.

Orbital Longitude Vector

The orbital longitude vector of a satellite is a vector that points from the center of the planet to the point on the planet where the satellite crosses the equator. It is determined by the mass of the planet, the radius of the orbit, and the gravitational constant.

Orbital Latitude Vector

The orbital latitude vector of a satellite is a vector that points from the center of the planet to the point on the planet where the satellite crosses the equator. It is determined by the mass of the planet, the radius of the orbit, and the gravitational constant.

Orbital Plane Vector

The orbital plane vector of a satellite is a vector that points from the center of the planet to the plane of the orbit. It is determined by the mass of the planet, the radius of the orbit, and the gravitational constant.

Orbital Node Vector

The orbital node vector of a satellite is a vector that points from the center of the planet to the node of the satellite. It is determined by the mass of the planet, the radius of the orbit, and the gravitational constant.

Orbital Periapsis Vector

The orbital periapsis vector of a satellite is a vector that points from the center of the planet to the periapsis of the satellite. It is determined by the mass of the planet, the radius of the orbit, and the gravitational constant.

Orbital Apoapsis Vector

The orbital apoapsis vector of a satellite is a vector that points from the center of the planet to the apoapsis of the satellite. It is determined by the mass of the planet, the radius of the orbit, and the gravitational constant.

Orbital Semi-major Axis Vector

The orbital semi-major axis vector of a satellite is a vector that points from the center of the planet to the semi-major axis of the satellite. It is determined by the mass of the planet, the radius of the orbit, and the gravitational constant.

Orbital Semi-minor Axis Vector

The orbital semi-minor axis vector of a
Q&A: Orbital Mechanics and Satellite Orbits

Q: What is orbital mechanics?

A: Orbital mechanics is the study of the motion of objects in space, including the gravitational forces that act on those objects and the motion of those objects in response to those forces.

Q: What is the orbital period of a satellite?

A: The orbital period of a satellite is the time it takes for the satellite to complete one orbit around a planet. It is determined by the mass of the planet, the radius of the orbit, and the gravitational constant.

Q: What is the orbital radius of a satellite?

A: The orbital radius of a satellite is the distance between the satellite and the center of the planet. It is determined by the mass of the planet, the velocity of the satellite, and the gravitational constant.

Q: What is the orbital velocity of a satellite?

A: The orbital velocity of a satellite is the velocity at which the satellite moves in its orbit around a planet. It is determined by the mass of the planet, the radius of the orbit, and the gravitational constant.

Q: What is the orbital energy of a satellite?

A: The orbital energy of a satellite is the energy required to move the satellite from one point in its orbit to another. It is determined by the mass of the planet, the radius of the orbit, and the gravitational constant.

Q: What is the orbital eccentricity of a satellite?

A: The orbital eccentricity of a satellite is a measure of how elliptical its orbit is. It is determined by the mass of the planet, the radius of the orbit, and the gravitational constant.

Q: What is the orbital inclination of a satellite?

A: The orbital inclination of a satellite is the angle between the plane of its orbit and the equatorial plane of the planet. It is determined by the mass of the planet, the radius of the orbit, and the gravitational constant.

Q: What is the orbital longitude of a satellite?

A: The orbital longitude of a satellite is the angle between the point on the planet where the satellite crosses the equator and the point where the satellite is at its highest or lowest point in its orbit. It is determined by the mass of the planet, the radius of the orbit, and the gravitational constant.

Q: What is the orbital latitude of a satellite?

A: The orbital latitude of a satellite is the angle between the point on the planet where the satellite crosses the equator and the point where the satellite is at its highest or lowest point in its orbit. It is determined by the mass of the planet, the radius of the orbit, and the gravitational constant.

Q: What is the orbital plane of a satellite?

A: The orbital plane of a satellite is the plane in which the satellite moves in its orbit around a planet. It is determined by the mass of the planet, the radius of the orbit, and the gravitational constant.

Q: What is the orbital node of a satellite?

A: The orbital node of a satellite is the point where the satellite crosses the equator of the planet. It is determined by the mass of the planet, the radius of the orbit, and the gravitational constant.

Q: What is the orbital periapsis of a satellite?

A: The orbital periapsis of a satellite is the point in its orbit where it is closest to the planet. It is determined by the mass of the planet, the radius of the orbit, and the gravitational constant.

Q: What is the orbital apoapsis of a satellite?

A: The orbital apoapsis of a satellite is the point in its orbit where it is farthest from the planet. It is determined by the mass of the planet, the radius of the orbit, and the gravitational constant.

Q: What is the orbital semi-major axis of a satellite?

A: The orbital semi-major axis of a satellite is the average distance between the satellite and the center of the planet. It is determined by the mass of the planet, the radius of the orbit, and the gravitational constant.

Q: What is the orbital semi-minor axis of a satellite?

A: The orbital semi-minor axis of a satellite is the average distance between the satellite and the center of the planet, perpendicular to the plane of the orbit. It is determined by the mass of the planet, the radius of the orbit, and the gravitational constant.

Q: What is the orbital eccentricity vector of a satellite?

A: The orbital eccentricity vector of a satellite is a vector that points from the center of the planet to the periapsis of the satellite. It is determined by the mass of the planet, the radius of the orbit, and the gravitational constant.

Q: What is the orbital inclination vector of a satellite?

A: The orbital inclination vector of a satellite is a vector that points from the center of the planet to the node of the satellite. It is determined by the mass of the planet, the radius of the orbit, and the gravitational constant.

Q: What is the orbital longitude vector of a satellite?

A: The orbital longitude vector of a satellite is a vector that points from the center of the planet to the point on the planet where the satellite crosses the equator. It is determined by the mass of the planet, the radius of the orbit, and the gravitational constant.

Q: What is the orbital latitude vector of a satellite?

A: The orbital latitude vector of a satellite is a vector that points from the center of the planet to the point on the planet where the satellite crosses the equator. It is determined by the mass of the planet, the radius of the orbit, and the gravitational constant.

Q: What is the orbital plane vector of a satellite?

A: The orbital plane vector of a satellite is a vector that points from the center of the planet to the plane of the orbit. It is determined by the mass of the planet, the radius of the orbit, and the gravitational constant.

Q: What is the orbital node vector of a satellite?

A: The orbital node vector of a satellite is a vector that points from the center of the planet to the node of the satellite. It is determined by the mass of the planet, the radius of the orbit, and the gravitational constant.

Q: What is the orbital periapsis vector of a satellite?

A: The orbital periapsis vector of a satellite is a vector that points from the center of the planet to the periapsis of the satellite. It is determined by the mass of the planet, the radius of the orbit, and the gravitational constant.

Q: What is the orbital apoapsis vector of a satellite?

A: The orbital apoapsis vector of a satellite is a vector that points from the center of the planet to the apoapsis of the satellite. It is determined by the mass of the planet, the radius of the orbit, and the gravitational constant.

Q: What is the orbital semi-major axis vector of a satellite?

A: The orbital semi-major axis vector of a satellite is a vector that points from the center of the planet to the semi-major axis of the satellite. It is determined by the mass of the planet, the radius of the orbit, and the gravitational constant.

Q: What is the orbital semi-minor axis vector of a satellite?

A: The orbital semi-minor axis vector of a satellite is a vector that points from the center of the planet to the semi-minor axis of the satellite. It is determined by the mass of the planet, the radius of the orbit, and the gravitational constant.

Q: How do I calculate the orbital period of a satellite?

A: To calculate the orbital period of a satellite, you can use the following equation:

T = 2πr^(3/2) / (G^(1/2) * M^(1/2))

where:

T is the orbital period
r is the radius of the orbit
G is the gravitational constant
M is the mass of the planet

Q: How do I calculate the orbital radius of a satellite?

A: To calculate the orbital radius of a satellite, you can use the following equation:

r = 2.43 × 10^6 m + 265,000 m

where:

r is the orbital radius
2.43 × 10^6 m is the radius of Mercury
265,000 m is the height of the satellite above the surface of Mercury

Q: How do I calculate the orbital velocity of a satellite?

A: To calculate the orbital velocity of a satellite, you can use the following equation:

v = √(G * M / r)

where:

v is the orbital velocity
G is the gravitational constant
M is the mass of the planet
r is the radius of the orbit

Q: How do I calculate the orbital energy of a satellite?

A: To calculate the orbital energy of a satellite, you can use the following equation:

E = -G * M / (2 * r)

where:

E is the orbital energy
G is the gravitational constant
M is the mass of the planet
r is the radius of the orbit

Q: How do I calculate the orbital eccentricity of a satellite?

A: To calculate the orbital eccentricity of a satellite, you can use the following equation:

e = √(1 - (b^2 / a^2))

where:

e is the orbital eccentricity
b is the semi-minor axis
a is the semi-major