- In astronomy, a circular orbit refers to an object (such as a planet or a star) which orbits around a central body in a fixed, circular motion. This motion follows Kepler's Laws.A circular orbit occurs when the eccentricity of its orbit is equal to 0. Objects with a circular orbit are uncommon. The Moon moves in an elliptical orbit around the Earth, and the planets move in an elliptical orbit.
- As previously mentioned, the circular orbit is a special case of the elliptical orbit with e = 0. Therefore, the radial distance is r = a = constant. An expression for the circular orbit speed can be obtained by combining Eqs.
- Triton's orbit is almost perfectly circular despite allegedly being a captured dwarf planet. Is it possible for spacecraft to achieve an even more perfect orbit or a perfect one with no orbital eccentricity? Please don't say geostationary satellites since they don't orbit the Earth relative to its surface.
Energy in a Circular Orbit
Imagine that we have an object of mass m in a circular orbit around an object of mass M. An example could be a satellite orbiting the Earth. What is the total energy associated with this object in its circular orbit?
As usual, E = U + K.
A circular orbit is depicted in the top-left quadrant of this diagram, where the gravitational potential well of the central mass shows potential energy, and the kinetic energy of the orbital speed is shown in red. The height of the kinetic energy remains constant throughout the constant speed circular orbit.
U = -GmM/r and K = ½ mv2.
The only force acting on the object is the force of gravity. Applying Newton's Second Law gives:
ΣF = ma
GmM/r2 = mv2/r
Therefore mv2 = GmM/r and K = ½ mv2 = ½GmM/r
The kinetic energy is positive, and half the size of the potential energy.
E = -GmM/r + ½GmM/r = -½GmM/r
A negative total energy tells us that this is a bound system. Much like an electron is bound to a proton in a hydrogen atom with a negative binding energy, the satellite is bound to the Earth - energy would have to be added to each system to remove the electron or the satellite.
Orbits and Energy
Again we have two masses, m and M, with m << M. The smaller mass will be placed at a particular distance from the larger one and given an initial velocity directed perpendicular to the line joining the masses. We'll examine a few different cases, giving the mass initial velocities of various speeds and seeing what kind of orbit we get in each case.
Case 1: A circular orbit. Let's say this happens to require an initial velocity of 1 unit.
Case 2: v < 1.0. With even less kinetic energy, the mass follows an elliptical path. The starting point is the aphelion, the point furthest from the Sun.
Case 3: v = 0.0. The object simply gets sucked in to the large mass.
Case 4: v > 1.0 but the total energy is still negative. We still have a bound system. The orbit is elliptical again, but this time the starting point is the perihelion - the point closest to the Sun.
Case 5: v is larger by a factor of the square root of two than the speed needed to go in a circle. This is actually the escape speed - the orbit is parabolic, and the object never comes back.
Case 6: v is larger than the escape speed, so the total energy is positive. The orbit is hyperbolic - note that it's much straighter than the parabolic curve.
If the object is constrained to move in a circle and the total tangential force acting on the object is zero, (F_{theta}^{text {total }}=0) then (Newton’s Second Law), the tangential acceleration is zero,
[a_{theta}=0]
This means that the magnitude of the velocity (the speed) remains constant. This motion is known as uniform circular motion. The acceleration is then given by only the acceleration radial component vector
(overrightarrow{mathbf{a}}_{r}(t)=-r omega^{2}(t) hat{mathbf{r}}(t)) uniform circular motion .
Because the speed (v=r|omega|) is constant, the amount of time that the object takes to complete one circular orbit of radius r is also constant. This time interval, T , is called the period. In one period the object travels a distance s = vT equal to the circumference, (s=2 pi r); thus
[s=2 pi r=v T]
The period T is then given by
[T=frac{2 pi r}{v}=frac{2 pi r}{r omega}=frac{2 pi}{omega}]
The frequency f is defined to be the reciprocal of the period,
[f=frac{1}{T}=frac{omega}{2 pi}]
The SI unit of frequency is the inverse second, which is defined as the hertz, (left[mathrm{s}^{-1}right] equiv[mathrm{Hz}])
The magnitude of the radial component of the acceleration can be expressed in several equivalent forms since both the magnitudes of the velocity and angular velocity are related by v = rω . Thus we have several alternative forms for the magnitude of the centripetal acceleration. The first is that in Equation (6.5.3). The second is in terms of the radius and the angular velocity,
[left|a_{r}right|=r omega^{2}]
The third form expresses the magnitude of the centripetal acceleration in terms of the speed and radius,
[left|a_{r}right|=frac{v^{2}}{r}]
Recall that the magnitude of the angular velocity is related to the frequency by (omega=2 pi f), so we have a fourth alternate expression for the magnitude of the centripetal acceleration in terms of the radius and frequency,
[left|a_{r}right|=4 pi^{2} r f^{2}]
A fifth form commonly encountered uses the fact that the frequency and period are related by (f=1 / T=omega / 2 pi). Thus we have the fourth expression for the centripetal acceleration in terms of radius and period,
[left|a_{r}right|=frac{4 pi^{2} r}{T^{2}}]
Other forms, such as (4 pi^{2} r^{2} f / T) or (2 pi r omega f), while valid, are uncommon.
Often we decide which expression to use based on information that describes the orbit. A convenient measure might be the orbit’s radius. We may also independently know the period, or the frequency, or the angular velocity, or the speed. If we know one, we can calculate the other three but it is important to understand the meaning of each quantity.
Circular Orbit Velocity Formula
Geometric Interpretation for Radial Acceleration for Uniform Circular Motion
An object traveling in a circular orbit is always accelerating towards the center. Any radial inward acceleration is called centripetal acceleration. Recall that the direction of the velocity is always tangent to the circle. Therefore the direction of the velocity is constantly changing because the object is moving in a circle, as can be seen in Figure 6.4. Because the velocity changes direction, the object has a nonzero acceleration.
The calculation of the magnitude and direction of the acceleration is very similar to the calculation for the magnitude and direction of the velocity for circular motion, but the change in velocity vector, (Delta overrightarrow{mathbf{v}}) is more complicated to visualize. The change in velocity (Delta overrightarrow{mathbf{v}}=overrightarrow{mathbf{v}}(t+Delta t)-overrightarrow{mathbf{v}}(t)) is depicted in Figure 6.5. The velocity vectors have been given a common point for the tails, so that the change in velocity, (Delta overrightarrow{mathbf{v}}) can be visualized. The length (|Delta overrightarrow{mathbf{v}}|) of the vertical vector can be calculated in exactly the same way as the displacement (|Delta overrightarrow{mathbf{r}}|). The magnitude of the change in velocity is
[|Delta overrightarrow{mathbf{v}}|=2 v sin (Delta theta / 2)]
We can use the small angle approximation (sin (Delta theta / 2) cong Delta theta / 2) to approximate the magnitude of the change of velocity,
[|Delta overrightarrow{mathbf{v}}| cong v|Delta theta|]
The magnitude of the radial acceleration is given by
[left|a_{r}right|=lim _{Delta t rightarrow 0} frac{|Delta overrightarrow{mathbf{v}}|}{Delta t}=lim _{Delta t rightarrow 0} frac{v|Delta theta|}{Delta t}=v lim _{Delta t rightarrow 0} frac{|Delta theta|}{Delta t}=vleft|frac{d theta}{d t}right|=v|omega|]
Circular Orbit Calculator
The direction of the radial acceleration is determined by the same method as the direction of the velocity; in the limit (Delta theta rightarrow 0, Delta overrightarrow{mathbf{v}} perp overrightarrow{mathbf{v}}) and so the direction of the acceleration radial component vector (overrightarrow{mathbf{a}}_{r}(t)) at time t is perpendicular to position vector (overrightarrow{mathbf{v}}(t)) and directed inward, in the (-hat{mathbf{r}})-direction.