GyroscopeA gyroscope

A gyroscope is a device for measuring or maintaining orientation, based on the principles of angular momentum.[1][2] The device is a spinning wheel or disk whose axle is free to take any orientation. This orientation changes much less in response to a given external torque than it would without the large angular momentum associated with the gyroscope's high rate of spin. Since external torque is minimized by mounting the device in gimbals, its orientation remains nearly fixed, regardless of any motion of the platform on which it is mounted.

Description and diagram

Diagram of a gyro wheel. Reaction arrows about the output axis (blue) correspond to forces applied about the input axis (green), and vice versa.

Within mechanical systems or devices, a conventional gyroscope is a mechanism comprising a rotor journaled to spin about one axis, the journals of the rotor being mounted in an inner gimbal or ring, the inner gimbal being journaled for oscillation in an outer gimbal which in turn is journaled for oscillation relative to a support. The outer gimbal or ring is mounted so as to pivot about an axis in its own plane determined by the support. The outer gimbal possesses one degree of rotational freedom and its axis possesses none. The inner gimbal is mounted in the outer gimbal so as to pivot about an axis in its own plane that is always perpendicular to the pivotal axis of the outer gimbal.

The axle of the spinning wheel defines the spin axis. The inner gimbal possesses two degrees of rotational freedom and its axis possesses one. The rotor is journaled to spin about an axis which is always perpendicular to the axis of the inner gimbal. So, the rotor possesses three degrees of rotational freedom and its axis possesses two. The wheel responds to a force applied about the input axis by a reaction force about the output axis.

The behaviour of a gyroscope can be most easily appreciated by consideration of the front wheel of a bicycle. If the wheel is leaned away from the vertical so that the top of the wheel moves to the left, the forward rim of the wheel also turns to the left. In other words, rotation on one axis of the turning wheel produces rotation of the third axis.

A gyroscope flywheel will roll or resist about the output axis depending upon whether the output gimbals are of a free- or fixed- configuration. Examples of some free-output-gimbal devices would be the attitude reference gyroscopes used to sense or measure the pitch, roll and yaw attitude angles in a spacecraft or aircraft.

Animation of a gyro wheel in action

The center of gravity of the rotor can be in a fixed position. The rotor simultaneously spins about one axis and is capable of oscillating about the two other axes, and thus, except for its inherent resistance due to rotor spin, it is free to turn in any direction about the fixed point. Some gyroscopes have mechanical equivalents substituted for one or more of the elements, e.g., the spinning rotor may be suspended in a fluid, instead of being pivotally mounted in gimbals. A control moment gyroscope (CMG) is an example of a fixed-output-gimbal device

that is used on spacecraft to hold or maintain a desired attitude angle or pointing direction using the gyroscopic resistance force.

In some special cases, the outer gimbal (or its equivalent) may be omitted so that the rotor has only two degrees of freedom. In other cases, the center of gravity of the rotor may be offset from the axis of oscillation, and thus the center of gravity of the rotor and the center of suspension of the rotor may not coincide.

History

Gyroscope invented by Léon Foucault, and built by Dumoulin-Froment, 1852. National Conservatory of Arts and Crafts museum, Paris.

The earliest known gyroscope was made by Johann Bohnenberger in 1817, although he called it simply the "Machine". The French mathematician Pierre-Simon Laplace, working at the École Polytechnique in Paris, recommended the machine for use as a teaching aid, and thus it came to the attention of Léon Foucault.[3] In 1852, Foucault used it in an experiment involving the rotation of the Earth. It was Foucault who gave the device its modern name, in an experiment to see (Greek skopeein, to see) the Earth's rotation (Greek gyros, circle or rotation), although the experiment was unsuccessful due to friction, which effectively limited each trial to 8 to 10 minutes, too short a time to observe significant movement.

In the 1860s, electric motors made the concept feasible, leading to the first prototype gyrocompasses; the first functional marine gyrocompass was developed between 1905 and 1908 by German inventor Hermann Anschütz-Kaempfe. The American Elmer Sperry followed with his own design in 1910, and other nations soon realized the military importance of the invention—in an age in which naval might was the most significant measure of military power—and created their own gyroscope industries. The Sperry Gyroscope Company quickly expanded to provide aircraft and naval stabilizers as well, and other gyroscope developers followed suit.[4]

In 1917, the Chandler Company of Indianapolis, Indiana created the "Chandler gyroscope," a toy gyroscope with a pull string and pedestal. It has been in continuous production ever since and is considered a classic American toy.

MEMS gyroscopes take the idea of the Foucault pendulum and use a vibrating element, known as a MEMS (Micro Electro-Mechanical System). The MEMS-based gyro was initially made practical and produceable by Systron Donner Inertial (SDI). Today, SDI is a large manufacturer of MEMS gyroscopes.

In the first several decades of the 20th century, other inventors attempted (unsuccessfully) to use gyroscopes as the basis for early black box navigational systems by creating a stable platform from which accurate acceleration measurements could be performed (in order to bypass the need for star sightings to calculate position). Similar principles were later employed in the development of inertial guidance systems for ballistic missiles.[5]

Properties

A gyroscope in operation with freedom in all three axes. The rotor will maintain its spin axis direction regardless of the orientation of the outer frame.

A gyroscope exhibits a number of behaviours including precession and nutation. Gyroscopes can be used to construct gyrocompasses which complement or replace magnetic compasses (in ships, aircraft and spacecraft, vehicles in general), to assist in stability (bicycle, Hubble Space Telescope, ships, vehicles in general) or be used as part of an inertial guidance system. Gyroscopic effects are used in toys like tops, yo-yos, and Powerballs. Many other rotating devices, such as flywheels, behave gyroscopically although the gyroscopic effect is not used.

The fundamental equation describing the behavior of the gyroscope is:

where the vectors and are, respectively, the torque on the gyroscope and its angular momentum, the scalar is its moment of inertia, the vector is its angular velocity, and the vector is its angular acceleration.

It follows from this that a torque applied perpendicular to the axis of rotation, and therefore perpendicular to , results in a rotation about an axis perpendicular to both and . This motion is called precession. The angular velocity of precession is given by the cross product:

Precession on a gyroscope

Precession can be demonstrated by placing a spinning gyroscope with its axis horizontal and supported loosely (frictionless toward precession) at one end. Instead of falling, as might be expected, the gyroscope appears to defy gravity by remaining with its axis horizontal, when the other end of the axis is left unsupported and the free end of the axis slowly describes a circle in a horizontal plane, the resulting precession turning. This effect is explained by the above equations. The torque on the gyroscope is supplied by a couple of forces: gravity acting downwards on the device's centre of mass, and an equal force acting upwards to support one end of the device. The rotation resulting from this torque is not downwards, as might be intuitively expected, causing the device to fall, but perpendicular to both the gravitational torque (horizontal and perpendicular to the axis of rotation) and the axis of rotation (horizontal and outwards from the point of support), i.e. about a vertical axis, causing the device to rotate slowly about the supporting point.

As the second equation shows, under a constant torque, the gyroscope's speed of precession is inversely proportional to its angular momentum. This means that, for instance, if friction causes the gyroscope's spin to slow down, the rate of precession increases. This continues until the device is unable to rotate fast enough to support its own weight, when it stops precessing and falls off its support, mostly because friction against precession cause another precession that goes to cause the fall.

By convention, these three vectors, torque, spin, and precession, are all oriented with respect to each other according to the right-hand rule.

To easily ascertain the direction of gyro effect, simply remember that a rolling wheel tends, when entering a corner, to turn over to the inside.

Gyrostat

A gyrostat is a variant of the gyroscope. The first gyrostat was designed by Lord Kelvin to illustrate the more complicated state of motion of a spinning body when free to wander about on a horizontal plane, like a top spun on the pavement, or a hoop or bicycle on the road. It consists of a massive flywheel concealed in a solid casing. Its behaviour on a table, or with various modes of suspension or support, serves to illustrate the curious reversal of the ordinary laws of static equilibrium due to the gyrostatic behaviour of the interior invisible flywheel when rotated rapidly.

Moment of inertiaMoment of inertia, also called mass moment of inertia or the angular mass, (SI units kg m2) is a measure of an object's resistance to changes in its rotation rate. It is the rotational analog of mass. That is, it is the inertia of a rigid rotating body with respect to its rotation. The moment of inertia plays much the same role in rotational dynamics as mass does in basic dynamics, determining the relationship between angular momentum and angular velocity, torque and angular acceleration, and several other quantities. While a simple scalar treatment of the moment of inertia suffices for many situations, a more advanced tensor treatment allows the analysis of such complicated systems as spinning tops and gyroscopic motion.

The symbol I and sometimes J are usually used to refer to the moment of inertia.

The concept was introduced by Euler in his book a Theoria motus corporum solidorum seu rigidorum in 1730.[1] In this book, he discussed the moment of inertia and many related concepts, such as the principal axis of inertia.

Overview

The moment of inertia of an object about a given axis describes how difficult it is to change its angular motion about that axis. For example, consider two discs, A and B, made of the same material and of equal mass. Disc A is larger in diameter but thinner than B. It requires more effort to accelerate disc A (change its angular velocity) because its mass is distributed farther from its axis of rotation: mass that is farther out from that axis must, for a given angular velocity, move more quickly than mass closer in. In this case, disc A has a larger moment of inertia than disc B.

The moment of inertia of an object can change if its shape changes. A figure skater who begins a spin with arms outstretched provides a striking example. By pulling in her arms, she reduces her moment of inertia, causing her to spin faster (by the conservation of angular momentum).

The moment of inertia has two forms, a scalar form I (used when the axis of rotation is known) and a more general tensor form that does not require knowing the axis of rotation. The scalar moment of inertia I (often called simply the "moment of inertia") allows a succinct analysis of many simple problems in rotational dynamics, such as objects rolling down inclines and the behavior of pulleys. For instance, while a block of any shape will slide down a frictionless decline at the same rate, rolling objects may descend at different rates, depending on their moments of inertia. A hoop will descend more slowly than a solid disk of equal mass and radius because more of its mass is located far from the axis of rotation, and thus needs to move faster if the hoop rolls at the same angular velocity. However, for (more complicated) problems in which the axis of rotation can change, the scalar treatment is inadequate, and the tensor treatment must be used (although shortcuts are possible in special situations). Examples requiring such a treatment include gyroscopes, tops, and even satellites, all objects whose alignment can change.

The moment of inertia can also be called the mass moment of inertia (especially by mechanical engineers) to avoid confusion with the second moment of area, which is sometimes called the moment of inertia (especially by structural engineers) and denoted by the same symbol I. The easiest way to differentiate these quantities is through their units. In addition, the moment of inertia should not be confused with the polar moment of inertia, which is a measure of an object's ability to resist torsion (twisting).

Scalar moment of inertia

Definition

A simple definition of the moment of inertia (with respect to a given axis of rotation) of any object, be it a point mass or a 3D-structure, is given by:

where m is mass and r is the perpendicular distance to the axis of rotation.

Detailed Analysis

The (scalar) moment of inertia of a point mass rotating about a known axis is defined by

The moment of inertia is additive. Thus, for a rigid body consisting of N point masses mi with distances ri to the rotation axis, the total moment of inertia equals the sum of the point-mass moments of inertia:

For a solid body described by a mass density function, ρ(r), the moment of inertia about a known axis can be calculated by integrating the square of the distance (weighted by the mass density) from a point in the body to the rotation axis:

where

V is the volume occupied by the object.ρ is the spatial density function of the object, andr = (r,θ,φ), (x,y,z), or (r,θ,z) are coordinates of a point inside the body.

Diagram for the calculation of a disk's moment of inertia. Here k is 1/2 and is the radius used in determining the moment.

Based on dimensional analysis alone, the moment of inertia of a non-point object must take the form:

where

M is the massR is the radius of the object from the center of mass (in some cases, the length of the object is used instead.)k is a dimensionless constant called the inertia constant that varies with the object in consideration.

Inertial constants are used to account for the differences in the placement of the mass from the center of rotation. Examples include:

k = 1, thin ring or thin-walled cylinder around its center, k = 2/5, solid sphere around its center k = 1/2, solid cylinder or disk around its center.

Parallel axis theorem

Once the moment of inertia has been calculated for rotations about the center of mass of a rigid body, one can conveniently recalculate the moment of inertia for all parallel rotation axes as well, without having to resort to the formal definition. If the axis of rotation is displaced by a distance R from the center of mass axis of rotation (e.g. spinning a disc about a point on its periphery, rather than through its center,) the displaced and center-moment of inertia are related as follows:

This theorem is also known as the parallel axes rule and is a special case of Steiner's parallel-axis theorem.

Composite bodies

If a body can be decomposed (either physically or conceptually) into several constituent parts, then the moment of inertia of the body about a given axis is obtained by summing the moments of inertia of each constituent part around the same given axis.[2]

Equations involving the moment of inertia

The rotational kinetic energy of a rigid body can be expressed in terms of its moment of inertia. For a system with N point masses mi moving with speeds vi, the rotational kinetic energy T equals

where ω is the common angular velocity (in radians per second). The final expression I ω2 / 2 also holds for a mass density function with a generalization of the above derivation from a discrete summation to an integration.

In the special case where the angular momentum vector is parallel to the angular velocity vector, one can relate them by the equation

where L is the angular momentum and ω is the angular velocity. However, this equation does not hold in many cases of interest, such as the torque-free precession of a rotating object, although its more general tensor form is always correct.

When the moment of inertia is constant, one can also relate the torque on an object and its angular acceleration in a similar equation:

where τ is the torque and α is the angular acceleration.

Moment of inertia tensor

For the same object, different axes of rotation will have different moments of inertia about those axes. In general, the moments of inertia are not equal unless the object is symmetric about all axes. The moment of inertia tensor is a convenient way to summarize all moments of inertia of an object with one quantity. It may be calculated with respect to any point in space, although for practical purposes the center of mass is most commonly used.

Definition

For a rigid object of N point masses mk, the moment of inertia tensor is given by

.

Its components are defined as

where

i, j equal 1, 2, or 3 for x, y, and z, respectively,r=(x1, x2, x3) is the vector to the mass element dM from the point about which the tensor is calculated, r=||x||, andδij is the Kronecker delta.

Note that, by the definition, I is a symmetric tensor.

The diagonal elements, also called the principal moments of inertia, are more succinctly written as

while the off-diagonal elements, also called the products of inertia, are

and

Here Ixx denotes the moment of inertia around the x-axis when the objects are rotated around the x-axis, Ixy denotes the moment of inertia around the y-axis when the objects are rotated around the x-axis, and so on.

These quantities can be generalized to an object with distributed mass, described by a mass density function, in a similar fashion to the scalar moment of inertia. One then has

where is their outer product, E3 is the 3 × 3 identity matrix, and V is a region of space completely containing the object.

Derivation of the tensor components

The distance r of a particle at from the axis of rotation passing through the origin in the direction is . By using the formula I = mr2 (and some simple vector algebra) it can be seen that the moment of inertia of this particle (about the axis of rotation passing through the

origin in the direction) is This is a quadratic form in and, after a bit more algebra, this leads to a tensor formula for the moment of inertia

.

This is exactly the formula given below for the moment of inertia in the case of a single particle. For multiple particles we need only recall that the moment of inertia is additive in order to see that this formula is correct.

Reduction to scalar

For any axis , represented as a column vector with elements ni, the scalar form I can be calculated from the tensor form I as

The range of both summations correspond to the three Cartesian coordinates.

The following equivalent expression avoids the use of transposed vectors which are not supported in maths libraries because internally vectors and their transpose are stored as the same linear array,

However it should be noted that although this equation is mathematically equivalent to the equation above for any matrix, inertia tensors are symmetrical. This means that it can be further simplified to:

Principal moments of inertia

By the spectral theorem, since the moment of inertia tensor is real and symmetric, it is possible to find a Cartesian coordinate system in which it is diagonal, having the form

where the coordinate axes are called the principal axes and the constants I1, I2 and I3 are called the principal moments of inertia. The unit vectors along the principal axes are usually denoted as (e1, e2, e3).

When all principal moments of inertia are distinct, the principal axes are uniquely specified. If two principal moments are the same, the rigid body is called a symmetrical top and there is no unique choice for the two corresponding principal axes. If all three principal moments are the same, the rigid body is called a spherical top (although it need not be spherical) and any axis can be considered a principal axis, meaning that the moment of inertia is the same about any axis.

The principal axes are often aligned with the object's symmetry axes. If a rigid body has an axis of symmetry of order m, i.e., is symmetrical under rotations of 360°/m about a given axis, the symmetry axis is a principal axis. When m > 2, the rigid body is a symmetrical top. If a rigid body has at least two symmetry axes that are not parallel or perpendicular to each other, it is a spherical top, e.g., a cube or any other Platonic solid. A practical example of this mathematical phenomenon is the routine automotive task of balancing a tire, which basically means adjusting the distribution of mass of a car wheel such that its principal axis of inertia is aligned with the axle so the wheel does not wobble.

Parallel axis theorem

Once the moment of inertia tensor has been calculated for rotations about the center of mass of the rigid body, there is a useful labor-saving method to compute the tensor for rotations offset from the center of mass.

If the axis of rotation is displaced by a vector R from the center of mass, the new moment of inertia tensor equals

where M is the total mass of the rigid body, E3 is the 3 × 3 identity matrix, and is the outer product.

Rotational symmetry

For bodies with rotational symmetry around an axis , the moments of inertia for rotation around two perpendicular axes and are

where we have defined

Using the above equation to express all moments of inertia in terms of integrals of variables either along or perpendicular to the axis of symmetry usually simplifies the calculation of these moments considerably.

Other mechanical quantities

Using the tensor I, the kinetic energy can be written as a quadratic form

and the angular momentum can be written as a product

Taken together, one can express the rotational kinetic energy in terms of the angular momentum (L1,L2,L3) in the principal axis frame as

The rotational kinetic energy and the angular momentum are constants of the motion (conserved quantities) in the absence of an overall torque. The angular velocity ω is not constant; even without a torque, the endpoint of this vector may move in a plane (see Poinsot's construction).

Comparison with covariance matrix

The moment of inertia tensor about the center of mass of a 3 dimensional rigid body is related to the covariance matrix of a trivariate random vector whose probability density function is proportional to the pointwise density of the rigid body by

where n is the number of points.

The structure of the moment-of-intertia tensor comes from the fact that it is to be used as a bilinear form on rotation vectors in the form

Each element of mass has a kinetic energy of

The velocity of each element of mass is where r is a vector from the center of rotation to that element of mass. The cross product can be converted to matrix multiplication so that

and similarly

Thus

plugging in the definition of the term leads directly to the structure of the moment tensor.

Relativistic Doppler Effect

A source of light waves moving to the right with velocity 0.7c. The frequency is higher on the right, and lower on the left.

The relativistic Doppler effect is the change in frequency (and wavelength) of light, caused by the relative motion of the source and the observer (as in the classical Doppler effect), when taking into account effects of the special theory of relativity.

The relativistic Doppler effect is different from the non-relativistic Doppler effect as the equations include the time dilation effect of special relativity and do not involve the medium of propagation as a reference point. They describe the total difference in observed frequencies and possess the required Lorentz symmetry.

Motion along the line of sight

Assume the observer and the source are moving away from each other with a relative velocity ( is negative if the observers are moving toward each other). Let us consider the problem in the reference frame of the source.

Suppose one wavefront arrives at the observer. The next wavefront is then at a distance away from him (where is the wavelength,

is the frequency of the wave the source emitted, and is the speed of light). Since the wavefront moves with velocity and the observer escapes with velocity v, the time observed between crests is

However, due to the relativistic time dilation, the observer will measure this time to be

where , so the corresponding observed frequency is

The ratio is called the Doppler factor of the source relative to the observer. (This terminology is particularly prevalent in the subject of astrophysics: see relativistic beaming.)

The corresponding wavelengths are related by

and the resulting redshift can be written as

In the non-relativistic limit—i.e. when —the approximate expressions are

Transverse Doppler effect

The transverse Doppler effect is the nominal red shift component associated with transverse (i.e. lateral) observation resulting from special relativity effects, and is important both theoretically and experimentally.

If the predictions of special relativity are compared to those of a simple flat nonrelativistic light medium that is stationary in the observer’s frame (“classical theory”), SR’s physical predictions of what an observer sees are always “redder”, by the Lorentz factor

The transverse Doppler effect is a direct consequence of the relativistic Doppler effect.

In the particular case when θo = π / 2, one obtains the transverse Doppler effect

For receding or approaching objects, the redshift factor modifies the redshift or blueshift predictions of "classical theory". Where the two effects act against each other, the propagation-based effects are stronger. But for the case of an object passing directly across the observer’s line of sight, special relativity’s predictions are qualitatively different from "classical theory" – a redshift where the “classical theory” reference model would have predicted no shift effect at all for the case that the observer is at rest in the aether.

Because of this, the transverse Doppler effect is sometimes held up as one of the main new predictions of the special theory. As Einstein put it in 1907: according to special relativity the moving object's emitted frequency is reduced by the Lorentz factor, so that - in addition to the classical Doppler effect - the received frequency is reduced by the same factor.

Reciprocity

Sometimes the question arises as to how the transverse Doppler effect can lead to a redshift as seen by the "observer" whilst another observer moving with the emitter would also see a redshift of light sent (perhaps accidentally) from the receiver.

It is essential to understand that the concept "transverse" is not reciprocal. Each participant understands that when the light reaches her/him transversely as measured in terms of that person's rest frame, the other had emitted the light aftward as measured in the other person's rest frame. In addition, each participant measures the other's frequency as reduced ("time dilation"). These effects combined make the observations fully reciprocal, thus obeying the principle of relativity.

Experimental verification

In practice, experimental verification of the transverse effect usually involves looking at the longitudinal changes in frequency or wavelength due to motion for approach and recession: by comparing these two ratios together we can rule out the relationships of "classical theory" and prove that the real relationships are "redder" than those predictions.

Longitudinal tests

The first of these experiments was carried out by Ives and Stilwell in (1938) and although the accuracy of this experiment has since been questioned,[citation needed] many other longitudinal tests have been performed since with much higher precision [http://* Herbert E. Ives and G.R. Stilwell, “An experimental study of the rate of a moving clock”

J. Opt. Soc. Am 28 215-226 (1938) and part II. J. Opt. Soc. Am. 31, 369-374 (1941)

Transverse Tests

To date, only one inertial experiment seems to have verified the redshift effect for a detector actually aimed at 90 degrees to the object.

D. Hasselkamp, E. Mondry, and A. Scharmann, "Direct Observation of the Transversal Doppler-Shift"

Z. Physik A 289, 151-155 (1979).

Motion in an arbitrary direction

If, in the reference frame of the observer, the source is moving away with velocity at an angle relative to the direction from the observer to the source (at the time when the light is emitted), the frequency changes as

(1)

In the particular case when and one obtains the transverse Doppler effect:

.

It should be noted that, due to the finite speed of light, the light ray (or photon, if you like) perceived by the observer as coming at angle , was, in

the reference frame of the source, emitted at a different angle . and are tied to each other via the relativistic aberration formula:

Therefore, Eq. (1) can be rewritten as

(2)

For example, a photon emitted at the right angle in the reference frame of the emitter ( ) would be seen blue-shifted by the observer:

In the non-relativistic limit, both formulæ (1) and (2) give

Visualization

Diagram 1. Demonstration of aberration of light and relativistic Doppler effect.

In diagram 1, the blue point represents the observer. The x,y-plane is represented by yellow graph paper. As the observer accelerates, he sees the

graph paper change colors. Also he sees the distortion of the x,y-grid due to the aberration of light. The black vertical line is the y-axis. The observer accelerates along the x-axis. If the observer looks to the left, (behind him) the lines look closer to him, and since he is accelerating away from the left side, the left side looks red to him (redshift). When he looks to the right (in front of him) because he is moving towards the right side, he sees the right side as green, blue, and violet, respectively as he accelerates (blueshift). Note that the distorted grid is just the observer's perspective, it is all still a consistent yellow graph, but looks more colored and distorted as the observer changes speed.

Aberration of lightThe aberration of light (also referred to as astronomical aberration or stellar aberration) is an astronomical phenomenon which produces an apparent motion of celestial objects about their real locations. It was discovered and later explained by the third Astronomer Royal, James Bradley, in 1725, who attributed it to the finite speed of light and the motion of Earth in its orbit around the Sun.[1]

At the instant of any observation of an object, the apparent position of the object is displaced from its true position by an amount which depends solely upon the transverse component of the velocity of the observer, with respect to the vector of the incoming beam of light (i.e., the line actually taken by the light on its path to the observer). The result is a tilting of the direction of the incoming light which is independent of the distance between object and observer.

In the case of an observer on Earth, the direction of a star's velocity varies during the year as Earth revolves around the Sun (or strictly speaking, the barycenter of the solar system), and this in turn causes the apparent position of the star to vary. This particular effect is known as annual aberration or stellar aberration, because it causes the apparent position of a star to vary periodically over the course of a year. The maximum amount of the aberrational displacement of a star is approximately 20 arcseconds in right ascension or declination. Although this is a relatively small value, it was well within the observational capability of the instruments available in the early eighteenth century.

Aberration should not be confused with stellar parallax, although it was an initially fruitless search for parallax that first led to its discovery. [1] Parallax is caused by a change in the position of the observer looking at a relatively nearby object, as measured against more distant objects, and is therefore dependent upon the distance between the observer and the object.[1]

In contrast, stellar aberration is independent of the distance of a celestial object from the observer, and depends only on the observer's instantaneous transverse velocity with respect to the incoming light beam, at the moment of observation. The light beam from a distant object cannot itself have any transverse velocity component, or it could not (by definition) be seen by the observer, since it would miss the observer. Thus, any transverse velocity of the emitting source plays no part in aberration. Another way to state this is that the emitting object may have a transverse velocity with respect to the observer, but any light beam emitted from it which reaches the observer, cannot, for it must have been previously emitted in such a direction that its transverse component has been "corrected" for. Such a beam must come "straight" to the observer along a line which connects the observer with the position of the object when it emitted the light.[1]

Aberration should also be distinguished from light-time correction, which is due to the motion of the observed object, like a planet, through space during the time taken by its light to reach an observer on Earth. Light-time correction depends upon the velocity and distance of the emitting object during the time it takes for its light to travel to Earth. Light-time correction does not depend on the motion of the Earth—it only depends on Earth's position at the instant when the light is observed. Aberration is usually larger than a planet's light-time correction except when the planet is near quadrature (90° from the Sun), where aberration drops to zero because then the Earth is directly approaching or receding from the planet. At opposition to or conjunction with the Sun, aberration is 20.5" while light-time correction varies from 4" for Mercury to 0.37" for Neptune (the Sun's light-time correction is less than 0.03").

Fermat's Principle: ReflectionFermat's Principle: Light follows the path of least time. The law of reflection can be derived from this principle as follows:

The path length L from A to B is

Since the speed is constant, the minimum time path is simply the minimum distance path. This may be found by setting the derivative of L with respect to x equal to zero.

Fermat's Principle and Refraction Fermat's Principle: Light follows the path of least time. Snell's Law can be derived from this by setting the derivative of the time =0. It shows how the refraction of light at an interface depends upon the indices of refraction.