{Note: If your browser does not distinguish between "a,b" and "α, β" (the Greek letters "alpha, beta") then I am afraid you will not be able to make much sense of the equations on this page.}
Early attempts to measure the distances of the stars,
by observing their parallactic ellipses,
were unsuccessful because the stars are so far away,
and their parallaxes are extremely small.
However, another effect was discovered instead: aberration.
This is caused by the fact that light moves at a finite velocity, c.
The apparent direction that light comes to us from a star
is a combination of its true direction
and the direction the Earth is moving.
Stars appear to be shifted slightly
in the direction of the Earth's motion.
(This is analogous to the way a person walking through the rain
has to hold their umbrella tilted forwards.)
Take the Earth's velocity as v.
During a time-interval t,
Earth moves a distance vt,
while light travels a distance ct down the telescope.
By plane trigonometry,
sin(θ-θ')/vt = sin(θ')/ct
where θ is the true angle
between the direction to the star, and the direction the Earth is moving around
the Sun,
and θ' is the observed angle.
Since vt is very small compared to ct,
θ' is very nearly equal to θ.
So we may write sin(θ-θ')/vt = sin(θ)/ct
i.e. sin(θ-θ') = sin(θ) v/c
Because the ratio v/c is very small,
sin(θ-θ') is approximately equal to θ-θ' (in radians),
so we may write:
θ-θ' = sin(θ) v/c = k sin(θ)
where k, the constant of aberration, is 20.5 arc-seconds.
But
in which direction is the Earth moving?
Taking the Earth's orbit as circular,
the tangent is always at right-angles to the radius.
So the direction of the Earth's motion is always
at 90° to the direction of the Sun.
Thus F, the "apex of the Earth's way", is
on the ecliptic, 90° behind the Sun.
i.e. λF = λS – 90°.
The geometry is very similar to the
parallax problem, with the following differences:
(i) we must write λF instead of λS .
(ii) θ-θ' is now the aberrational shift k sin(θ),
not the parallactic shift Π sin(θ),
so we replace Π by k.
So we find:
Δλ cos(β) = k sin(λF-λ) = -k cos(λS-λ)
Δβ = -k cos(λF-λ) sin(β) = - k sin(β) sin(λS-λ)
Again this is the formula for an ellipse of the form:
x = a cos(θ), y = b sin(θ)
where θ is now temporary shorthand for (λS-λ).
The aberrational ellipse has
semi-major axis k, parallel to the ecliptic,
and semi-minor axis k sin(β), perpendicular to the ecliptic.
There are two important differences between the parallactic and aberrational
ellipses:
1) The aberrational ellipse is much bigger.
(k is 20.5 arc-seconds, whereas parallax is always less than 1 arc-second.)
Also the major axis of the aberrational ellipse
is the same for all stars,
whereas the major axis of the parallactic ellipse
depends on the star's distance.
2) The phase is different.
When the Sun has the same longitude as the star,
then the longitude shift is zero in the
parallactic ellipse,
but the latitude shift is zero in the
aberrational ellipse.
So far, we have been assuming that the Earth's orbit is circular,
and hence the value of k = v/c is constant;
in fact the orbit is elliptical, and this means the velocity v varies with time.
The
velocity ET in any elliptical orbit can be resolved into two components:
EF = h/p, perpendicular to the radius vector,
EG = eh/p, perpendicular to the major axis of the ellipse.
The values of EF and EG are both constant.
It's the changing angle between these two constant components
which causes the orbital velocity to vary (Kepler's Second Law).
Here, EF is the velocity for a circular orbit, as assumed above.
EG adds second-order terms, 0.3 arc-seconds or less,
which are independent of Earth’s position,
and depend only on the star’s position.
A star itself also has its own proper motion across the sky,
but this is always small and generally not known,
so it is often ignored.
However, for objects within the solar system,
the motion is usually known, and is too large to ignore.
So astrometric observations of a planet have to be corrected for light-time:
the time between the light leaving the planet, and being measured on Earth.
The planet may move a significant distance during this time.
Annual aberration and light-time are sometimes grouped together
and they are called planetary aberration,
in which case annual aberration alone is called stellar aberration.
Exercise:
A star’s true position is
Right Ascension 6h 0m 0s, declination 0° 0' 0".
On the date of the Spring Equinox,
how far will it appear to be shifted by aberration,
and in what direction?
Click here for the answer.