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.
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Annual
parallax
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