• "الفلك الموضعي"

• مقدمة
• الكرة الأرضية
• هندسة المثلثات الكروية
• ملاحظات حول الهندسة الكروية
• إحداثيات أفقية "alt-az"
• إحداثيات استوائية "HA-dec"
• إحداثيات استوائية "RA-dec"
• الزمن النجمي
• الانتقال بين أفقي و استوائي
• إحداثيات مجرية
• إحداثيات بروجية
• الانتقال بين بروجي و استوائي
• حركة الشمس و ضبط الوقت
• القمر
• الإنكسار الجوي
• شروق و غروب الشمس، الشفق
• اختلاف المنظر المركزي اليومي
• اختلاف المنظر الموسمي
• الزيغ
• المبادرة و الترنح :اضطراب محور الأرض
• تقاويم
• الامتحان الأخير
• الإحداثيات الفلكية، خضر الأحمد
• الحركة الظاهرية للنجوم (تفاعلي)
• الأقطاب السماوية (تفاعلي)

الزيغ

English المواضيع

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. λ_F = λ_S – 90°.

The geometry is very similar to the parallax problem, with the following differences:
(i) we must write λ_F instead of λ_S .
(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(λ_F-λ) = -k cos(λ_S-λ)
Δβ = -k cos(λ_F-λ) sin(β) = - k sin(β) sin(λ_S-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 (λ_S-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.