The celestial sphere is the imaginary sphere with infinite radius that is centered at the...

The celestial sphere is the imaginary sphere with infinite radius that is centered at the center of the earth and on which all astronomical objects appear to move. We can also visualize it as a sphere of finite radius where the earth, placed at its center O, has zero radius. The north celestial pole N is the point on the celestial sphere in which the ray from earth’s center through its north pole intersects the celestial sphere. The south celestial pole S is defined similarly. The celestial equator is the great circle on the celestial sphere obtained by projecting earth’s equator onto it from earth’s center.

Very distant objects (stars and galaxies, objects outside the solar system) appear to move along circles of constant latitude on the celestial sphere, but the motion of the sun and planets is more complicated. Because the axis of earth’s rotation is inclined about 23° from a perpendicular to the plane in which it revolves about the sun, the sun appears to move (more or less) along a great circle (called the ecliptic) that is inclined about 23° with the celestial equator and that intersects the equator in two points, the vernal equinox and the autumnal equinox. The vernal equinox lies in the constellation Aries; for this reason it is often symbolized by the capital Greek letter ϒ (upsilon), which serves as an effigy of a ram’s head.

A point P on the celestial sphere is located as follows: Construct the great circle through P and N, and let Q be its intersection with the celestial equator. The declination of P is the angle δ = ∠POQ measured in degrees as positive if P is in the northern hemisphere and negative if in the southern hemisphere. The right ascension of P is the angle α = ∠ϒOQ measured in hours, minutes, and seconds clockwise. (One hour of angle measure is of the angle measure of an entire circle.)

(a) Sketch the celestial sphere, labeling the celestial poles and equator, the ecliptic, and the vernal equinox. For a point P on the celestial sphere, sketch the right ascension and declination.

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(b) Suppose that an observer stands on earth with the vernal equinox straight overhead. Can she see a star whose declination is +48° and whose right ascension is 4 hr 23′ 28″?

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(c) By putting a “standard” coordinate system in the celestial sphere (origin at O, x-axis piercing ϒ, z-axis piercing N) write equations that give spherical coordinates θ and ϕ in terms of right ascension α and declination δ.