SkyEye

O my beloved, my sweet Hesperus!
My morning and my evening star of love!
My best and gentlest lady! even thus,
As that fair planet in the sky above,
Dost thou retire unto thy rest at night,
And from thy darkened window fades the light.
— Henry Wadsworth Longfellow, "The Evening Star", 1866

Venus

The Moon is not the only solar system body to exhibit phases as seen from Earth. The inferior planets, Mercury and Venus, and, to a lesser extent, the superior planet Mars, also show phases which depend on their relative positions to Earth and the Sun.

Galileo was the first person to see the phases of Venus, when he observed the planet in 1610 using the newly-invented astronomical telescope. (Warning! Great care must be taken when observing any object near the Sun as blindness can result.) The phases occur as Venus orbits the Sun. When Venus is between the Sun and Earth (inferior conjunction), it presents its unlit side to us, the same as a New Moon. As it moves around the Sun, it becomes a crescent and eventually is half lit. This occurs near greatest elongation west. The phase continues to grow into a gibbous disk and eventually is fully illuminated at superior conjunction, when it is on the opposite side of the Sun from Earth. Then it carries on, slowly waning through its gibbous phase, quarter phase (when it again is half lit) and crescent phase until it returns to inferior conjunction.

The fact that Venus shows a complete set of phases proves that the heliocentric model (the planets revolve around the Sun) is correct and that the geocentric model (the Sun and planets revolve around Earth) is not.

The term dichotomy refers to when Venus is exactly half lit. This always occurs near greatest elongation but not necessarily at exactly the same time. The observed instant of dichotomy can sometimes differ from the predicted date by several days, and this remains an unsolved mystery.

The orbit of Venus is inclined just over 3° with respect to the ecliptic so it rarely passes between the Sun and Earth (an event called a transit) or behind the Sun as seen from Earth (an event sometimes referred to as an anti-transit or secondary eclipse). The most recent transits of Venus occurred in 2004 and 2012, and the next pair won't happen until 2117 and 2125. (Transits usually occur in pairs separated by eight years.)

Venus is the brightest planet in the sky. It is around magnitude −3.9 at superior conjunction but can get brighter than −4.5 in its crescent phases. The brightness is a balance of the phase (the proportion of planet's disk that is illuminated from the viewpoint of Earth) and the distance from us. At inferior conjunction, Venus is less than 0.3 au from Earth whilst at superior conjunction, it is over 1.7 au distant. The planet's apparent angular size varies sixfold, from under 10 arc-seconds to over 60 arc-seconds during an apparition. The planet is actually brightest in the sky when it is in its crescent phase, just over a month either side of inferior conjunction.

The apparent magnitude V of a planet is given by the equation

V = V(1,0) + 5log(r⋅d) + Δm(i)

where V(1,0) is the magnitude of the planet as seen at a distance of 1 au from both the Sun and Earth and at a phase angle of 0° (a geometric configuration which is actually impossible!), r is the heliocentric distance of the planet (in au), d is the geocentric distance of the planet (again in au), and Δm(i) is the correction for the solar phase angle i. The expression Δm(i) is determined by fitting a polynomial function to observed magnitudes as a range of phase angles.

For Venus, the following formula, based on recent photometry of the planet, is employed in The Astronomical Almanac:

V(1,0) + Δm(i) = −4.47 + 1.03(i/100) + 0.57(i/100)² + 0.13(i/100)³ for 2.2° < i < 163.6° and

V(1,0) + Δm(i) = 0.98 − 1.02(i/100) for 163.6° < i < 170.2°.

Mercury

Mercury changes its appearance just as Venus does except that it completes three entire phase cycles every year. However, the brightness profile of Mercury is quite different. Mercury is faintest (magnitude +6) near inferior conjunction and brightest (at least magnitude −2) at superior conjunction. This is because Mercury does not show the same variation in angular size that Venus does. Mercury is only about 2.3 times larger at inferior conjunction than at superior conjunction.

The apparent magnitude of Mercury can be approximated by the formula given above, with

V(1,0) + Δm(i) = −0.60 + 4.98(i/100) − 4.88(i/100)² + 3.02(i/100)³.

Mars

The superior planets always appear full or nearly so. Only Mars takes on a distinctly gibbous aspect (more than half illuminated but less than full) near quadrature. In 2018, Mars reached west quadrature on 24 March, opposition on 27 July and east quadrature on 3 December. Notice the change in phase and size of the disk over the course of a year.