The 4-Point backyard diurnal parallax method

Solar System:
The Solar System consists of the Sun and the objects that orbit the Sun. These include eight planets and various secondary bodies, dwarf planets and small Solar System objects that orbit the Sun directly, as well as satellites (moons) that orbit many planets and smaller objects. The Solar System formed 4.6 billion years ago from the gravitational collapse of a giant molecular cloud. The vast majority of the system's mass is in the Sun, with most of the remaining mass contained in Jupiter. The four smaller inner planets, Mercury, Venus, Earth and Mars, also called the terrestrial planets, are primarily composed of rock and metal. The four outer planets, called the gas giants, are substantially bigger than the terrestrials. The two largest planets, Jupiter and Saturn, are composed mainly of hydrogen and helium. The two outermost planets, Uranus and Neptune, are composed largely of substances with relatively high melting points (compared with hydrogen and helium), called ices, such as water, ammonia and methane, and are often referred to separately as ‘ice giants’. All planets have almost circular orbits that lie within a nearly flat disc called the ecliptic plane.

A solar system refers to a star and all the objects that travel in orbit around it. Our Solar System consists of the Sun - our star - eight planets and their natural satellites (such as our moon); dwarf planets; asteroids and comets. Our Solar System is located in an outward spiral of the Milky Way galaxy.

CCD astro-imaging:
A charge-coupled device (CCD) is an electronic instrument for detecting light. In the case of an astronomical CCD camera, this light is very dim. This has certain implications on how the CCD operates. A CCD uses a thin silicon wafer chip. The chip is divided into thousands or millions of tiny light sensitive squares (or sometimes rectangles) called photosites. Each photosite corresponds to an individual pixel in the final image, and the photosites are often referred to simply as pixels. For clarity in this discussion, ‘photosites’ will refer to the CCD chip and ‘pixels’ will refer to an image. Each photosite is surrounded by a non-conductive boundary, which contains the charge that is collected during an exposure.

Parallax:
Parallax is a method of measuring the distance to an object. Just like how our binocular vision helps us determine distance, the direction to a distant point is slightly different from two separate observation positions. When observing a distant object at exactly the same time from two separated places, if the distance between the observation positions is known, and the angle between them can be measured, with the help of simple geometry, the distance to the object can be calculated. In practice, given that in most astronomical applications, the parallax angle is very small, the only way to take advantage of this technique for distance determination is by means of a very large baseline.

Diurnal Parallax Effect:

Figure 1. Geometry of the diurnal parallax effect. As the Earth rotates, any observer on its surface sees the target’s parallactic angle φ constantly varying, from a maximum value occurring when his Meridian is at right angle to the plane containing the Earth’s rotation axis and the target (some 6 hours prior or after transit time) to a null value whenever the target happens to be placed on his Meridian (target culmination).

Placed on the Earth’s surface, any observer actually moves along a perfectly circular track, several thousand kilometres in radius, making one complete circle each 23.93 hours – the sidereal period. At any given time, the observer is seeing the vault of heaven from a slightly different and continuously changing perspective compared to a fictitious observer placed at rest at the centre of the Earth. The apparent varying displacement of relatively nearby celestial objects with respect to distant ‘fixed’ stars as observed from any ground site, exclusively due to the Earth’s daily rotation, is known as the diurnal parallax effect. This effect materializes as a very subtle angular oscillation, ranging from a few hundreds of arcseconds for sufficiently close near-Earth objects (NEOs) to barely tenths of arcseconds for the furthest observable Solar System objects. The diurnal parallax effect is immeasurably small for celestial bodies beyond the Solar System.

The diurnal parallax effect has nothing to do with the fact that any given Solar System object is moving with respect to the centre of the Earth. Comparatively, the apparent celestial motion of any Solar System body due to its orbital revolution around the Sun becomes orders of magnitude larger than its corresponding diurnal parallax effect. Therefore, the measurement of diurnal parallax must not only deal with minute oscillating displacements with respect to ‘fixed’ stars, but at the same time such tiny effects are always encapsulated within much larger ‘proper motion’ displacements across the sky. Consequently, the first step in the diurnal parallax game requires accurately subtracting the dominant orbital trajectory across the sky from the actually observed apparent celestial motion of the target in question. If carefully and properly done, what remains after this sieving – usually referred to as ‘residuals’ – is only due to the diurnal parallax effect.

Assuming that for a time period as short as a couple of days the distance from the Earth’s centre to the target is basically constant – a generally appropriate assumption except for nearby NEO asteroids and the Moon – the time variability of the parallactic angle can be exclusively attributed to the observer’s changing position relative to a fixed direction in the equatorial plane. In other words, with respect to the plane that contains the axis of rotation of the Earth (i.e. in declination), the parallax angle remains basically invariant. This is why the diurnal parallax is also called the east-west parallax.

The diurnal parallax effect exactly corresponds to the variable parallactic angle – the angle that at any given moment is formed by the target at the vertex, the observer on one side and the centre of the Earth on the other side – projected to the equatorial plane. It’s clear that at the moment of the target’s culmination (when the target happens to be on the observer’s meridian), the variable diurnal parallax angle becomes null, while achieving its maximum theoretical value some 5.98 hours (¼ of sidereal period) either before or after transit. (It would be at exactly ¼ of the sidereal period if the target were not moving with respect to Earth’s centre.)

Therefore, by definition, at any given time t, the variable diurnal parallax angle φ (t) becomes

φ (t) = [RAtopo (t) − RAgeo (t)] cos δ (t) (1)

where RAtopo(t) is the object’s topocentric (measured from the observer location) right ascension at the considered moment, RAgeo(t) is the object’s geocentric (measured from the centre of the Earth) right ascension at the same considered moment and δ(t) is the object’s declination also at the same moment (assumed to be practically the same as measured either from the observer’s place or from the centre of the Earth).

For observations taken before transit, the parallactic projection of the target on the background star field makes it appear to the observer as displaced towards the east with respect to what would be observed at the same time from Earth’s centre – that is, before transit: RAtopo(t) > RAgeo(t). Conversely, the parallactic projection appears displaced towards the west for observations that take place after transit – that is, after transit: RAtopo(t) < RAgeo(t). Therefore, as the Earth rotates, the term RAtopo(t) – RAgeo(t) inside the square brackets continuosly varies as the observer is carried along by the Earth’s rotation, making the variable diurnal parallax angle φ(t) positive for observations taking place before transit, exactly zero at transit, and negative for observations taking place after the transit. Maximum diurnal parallax angles are theoretically reached whenever the observer and the target happen to be placed at right angles with respect to the Earth’s centre.

Meanwhile, taking into account that the target’s declination does vary – gradually – with time but totally independent of the Earth’s rotation, the term cos δ(t) changes only slightly during a short time period of a few days. Therefore, the variable diurnal parallax angle basically varies on a daily basis as a sinusoidal function, which at any time equals RAtopo(t) – RAgeo(t) times the almost constant cosine of the target’s declination, thus becoming:

φ (t) = [RAtopo (t) − RAgeo (t)] cos δ (t) = φmax sin (ω (T0 – t)) (2)

where φmax is the maximum amplitude of the variable diurnal parallax angle for the considered day, ω is the angular sidereal rotation rate of the Earth and T0 is the moment of the target culmination for the considered day. Once φmax is known, the distance D to the target can be readily obtained by making use of the small-angle approximation, thus becoming:

D = RE cos λ / φmax (3)

where RE is the radius of the Earth, and λ is the latitude of the observer (D and RE have to be expressed in the same units, while φmax has to be in radians).

The 4-Point Backyard Diurnal Parallax Method: The 4-Point Backyard Diurnal Parallax Method is a straightforward procedure for determining the key value of φmax from a single station, not requiring any data other than that strictly derived from direct observations. Just by using an amateur-level telescope and CCD on two consecutive nights, imaging continuously for 10 minutes twice each night – at the beginning and at culmination – one can determine φmax accurately, and then use it to compute the distance to the object in question.

The four required data points are the object’s topocentric right ascension measured on two consecutive nights, and corresponding times. On the first night, it should be measured at any time as far as possible from transit (RAtopo (t1)) and at exact transit time (RAtopo (T01)); on the second night, too, it should be measured at any time as far as possible from transit (RAtopo (t2)) and at exact transit time (RAtopo (T02). It doesn’t matter whether the measurements taken in addition to transit each night have the same time difference or not. Therefore, the four required data points actually are four pairs such as: On night #1: (1) RAtopo (t1) and corresponding t1; and (2) RAtopo (T01) and corresponding T01. On night #2: (3) RAtopo (t2) and corresponding t2; and (4) RAtopo (T02) and corresponding T02.

Given that at the moment of transit, the object’s topocentric and geocentric right ascension are exactly the same, a linear approximation for estimating at any other time the otherwise unknown RAgeo (t) can be readily obtained:

(4)

This linear approximation for RAgeo (t) helps determine the diurnal parallax angles φ (t1) and φ (t2) measured at times t1 and t2 far apart from transit on each night. On each night, the variation of the diurnal parallax angle can be represented by a sinusoidal curve that, besides becoming null at the transit time, happens to pass through the measured φ (t) point. Such sinusoidal curves are unique; therefore once such curves are found, the maximum amplitude of the variable diurnal parallax angle for that day (respectively, φmax1 and φmax2) is known. Considering that both φmax1 and φmax2 are derived from a linear approximation that necessarily makes one of them underestimated and the other overestimated (a happy circumstance duly explained in correspondent published papers), an accurate final φmax value results from averaging them:

φmax (φmax1 + φmax2) (5)

Figure 2: Opposition - The apparent motion in right ascension of a minor planet. Most of the time it displaces towards higher RA values (direct motion) but once a year, as it approaches the closest distance to Earth, it appears moving backwards to lower RA values for a while (retrograde motion).

Figure 3: Before Opposition - The real apparent retrograde motion in RA of a minor planet before opposition. The instant speed continues to increase until opposition is reached.

Figure 4: After Opposition - The real apparent retrograde motion in RA of a minor planet after opposition. The instant speed continues to decrease until the loop is ended and direct motion reassumed.