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Inconstant Moon

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<center> The Moon at Perigee and Apogee



sidebyside.jpg </center>

<hr> One of my favourite science fiction stories is Larry Niven's <cite>Inconstant Moon</cite>, about a night when the full Moon shone brighter than ever before. I won't say any more about the story so as not to spoil it for those who have yet to discover this most atypical gem in Niven's vast treasure chest. Find it; read it; enjoy!

Everybody notices the phases of the Moon, but to most people every full Moon is alike—the rising or setting Moon looks large due to perspective's playing tricks on the eye, but surely the full Moon high in the sky is always the same, right? Wrong.

One of the most spectacular phenomena in naked-eye astronomy escapes notice by the vast majority of people simply because the the eye and brain can't compare the size and brightness of objects observed on separate occasions. This page explores the inconstant Moon in our everyday sky. While not as dramatic as that conjured up by the imagination of Larry Niven, we'll discover in it a celestial phenomenon seen by everybody, yet observed by only a few individuals.

Earth's Eccentric Companion


The Moon's orbit around the Earth is elliptical, with a substantial eccentricity (as major Solar System bodies go) of 5.49%. In addition, the tidal effect of the Sun's gravitational field increases the eccentricity when the orbit's major axis is aligned with the Sun-Earth vector or, in other words, the Moon is full or new. The combined effects of orbital eccentricity and the Sun's tides result in a substantial difference in the apparent size and brightness of the Moon at perigee and apogee. Extreme values for perigee and apogee distance occur when perigee or apogee passage occurs close to new or full Moon, and long-term extremes are in the months near to Earth's perihelion passage (closest approach to the Sun, when the Sun's tidal effects are strongest) in the first few days of January.

The image above shows how strikingly different the Moon appears at a full-Moon perigee and apogee. Most people don't notice the difference because they see the Moon in a sky that offers no reference by which angular extent may be judged. To observe the difference, you have to either make a scale to measure the Moon, or else photograph the Moon at perigee and apogee and compare the pictures, as I've done here.

The following table shows larger images of perigean and apogean full Moons, with details of the position of the Moon at the moment the pictures were taken. If your screen can't display the images one above another, use the side by side image above to appreciate the difference in size.

.... colspan="2">Views from Mill Valley, CA, USA, 37°54' N 122°32' W; all times UTC.....> <table> <tbody><tr> </tr> <tr> <td valign="bottom">perigee.jpg</td> <td valign="top">




Date/time: 1987 August 10 08:00

Julian day: 2447017.83


<dl><dt>Moon:</dt><dd>Age: 15 Days, 19 Hours

Phase: 98%

Full: 1987 August 9 10:18

Perigee: 1987 August 8 19:00, 357643 km

</dd><dt>Geocentric:</dt><dd>Distance: 359861 km

Right ascension: 22h 12m

Declination: −14° 7.1'

</dd><dt>Topocentric:</dt><dd>Distance: 359000 km

Angle subtended: 0.5548°

Altitude: 60.16°

Azimuth −68.13°

</dd></dl> </td></tr> <tr> <td>


Click on images for full resolution picture.


</td> </tr> <tr> <td valign="top">apogee.jpg</td> <td valign="top">




Date/time: 1988 February 2 06:00

Julian day: 2447193.75


<dl><dt>Moon:</dt><dd>Age: 14 Days, 5 Hours

Phase: 99%

Full: 1988 February 2 20:52

Apogee: 1988 February 3 10:00, 406395 km

</dd><dt>Geocentric:</dt><dd>Distance: 405948 km

Right ascension: 8h 37m

Declination: +22° 30.1'

</dd><dt>Topocentric:</dt><dd>Distance: 404510 km

Angle subtended: 0.4923°

Altitude: 35.45°

Azimuth −22.01°

</dd></dl> </td></tr> </tbody></table> Are the Pictures Accurate?


Since we can determine the position of the Moon at the time the exposures were made, it's possible to verify whether the resulting images agree with our calculations. To do this, we first measure the size of the Moon's disc in the perigee and apogee images, then compute the size ratio. This should agree, within the accuracy of the measurement, with the ratio of angular sizes computed from the distance of the Moon when the respective photos were shot. To avoid errors due to the Moon's not being perfectly full in either of the images, we'll measure the vertical extent of the disc, which is fully illuminated. Due to its inherent roughness, resolution limits of the optics and film, and distortion caused by turbulence in the Earth's atmosphere (“seeing”), the Moon's limb is not perfectly sharp in these pictures, so some judgement enters into the measuring process. Trying to use a consistent ratio of brightness on the two images, I measure the Moon in the perigee image to be 363 pixels high and the Moon at apogee to be 323 pixels, yielding a perigee/apogee ratio of 1.1238. I believe these size estimates are correct within ±1 pixel, giving tolerance limits on the ratio of 1.1173 to 1.1304.

For sufficiently small angles, the sine of an angle is approximated closely by the angle in radians. The Moon's angular extent viewed from Earth is small enough that this approximation is adequate for this calculation, so we can simply use the ratio of viewing distances as a proxy for the Moon's angular size. Dividing the apogee distance by that of the perigee gives 405948/359861 = 1.1281, in close agreement with the ratio of image sizes.

But we can do better than this: the perigee and apogee distances are calculated based on the distance between the centres of the Earth and Moon. Now from any sufficiently distant viewpoint the distance to the Moon's limb is essentially the same as that to its centre, but an observer on the surface of the Earth is necessarily closer to the Moon than the centre of the Earth. While the Earth's surface is not an ideal place to do astronomy, it sure beats setting up your telescope at the Earth's core, where 6378 km of rock attenuates even the brightest moonlight something terrible! So, what we're really interested in is not how far the Moon was from the centre of the Earth (its geocentric coordinates), but how far it was from the telescope when each picture was made. This is not an insignificant consideration: an observer at the equator observing the Moon at zenith is 1.8% closer to the Moon's limb than an observer 90° east or west in longitude, watching the Moon set or rise at the same moment.

What we want, then, is the position of the Moon relative to the observer, its topographic coordinates for the observing site. An easy way to calculate this is to transform the Moon's position in the spherical geocentric coordinate system into rectangular (Cartesian, or XYZ) coordinates with the origin at the centre of the Earth. The observer's position in the same coordinate system is easily calculated from the latitude and longitude of the observing site and, if you want to be as precise as possible, the distance from the centre of the Earth to the observing site, taking into account the Earth's ellipsoidal shape and the site's altitude above mean sea level. Then the distance from the observer, (X<sub>O</sub>, Y<sub>O</sub>, Z<sub>O</sub>), to the Moon, (X<sub>M</sub>, Y<sub>M</sub>, Z<sub>M</sub>), can be calculated with the distance formula for rectangular coordinates:

<center> cartdist.gif </center> This calculation gives an observer to Moon's limb distance of 404510 kilometres for the apogee image and 359000 km for the perigee image, with a perigee to apogee ratio of 1.1268, even closer to the best estimate of the image size ratio, 1.1238.

A Sense of Scale


<center> toscale.gif

The Earth-Moon System to Scale, 650 km/pixel </center> Space is called “space” because there's so much space there. Astronauts who flew to the Moon were struck by how the Earth and Moon seemed tiny specks in an infinite, empty void. So large are the voids that separate celestial bodies that most illustrations exaggerate the size of the objects to avoid rendering them as invisible dots. Compared to most other moons in the Solar System (Pluto's moon Charon is a notable exception), the Earth's Moon is very large compared to the planet it orbits, so it is possible, just barely, to draw the Earth-Moon system to scale in a form that will fit on a typical computer screen. The image above shows the Earth at the left and the Moon at the right, as they would appear to an observer looking from the direction of the Sun when the Moon is at first quarter; both worlds are fully illuminated (as is always the case when viewing from sunward, of course), and the Moon is at its maximum elongation from the Earth. Earth's orbital motion is toward the left, with the arrow at the top showing how far the Earth and Moon travel along their common orbit about the Sun every hour.

On this scale, all human spaceflight with the exception of the Apollo lunar missions has been confined to a region of two pixels surrounding the Earth; seeing the Moon's orbit in its true scale brings home how extraordinary an undertaking the Apollo project was. Of all the human beings who have lived on Earth since the origin of our species, only 24 have ventured outside that thin shell surrounding our Home Planet. Even the orbit which geosynchronous communications satellites occupy is only a little more than a tenth of the way to the Moon.

The mean distance to the moon, 384401 km, is the semimajor axis of its elliptical orbit. The closest perigee in the years 1750 through 2125 was 356375 km on 4th January 1912; the most distant apogee in the same period will be 406720 km on 3rd February 2125 (have your camera ready!). These extrema are marked on the chart, although in reality extreme perigees and apogees always occur close to a new or full Moon, not at a quarter phase as illustrated here. The mean distance is not equidistant between the minimum and maximum because the Sun's gravity perturbs the orbit away from a true ellipse. Although the absolute extremes are separated by many years, almost every year has a perigee and apogee close enough to the absolute limits to be indistinguishable at this scale.

The Moon's orbit is inclined 5.145396° with regard to the ecliptic (the plane in which the Earth's orbit around the Sun lies or, more precisely, the plane in which the centre of gravity of the Earth-Moon system [its barycentre] orbits the Sun), so as seen from the centre of the Earth the Moon drifts up and down slightly more than five degrees in the course of each orbit. The dark grey wedge shows the limits of the Moon's excursion above and below the plane of the ecliptic.

The Moon's orbital inclination, combined with the inclination of the Earth's axis of rotation, causes the Moon's declination, as observed from the Earth, to vary between ±28.5° when the Moon's inclination adds to that of the Earth, and ±18° when the two inclinations oppose one another; the maxima and minima of declination repeat every 18.6 years, the period in which the ascending node of the Moon's orbit precesses through a full circle.

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