Vol. Science · Astronomy · Apparent size
A disc’s apparent size is an angle. For a sphere of physical diameter d at distance D from the observer to the center, that angle is δ = 2·arctan(d / 2D). The small-angle line beside it is d/D in radians. The Moon’s volumetric mean diameter, twice the Moon Fact Sheet radius 1,737.4 km, is 3,474.8 km. At mean perigee, 0.3633 × 106 km, the disc is 32.880′ (0.5480°). The Sun’s volumetric mean diameter, twice the Sun Fact Sheet radius 695,700 km, is 1,391,400 km. At Earth’s aphelion, 152.100 × 106 km on the Earth Fact Sheet, the solar disc is 31.448′ (0.5241°). That Moon is wider: ratio 1.0455, so a perfect lineup can be total. At mean apogee, 0.4055 × 106 km, the Moon is 29.458′ and a perihelion Sun is 32.518′. Ratio 0.9059. The Moon no longer covers it, and the lineup is annular. The tilt that makes most new Moons miss is the eclipses desk. The phase that night is the Moon desk. The peek past the limb is libration.
Classroom schematic. One angle, drawn as a disc scaled to δ. A dashed ring, when the angle is small enough to share the frame, is 0.5°. The discs are the angle, not a photograph and not a map. Fact-sheet presets use the published radii and distances named above. Thumb, coin, and the half-degree mark are approximate examples, not measurements. Not an ephemeris. Not a prediction of the next eclipse.
Disc scaled to the angle · dashed ring is 0.5°
Enter a diameter greater than zero.
Enter a distance greater than zero.
Fact sheet
Approximate examples
Same scale · side by side · then on one center
Moon distance · diameter fixed at 3,474.8 km
Sun distance · diameter fixed at 1,391,400 km
The formula
δ = 2·arctan(d / 2D). d is the physical diameter. D is the distance from the observer to the center. The result is in radians, then converted to degrees, arcminutes, and arcseconds. This is the exact angular diameter of a sphere seen from outside.
δ ≈ d/D. The small-angle line, in radians. For the Moon at mean perigee it sits 0.0008% above the exact angle. It is the right classroom shortcut near half a degree. It drifts when the object fills more of the view: if d = D, the exact angle is 53.1301° and d/D is 1 radian, 57.2958°, several percent high.
The inputs. Moon volumetric mean radius 1,737.4 km, so the diameter used here is 3,474.8 km. Equatorial radius 1,738.1 km is on the same sheet and is not the diameter in the presets. Mean perigee 0.3633 × 106 km. Mean apogee 0.4055 × 106 km. Semimajor axis 0.3844 × 106 km. Sun volumetric mean radius 695,700 km, so the diameter is 1,391,400 km. Earth perihelion 147.095 × 106 km. Aphelion 152.100 × 106 km. Semimajor axis 149.598 × 106 km. At the two semimajor axes the Moon is 31.075′ and the Sun is 31.974′. Ratio 0.9719, slightly under 1. They look the same size, and a mean lineup is annular.
Ratio greater than 1, and less than 1. Divide the Moon’s angular diameter by the Sun’s. Above 1, the Moon can cover the photosphere: a total solar eclipse along the center of a perfect lineup. Below 1, a ring of Sun remains: an annular eclipse. With these mean perigee and apogee distances, every perigee pairing on the card is total and every apogee pairing is annular. The discs match at 367,346 km when the Sun is at perihelion, and at 379,846 km when the Sun is at aphelion. Angular size does not say whether the new Moon is on the line. The orbit is tilted about 5°. That question is the eclipses desk.
Approximate examples. A thumb about 2 cm wide at about 57 cm is 2.0102°. A coin about 2.4 cm across at about 70 cm is 1.9642°. A 5 mm mark at about 57 cm is 0.5026°, the half-degree rule of thumb. 57 cm is the convenient length where 1 cm subtends about 1°, because 1° is about 1/57 of a radian. None of these is a measured hand or a mint specification. They are classroom marks, so a 2° thumb reads as about four times the Moon.
The sheet’s own apparent diameters. The Moon sheet prints 1,896″ at 378,000 km. This diameter gives 1,896.09″, which rounds to that line. The Sun sheet prints a maximum of 1,952″, a minimum of 1,887″, and 1,919″ at 1 AU. This radius at Earth’s perihelion, aphelion, and semimajor axis gives 1,951.08″, 1,886.88″, and 1,918.44″. The minimum rounds to the sheet’s 1,887″. The other two sit under the sheet’s integers by less than 1″. The presets use the radii and the distances. They do not adjust a radius to force the integer.
Literacy · an angle, not a forecast
Sources · public NASA pages
Classroom picture of angular size. δ = 2·arctan(d / 2D), and d/D beside it. Moon at mean perigee: 32.880′, 0.5480°, 1,972.81″. Moon at mean apogee: 29.458′. Sun at perihelion: 32.518′, 1,951.08″. Sun at aphelion: 31.448′, 1,886.88″. Perigee Moon over perihelion Sun is 1.0111, total. Apogee Moon over perihelion Sun is 0.9059, annular. The perigee disc is 11.616% wider than the apogee disc. Thumb, coin, and the 5 mm mark are approximate. Not an ephemeris, and not a forecast. Earthrise preview: NASA / Bill Anders (public domain).
Keep going
The shadow
Why a new or full Moon is not an eclipse. The orbit is tilted about 5°. This page only says whether a perfect lineup is total or annular.
The phase
New through full. Angular size is a separate question from how much of the disc is lit.
The limbs
The same face stays toward Earth. An eccentric orbit and a tilt let us peek past the edge.
Neighborhood
Where the Sun, Earth, and Moon sit. Still a schematic.
One angle, δ = 2·arctan(d / 2D), and the small-angle d/D beside it. Fact-sheet presets use NASA radii and distances. Thumb, coin, and the half-degree mark are approximate examples. A classroom schematic, not an ephemeris and not an eclipse forecast.
| Diameter d | Physical diameter. The slider runs from 1 mm to 2 million km, logarithmic, about 0.3% per step. The number field is the exact figure. Units: mm, cm, m, km. |
|---|---|
| Distance D | Distance from the observer to the center. The slider runs from 5 cm to 300 million km. Same units. A fact-sheet value stays exact in the field even when the slider step is nearby rather than on it. |
| Degrees | The exact angle, δ, in decimal degrees. |
| Arcminutes | The same angle in arcminutes. 60′ = 1°. The Moon and the Sun sit near 30′. |
| Arcseconds | The same angle in arcseconds. 60″ = 1′. 3,600″ = 1°. |
| Exact radians | 2·arctan(d / 2D). This is the angle the degrees, arcminutes, and arcseconds come from. |
| d/D | The small-angle approximation, in radians. For the Moon it is a fraction of a thousandth of a percent high. It drifts when d is a large share of D. |
| Moon perigee | Diameter 3,474.8 km. Distance 363,300 km, the sheet’s mean perigee, 0.3633 × 106 km. 32.880′. |
|---|---|
| Moon apogee | Same diameter. Distance 405,500 km, mean apogee, 0.4055 × 106 km. 29.458′. |
| Moon mean | Same diameter. Semimajor axis 384,400 km, 0.3844 × 106 km. 31.075′. |
| Sun perihelion | Diameter 1,391,400 km. Earth’s perihelion, 147.095 × 106 km. 32.518′. 1,951.08″. |
| Sun aphelion | Same solar diameter. Earth’s aphelion, 152.100 × 106 km. 31.448′. 1,886.88″, which rounds to the Sun sheet’s minimum of 1,887″. |
| Sun mean | Same solar diameter. Earth’s semimajor axis, 149.598 × 106 km. 31.974′. 1,918.44″. |
| Thumb, approx. | About 2 cm at about 57 cm. 2.0102°. Approximate example, not a measured hand. 57 cm is where 1 cm subtends about 1°. |
| Coin, approx. | About 2.4 cm at about 70 cm. 1.9642°. Approximate example, not a mint specification. Several times the Moon. |
| 0.5° rule | A 5 mm mark at about 57 cm. 0.5026°. Approximate example of the half-degree rule: about the Moon, about the Sun. |
| Two discs | Moon and Sun drawn at one scale, so a few percent of angular size is visible. Arcminutes are printed under each. |
|---|---|
| Same center | The third drawing puts them on one center. A dashed gold circle is the Sun’s edge. If the Moon is larger, it covers. If the Moon is smaller, a gold ring remains. |
| Ratio | Moon angular diameter divided by Sun angular diameter. Greater than 1: a perfect lineup can be total. Less than 1: annular. The four corners of mean perigee, mean apogee, perihelion, and aphelion are listed on the card. |
| Distances | Perigee, mean, and apogee for the Moon. Perihelion, mean, and aphelion for the Sun. Diameters stay at the fact-sheet values. These buttons do not move the sliders above. |
| Presets | Tab to a button. Enter or Space selects it. Left and Right step through the fact-sheet presets, or through the approximate examples, when focus is on one of those buttons. The pressed button is marked aria-pressed. |
|---|---|
| Sliders | Arrow keys nudge the focused slider. The number field is where a fact-sheet figure is typed exactly. Shift is not a special step. |
| Touch | Preset buttons, the question mark, Close, the number fields, and the unit menus are at least 44 pixels on a side. The drawing does not drag. |
| This dialog | The question mark opens it. Escape closes it. Tab stays inside until you close it. Click the dim page behind the card to close it. |
| 0.5° ring | Drawn when the angle is at most 2.5°, on the same scale as the disc. Wider angles scale to fit, and the caption says so. The readout stays the true angle. |
| Inside | If D is less than or equal to d/2, you are inside the sphere. The card says so. The formula is for an outside view. |