Conversion from Minutes of arc to Seconds of arc

Information about the Minute of arc (arcmin)

The minute of arc is a unit of plane angle equal to one sixtieth of a degree. Its symbol is arcmin, or a single prime after the number, and it is also called the arcminute or the minute of angle. A full circle contains 21,600 of them.

It is the reason the nautical mile has the length it does. One minute of arc measured along a meridian on the Earth's surface is almost exactly one nautical mile, so a navigator reading latitude off a chart can convert directly to distance without arithmetic. The relationship is not perfectly exact because the Earth is not a perfect sphere, but the modern nautical mile of 1852 metres was chosen to sit close to the mean value.

Human vision resolves detail down to roughly one minute of arc, which corresponds to the standard used in eye tests: the smallest letters on the twenty-twenty line subtend five minutes of arc in total, with each stroke one minute wide. That limit comes from the spacing of photoreceptors in the retina and from the diffraction limit of the pupil, and the two happen to agree closely.

Rifle shooters use the unit for accuracy, where a group of one minute of angle spreads about 1.047 inches at a hundred yards, commonly rounded to one inch. Astronomy uses it for the apparent sizes of objects: the Moon and the Sun both subtend about thirty minutes of arc, which is why a total solar eclipse is possible at all.

Astronomy used it as the limit of what could be achieved before the telescope. Tycho Brahe, working with large graduated quadrants in the 1580s, measured stellar positions to about one minute of arc, and that accuracy was precisely what allowed Kepler to show that the orbit of Mars is an ellipse rather than a circle. The unit also gives a convenient handle on the sky: the Sun and the Moon both subtend about 30 minutes of arc, so half a degree is the width of either disc, and a hand held at arm's length covers several degrees. Estimating an angle in the sky against the width of the Moon is a habit older than any instrument.

One minute of arc equals one sixtieth of a degree, 60 seconds of arc, or about 0.000290888 radians.

Information about the Second of arc (arcsec)

The second of arc is a unit of plane angle equal to one sixtieth of a minute of arc, or one three-thousand-six-hundredth of a degree. Its symbol is arcsec, or a double prime after the number, and it is also called the arcsecond. A full circle contains 1,296,000 of them.

It is the working unit of positional astronomy. Star positions, proper motions and the apparent diameters of planets are all quoted in arcseconds, and the best ground-based telescopes are limited by atmospheric turbulence to a resolution of about half an arcsecond on a good night. Space telescopes escape that limit; the Hubble Space Telescope resolves detail around one twentieth of an arcsecond.

The parsec, the standard unit of astronomical distance, is defined directly from it. A star one parsec away shows an annual parallax of exactly one arcsecond as the Earth moves from one side of its orbit to the other, and the name is a contraction of parallax and second. No star is that close, so every measured stellar parallax is smaller than one arcsecond, and the Gaia spacecraft measures some to a few millionths of one.

On Earth, an arcsecond of latitude is about 31 metres, which is why coordinates written in degrees, minutes and seconds need decimal fractions of a second to locate a building. Surveying instruments and telescope mounts are specified by how many arcseconds of error they carry, and precision optical work uses the unit for angular tolerances.

The atmosphere sets the practical limit for a telescope on the ground. Turbulence in the air smears a star into a blur typically one arcsecond across, a quantity astronomers call the seeing, and the finest mountain sites reach a few tenths of that on a good night. Beating it requires either going above the air or correcting for it in real time with a deformable mirror. Space telescopes and modern astrometry work far below the arcsecond: the Gaia spacecraft measures stellar positions to tens of microarcseconds, which is the angle a coin would subtend on the Moon, and radio interferometers spanning continents do better still.

One second of arc equals one sixtieth of a minute of arc, one three-thousand-six-hundredth of a degree, or about 4.84814 millionths of a radian.