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69-409: To measure the index, one observes the magnitude of an object successively through two different filters , such as U and B, or B and V, where U is sensitive to ultraviolet rays, B is sensitive to blue light, and V is sensitive to visible (green-yellow) light (see also: UBV system ). The set of passbands or filters is called a photometric system . The difference in magnitudes found with these filters

138-475: A black body gives the value for luminosity for a black body, an idealized object which is perfectly opaque and non-reflecting: L = σ A T 4 , {\displaystyle L=\sigma AT^{4},} where A is the surface area, T is the temperature (in kelvins) and σ is the Stefan–Boltzmann constant , with a value of 5.670 374 419 ... × 10  W⋅m ⋅K . Imagine

207-460: A radio source is measured in W Hz , to avoid having to specify a bandwidth over which it is measured. The observed strength, or flux density , of a radio source is measured in Jansky where 1 Jy = 10 W m Hz . For example, consider a 10   W transmitter at a distance of 1 million metres, radiating over a bandwidth of 1 MHz. By the time that power has reached the observer, the power

276-408: A 1 Jy signal from a radio source at a redshift of 1, at a frequency of 1.4 GHz. Ned Wright's cosmology calculator calculates a luminosity distance for a redshift of 1 to be 6701 Mpc = 2×10 m giving a radio luminosity of 10 × 4 π (2×10 ) / (1 + 1) = 6×10 W Hz . To calculate the total radio power, this luminosity must be integrated over the bandwidth of the emission. A common assumption

345-557: A certain Bigness, which is one of the Six; but rather in reality there are almost as many Orders of Stars , as there are Stars , few of them being exactly of the same Bigness and Lustre. And even among those Stars which are reckoned of the brightest Class, there appears a Variety of Magnitude; for Sirius or Arcturus are each of them brighter than Aldebaran or the Bull's Eye, or even than

414-469: A distance of 1 AU. These are referred to with a capital H symbol. Since these objects are lit primarily by reflected light from the Sun, an H magnitude is defined as the apparent magnitude of the object at 1 AU from the Sun and 1 AU from the observer. The following is a table giving apparent magnitudes for celestial objects and artificial satellites ranging from the Sun to the faintest object visible with

483-486: A few million years for the most extreme. In the Hertzsprung–Russell diagram , the x-axis represents temperature or spectral type while the y-axis represents luminosity or magnitude. The vast majority of stars are found along the main sequence with blue Class O stars found at the top left of the chart while red Class M stars fall to the bottom right. Certain stars like Deneb and Betelgeuse are found above and to

552-464: A less prominent star such as Mizar , which in turn appears larger than a truly faint star such as Alcor . In 1736, the mathematician John Keill described the ancient naked-eye magnitude system in this way: The fixed Stars appear to be of different Bignesses, not because they really are so, but because they are not all equally distant from us. Those that are nearest will excel in Lustre and Bigness;

621-438: A logarithmic scale of √ 100 ≈ 2.512 be adopted between magnitudes, so five magnitude steps corresponded precisely to a factor of 100 in brightness. Every interval of one magnitude equates to a variation in brightness of √ 100 or roughly 2.512 times. Consequently, a magnitude 1 star is about 2.5 times brighter than a magnitude 2 star, about 2.5 times brighter than a magnitude 3 star, about 2.5 times brighter than

690-428: A luminosity around 200,000 L ⊙ , a spectral type of A2, and an effective temperature around 8,500 K, meaning it has a radius around 203  R ☉ (1.41 × 10   m ). For comparison, the red supergiant Betelgeuse has a luminosity around 100,000 L ⊙ , a spectral type of M2, and a temperature around 3,500 K, meaning its radius is about 1,000  R ☉ (7.0 × 10   m ). Red supergiants are

759-431: A magnitude 4 star, and so on. This is the modern magnitude system, which measures the brightness, not the apparent size, of stars. Using this logarithmic scale, it is possible for a star to be brighter than “first class”, so Arcturus or Vega are magnitude 0, and Sirius is magnitude −1.46. As mentioned above, the scale appears to work 'in reverse', with objects with a negative magnitude being brighter than those with

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828-509: A point source of light of luminosity L {\displaystyle L} that radiates equally in all directions. A hollow sphere centered on the point would have its entire interior surface illuminated. As the radius increases, the surface area will also increase, and the constant luminosity has more surface area to illuminate, leading to a decrease in observed brightness. F = L A , {\displaystyle F={\frac {L}{A}},} where The surface area of

897-556: A positive magnitude. The more negative the value, the brighter the object. Objects appearing farther to the left on this line are brighter, while objects appearing farther to the right are dimmer. Thus zero appears in the middle, with the brightest objects on the far left, and the dimmest objects on the far right. Two of the main types of magnitudes distinguished by astronomers are: The difference between these concepts can be seen by comparing two stars. Betelgeuse (apparent magnitude 0.5, absolute magnitude −5.8) appears slightly dimmer in

966-531: A reference point, and these are called "absolute" reference systems. Current absolute reference systems include the AB magnitude system, in which the reference is a source with a constant flux density per unit frequency, and the STMAG system, in which the reference source is instead defined to have constant flux density per unit wavelength. Another logarithmic measure for intensity is the level, in decibel . Although it

1035-400: A sphere with radius r is A = 4 π r 2 {\displaystyle A=4\pi r^{2}} , so for stars and other point sources of light: F = L 4 π r 2 , {\displaystyle F={\frac {L}{4\pi r^{2}}}\,,} where r {\displaystyle r} is the distance from the observer to

1104-568: A star independent of distance. The concept of magnitude, on the other hand, incorporates distance. The apparent magnitude is a measure of the diminishing flux of light as a result of distance according to the inverse-square law . The Pogson logarithmic scale is used to measure both apparent and absolute magnitudes, the latter corresponding to the brightness of a star or other celestial body as seen if it would be located at an interstellar distance of 10 parsecs (3.1 × 10 metres ). In addition to this brightness decrease from increased distance, there

1173-532: A star that was larger for brighter stars and smaller for fainter ones. Astronomers from Galileo to Jaques Cassini mistook these spurious disks for the physical bodies of stars, and thus into the eighteenth century continued to think of magnitude in terms of the physical size of a star. Johannes Hevelius produced a very precise table of star sizes measured telescopically, but now the measured diameters ranged from just over six seconds of arc for first magnitude down to just under 2 seconds for sixth magnitude. By

1242-609: A subscript to indicate the passband. For example, M V is the magnitude at 10 parsecs in the V passband. A bolometric magnitude (M bol ) is an absolute magnitude adjusted to take account of radiation across all wavelengths; it is typically smaller (i.e. brighter) than an absolute magnitude in a particular passband, especially for very hot or very cool objects. Bolometric magnitudes are formally defined based on stellar luminosity in watts , and are normalised to be approximately equal to M V for yellow stars. Absolute magnitudes for Solar System objects are frequently quoted based on

1311-533: Is a measure of the brightness of an object , usually in a defined passband . An imprecise but systematic determination of the magnitude of objects was introduced in ancient times by Hipparchus . Magnitude values do not have a unit. The scale is logarithmic and defined such that a magnitude 1 star is exactly 100 times brighter than a magnitude 6 star. Thus each step of one magnitude is 100 5 ≈ 2.512 {\displaystyle {\sqrt[{5}]{100}}\approx 2.512} times brighter than

1380-486: Is also used in relation to particular passbands such as a visual luminosity of K-band luminosity. These are not generally luminosities in the strict sense of an absolute measure of radiated power, but absolute magnitudes defined for a given filter in a photometric system . Several different photometric systems exist. Some such as the UBV or Johnson system are defined against photometric standard stars, while others such as

1449-526: Is an extra decrease of brightness due to extinction from intervening interstellar dust. By measuring the width of certain absorption lines in the stellar spectrum , it is often possible to assign a certain luminosity class to a star without knowing its distance. Thus a fair measure of its absolute magnitude can be determined without knowing its distance nor the interstellar extinction. In measuring star brightnesses, absolute magnitude, apparent magnitude, and distance are interrelated parameters—if two are known,

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1518-674: Is called the U−B or B−V color index respectively. In principle, the temperature of a star can be calculated directly from the B−V index, and there are several formulae to make this connection. A good approximation can be obtained by considering stars as black bodies , using Ballesteros' formula (also implemented in the PyAstronomy package for Python): Color indices of distant objects are usually affected by interstellar extinction , that is, they are redder than those of closer stars. The amount of reddening

1587-485: Is characterized by color excess , defined as the difference between the observed color index and the normal color index (or intrinsic color index ), the hypothetical true color index of the star, unaffected by extinction. For example, in the UBV photometric system we can write it for the B−V color: The passbands most optical astronomers use are the UBVRI filters, where the U, B, and V filters are as mentioned above,

1656-612: Is measured either in the SI units, watts , or in terms of solar luminosities ( L ☉ ). A bolometer is the instrument used to measure radiant energy over a wide band by absorption and measurement of heating. A star also radiates neutrinos , which carry off some energy (about 2% in the case of the Sun), contributing to the star's total luminosity. The IAU has defined a nominal solar luminosity of 3.828 × 10  W to promote publication of consistent and comparable values in units of

1725-401: Is more commonly used for sound intensity, it is also used for light intensity. It is a parameter for photomultiplier tubes and similar camera optics for telescopes and microscopes. Each factor of 10 in intensity corresponds to 10 decibels. In particular, a multiplier of 100 in intensity corresponds to an increase of 20 decibels and also corresponds to a decrease in magnitude by 5. Generally,

1794-401: Is most likely to match those measurements. In some cases, the process of estimation is extreme, with luminosities being calculated when less than 1% of the energy output is observed, for example with a hot Wolf-Rayet star observed only in the infrared. Bolometric luminosities can also be calculated using a bolometric correction to a luminosity in a particular passband. The term luminosity

1863-475: Is present, a condition that usually arises because of gas and dust present in the interstellar medium (ISM), the Earth's atmosphere , and circumstellar matter . Consequently, one of astronomy's central challenges in determining a star's luminosity is to derive accurate measurements for each of these components, without which an accurate luminosity figure remains elusive. Extinction can only be measured directly if

1932-522: Is really a measure of illuminance , which can also be measured in photometric units such as lux . The Greek astronomer Hipparchus produced a catalogue which noted the apparent brightness of stars in the second century BCE. In the second century CE the Alexandrian astronomer Ptolemy classified stars on a six-point scale, and originated the term magnitude . To the unaided eye, a more prominent star such as Sirius or Arcturus appears larger than

2001-411: Is spread over the surface of a sphere with area 4 πr or about 1.26×10 m , so its flux density is 10 / 10 / (1.26×10 ) W m Hz = 8×10 Jy . More generally, for sources at cosmological distances, a k-correction must be made for the spectral index α of the source, and a relativistic correction must be made for the fact that the frequency scale in the emitted rest frame is different from that in

2070-399: Is synonymous with the radiant power emitted by a light-emitting object. In astronomy , luminosity is the total amount of electromagnetic energy emitted per unit of time by a star , galaxy , or other astronomical objects . In SI units, luminosity is measured in joules per second, or watts . In astronomy, values for luminosity are often given in the terms of the luminosity of

2139-415: Is the absolute magnitude. If the line of sight between the object and observer is affected by extinction due to absorption of light by interstellar dust particles , then the object's apparent magnitude will be correspondingly fainter. For A magnitudes of extinction, the relationship between apparent and absolute magnitudes becomes Stellar absolute magnitudes are usually designated with a capital M with

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2208-447: Is the luminosity in W Hz , S obs is the observed flux density in W m Hz , D L is the luminosity distance in metres, z is the redshift, α is the spectral index (in the sense I ∝ ν α {\displaystyle I\propto {\nu }^{\alpha }} , and in radio astronomy, assuming thermal emission the spectral index is typically equal to 2. ) For example, consider

2277-511: Is to set the bandwidth to the observing frequency, which effectively assumes the power radiated has uniform intensity from zero frequency up to the observing frequency. In the case above, the total power is 4×10 × 1.4×10 = 5.7×10 W . This is sometimes expressed in terms of the total (i.e. integrated over all wavelengths) luminosity of the Sun which is 3.86×10 W , giving a radio power of 1.5×10 L ⊙ . The Stefan–Boltzmann equation applied to

2346-408: The AB system are defined in terms of a spectral flux density . A star's luminosity can be determined from two stellar characteristics: size and effective temperature . The former is typically represented in terms of solar radii , R ⊙ , while the latter is represented in kelvins , but in most cases neither can be measured directly. To determine a star's radius, two other metrics are needed:

2415-555: The James Webb Space Telescope (JWST) : Any magnitude systems must be calibrated to define the brightness of magnitude zero. Many magnitude systems, such as the Johnson UBV system, assign the average brightness of several stars to a certain number to by definition, and all other magnitude measurements are compared to that reference point. Other magnitude systems calibrate by measuring energy directly, without

2484-614: The Star in Spica ; and yet all these Stars are reckoned among the Stars of the first Order: And there are some Stars of such an intermedial Order, that the Astronomers have differed in classing of them; some putting the same Stars in one Class, others in another. For Example: The little Dog was by Tycho placed among the Stars of the second Magnitude, which Ptolemy reckoned among

2553-454: The Stars of the first Class: And therefore it is not truly either of the first or second Order, but ought to be ranked in a Place between both. Note that the brighter the star, the smaller the magnitude: Bright "first magnitude" stars are "1st-class" stars, while stars barely visible to the naked eye are "sixth magnitude" or "6th-class". The system was a simple delineation of stellar brightness into six distinct groups but made no allowance for

2622-406: The list of brightest stars ). For example, Sirius is magnitude −1.46, Arcturus is −0.04, Aldebaran is 0.85, Spica is 1.04, and Procyon is 0.34. Under the ancient magnitude system, all of these stars might have been classified as "stars of the first magnitude". Magnitudes can also be calculated for objects far brighter than stars (such as the Sun and Moon ), and for objects too faint for

2691-485: The R filter passes red light, and the I filter passes infrared light. This system of filters is sometimes called the Johnson–Kron–Cousins filter system, named after the originators of the system (see references). These filters were specified as particular combinations of glass filters and photomultiplier tubes . M. S. Bessell specified a set of filter transmissions for a flat response detector, thus quantifying

2760-550: The Sun , L ⊙ . Luminosity can also be given in terms of the astronomical magnitude system: the absolute bolometric magnitude ( M bol ) of an object is a logarithmic measure of its total energy emission rate, while absolute magnitude is a logarithmic measure of the luminosity within some specific wavelength range or filter band . In contrast, the term brightness in astronomy is generally used to refer to an object's apparent brightness: that is, how bright an object appears to an observer. Apparent brightness depends on both

2829-572: The absolute magnitude scale is actually defined as a fixed luminosity of 3.0128 × 10  W . Therefore, the absolute magnitude can be calculated from a luminosity in watts: M b o l = − 2.5 log 10 ⁡ L ∗ L 0 ≈ − 2.5 log 10 ⁡ L ∗ + 71.1974 {\displaystyle M_{\mathrm {bol} }=-2.5\log _{10}{\frac {L_{*}}{L_{0}}}\approx -2.5\log _{10}L_{*}+71.1974} where L 0

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2898-403: The actual and observed luminosities are both known, but it can be estimated from the observed colour of a star, using models of the expected level of reddening from the interstellar medium. In the current system of stellar classification , stars are grouped according to temperature, with the massive, very young and energetic Class O stars boasting temperatures in excess of 30,000  K while

2967-445: The brightest visible star in the night sky, −1.46. Venus at its brightest is -5. The International Space Station (ISS) sometimes reaches a magnitude of −6. Amateur astronomers commonly express the darkness of the sky in terms of limiting magnitude , i.e. the apparent magnitude of the faintest star they can see with the naked eye. At a dark site, it is usual for people to see stars of 6th magnitude or fainter. Apparent magnitude

3036-405: The calculation of the color indices. For precision, appropriate pairs of filters are chosen depending on the object's color temperature: B−V are for mid-range objects, U−V for hotter objects, and R−I for cool ones. Color indices can also be determined for other celestial bodies, such as planets and moons: The common color labels (e.g. red supergiant) are subjective and taken using the star Vega as

3105-437: The change in level is related to a change in magnitude by For example, an object that is 1 magnitude larger (fainter) than a reference would produce a signal that is 4 dB smaller (weaker) than the reference, which might need to be compensated by an increase in the capability of the camera by as many decibels. Luminosity Luminosity is an absolute measure of radiated electromagnetic energy per unit time, and

3174-401: The diameter of the full moon), with second through sixth magnitude stars measuring 1 + 1 ⁄ 2 ′, 1 + 1 ⁄ 12 ′, 3 ⁄ 4 ′, 1 ⁄ 2 ′, and 1 ⁄ 3 ′, respectively. The development of the telescope showed that these large sizes were illusory—stars appeared much smaller through the telescope. However, early telescopes produced a spurious disk-like image of

3243-399: The distances to stars via stellar parallax , and so understood that stars are so far away as to essentially appear as point sources of light. Following advances in understanding the diffraction of light and astronomical seeing , astronomers fully understood both that the apparent sizes of stars were spurious and how those sizes depended on the intensity of light coming from a star (this is

3312-423: The effective temperature is merely a number that represents the temperature of a black body that would reproduce the luminosity, it obviously cannot be measured directly, but it can be estimated from the spectrum. An alternative way to measure stellar luminosity is to measure the star's apparent brightness and distance. A third component needed to derive the luminosity is the degree of interstellar extinction that

3381-407: The human eye to see (such as Pluto ). Often, only apparent magnitude is mentioned since it can be measured directly. Absolute magnitude can be calculated from apparent magnitude and distance from: because intensity falls off proportionally to distance squared. This is known as the distance modulus , where d is the distance to the star measured in parsecs , m is the apparent magnitude, and M

3450-434: The intrinsic luminosity emitted by an object and is defined to be equal to the apparent magnitude that the object would have if it were placed at a certain distance, 10 parsecs for stars. A more complex definition of absolute magnitude is used for planets and small Solar System bodies , based on its brightness at one astronomical unit from the observer and the Sun. The Sun has an apparent magnitude of −27 and Sirius ,

3519-472: The largest type of star, but the most luminous are much smaller and hotter, with temperatures up to 50,000 K and more and luminosities of several million L ⊙ , meaning their radii are just a few tens of R ⊙ . For example, R136a1 has a temperature over 46,000 K and a luminosity of more than 6,100,000 L ⊙ (mostly in the UV), it is only 39  R ☉ (2.7 × 10   m ). The luminosity of

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3588-451: The less massive, typically older Class M stars exhibit temperatures less than 3,500 K. Because luminosity is proportional to temperature to the fourth power, the large variation in stellar temperatures produces an even vaster variation in stellar luminosity. Because the luminosity depends on a high power of the stellar mass, high mass luminous stars have much shorter lifetimes. The most luminous stars are always young stars, no more than

3657-410: The light source. For stars on the main sequence , luminosity is also related to mass approximately as below: L L ⊙ ≈ ( M M ⊙ ) 3.5 . {\displaystyle {\frac {L}{L_{\odot }}}\approx {\left({\frac {M}{M_{\odot }}}\right)}^{3.5}.} Luminosity is an intrinsic measurable property of

3726-435: The luminosity of the object and the distance between the object and observer, and also on any absorption of light along the path from object to observer. Apparent magnitude is a logarithmic measure of apparent brightness. The distance determined by luminosity measures can be somewhat ambiguous, and is thus sometimes called the luminosity distance . When not qualified, the term "luminosity" means bolometric luminosity, which

3795-467: The magnitude 1 higher. The brighter an object appears, the lower the value of its magnitude, with the brightest objects reaching negative values. Astronomers use two different definitions of magnitude: apparent magnitude and absolute magnitude . The apparent magnitude ( m ) is the brightness of an object and depends on an object's intrinsic luminosity , its distance , and the extinction reducing its brightness. The absolute magnitude ( M ) describes

3864-540: The more remote Stars will give a fainter Light, and appear smaller to the Eye. Hence arise the Distribution of Stars , according to their Order and Dignity, into Classes ; the first Class containing those which are nearest to us, are called Stars of the first Magnitude; those that are next to them, are Stars of the second Magnitude ... and so forth, 'till we come to the Stars of the sixth Magnitude, which comprehend

3933-711: The observed visible brightness from Earth which depends on the distance of the object. The absolute magnitude is the apparent magnitude at a distance of 10  pc (3.1 × 10   m ), therefore the bolometric absolute magnitude is a logarithmic measure of the bolometric luminosity. The difference in bolometric magnitude between two objects is related to their luminosity ratio according to: M bol1 − M bol2 = − 2.5 log 10 ⁡ L 1 L 2 {\displaystyle M_{\text{bol1}}-M_{\text{bol2}}=-2.5\log _{10}{\frac {L_{\text{1}}}{L_{\text{2}}}}} where: The zero point of

4002-420: The observer's rest frame . So the full expression for radio luminosity, assuming isotropic emission, is L ν = S o b s 4 π D L 2 ( 1 + z ) 1 + α {\displaystyle L_{\nu }={\frac {S_{\mathrm {obs} }4\pi {D_{L}}^{2}}{(1+z)^{1+\alpha }}}} where L ν

4071-467: The reference. However, these labels, which have a quantifiable basis, do not reflect how the human eye would perceive the colors of these stars. For instance, Vega has a bluish white color, while the Sun, from outer space, would look like a neutral white somewhat warmer than the illuminant D65 (which may be considered a slightly cool white). "Green" stars would be perceived as white by the human eye. Magnitude (astronomy) In astronomy , magnitude

4140-474: The right of the main sequence, more luminous or cooler than their equivalents on the main sequence. Increased luminosity at the same temperature, or alternatively cooler temperature at the same luminosity, indicates that these stars are larger than those on the main sequence and they are called giants or supergiants. Blue and white supergiants are high luminosity stars somewhat cooler than the most luminous main sequence stars. A star like Deneb , for example, has

4209-512: The sky than Alpha Centauri A (apparent magnitude 0.0, absolute magnitude 4.4) even though it emits thousands of times more light, because Betelgeuse is much farther away. Under the modern logarithmic magnitude scale, two objects, one of which is used as a reference or baseline, whose flux (i.e., brightness, a measure of power per unit area) in units such as watts per square metre (W m ) are F 1 and F ref , will have magnitudes m 1 and m ref related by Astronomers use

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4278-517: The smallest Stars that can be discerned with the bare Eye. For all the other Stars , which are only seen by the Help of a Telescope, and which are called Telescopical, are not reckoned among these six Orders. Altho' the Distinction of Stars into six Degrees of Magnitude is commonly received by Astronomers ; yet we are not to judge, that every particular Star is exactly to be ranked according to

4347-465: The solar luminosity. While bolometers do exist, they cannot be used to measure even the apparent brightness of a star because they are insufficiently sensitive across the electromagnetic spectrum and because most wavelengths do not reach the surface of the Earth. In practice bolometric magnitudes are measured by taking measurements at certain wavelengths and constructing a model of the total spectrum that

4416-430: The star's angular diameter and its distance from Earth. Both can be measured with great accuracy in certain cases, with cool supergiants often having large angular diameters, and some cool evolved stars having masers in their atmospheres that can be used to measure the parallax using VLBI . However, for most stars the angular diameter or parallax, or both, are far below our ability to measure with any certainty. Since

4485-473: The star's apparent brightness, which can be measured in units such as watts per square metre) so that brighter stars appeared larger. Early photometric measurements (made, for example, by using a light to project an artificial “star” into a telescope's field of view and adjusting it to match real stars in brightness) demonstrated that first magnitude stars are about 100 times brighter than sixth magnitude stars. Thus in 1856 Norman Pogson of Oxford proposed that

4554-546: The term "flux" for what is often called "intensity" in physics, in order to avoid confusion with the specific intensity . Using this formula, the magnitude scale can be extended beyond the ancient magnitude 1–6 range, and it becomes a precise measure of brightness rather than simply a classification system. Astronomers now measure differences as small as one-hundredth of a magnitude. Stars that have magnitudes between 1.5 and 2.5 are called second-magnitude; there are some 20 stars brighter than 1.5, which are first-magnitude stars (see

4623-449: The third can be determined. Since the Sun's luminosity is the standard, comparing these parameters with the Sun's apparent magnitude and distance is the easiest way to remember how to convert between them, although officially, zero point values are defined by the IAU. The magnitude of a star, a unitless measure, is a logarithmic scale of observed visible brightness. The apparent magnitude is

4692-450: The time of William Herschel astronomers recognized that the telescopic disks of stars were spurious and a function of the telescope as well as the brightness of the stars, but still spoke in terms of a star's size more than its brightness. Even into the early nineteenth century, the magnitude system continued to be described in terms of six classes determined by apparent size. However, by the mid-nineteenth century astronomers had measured

4761-422: The variations in brightness within a group. Tycho Brahe attempted to directly measure the "bigness" of the stars in terms of angular size, which in theory meant that a star's magnitude could be determined by more than just the subjective judgment described in the above quote. He concluded that first magnitude stars measured 2 arc minutes (2′) in apparent diameter ( 1 ⁄ 30 of a degree, or 1 ⁄ 15

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