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Absolute Magnitude to Compare the Intrinsic Brightness of Stars, One Can Adjust the Apparent Magnitude to Account for the Measured Distance

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Absolute Magnitude to Compare the Intrinsic Brightness of Stars, One Can Adjust the Apparent Magnitude to Account for the Measured Distance Astronomy 218 How Bright is that star? Apparent Brightness When you look at the stars in the night sky, one Meissa of the most noticeable Betelgeuse features is the difference Bellatrix in brightness. Looking deeper reveals Mintaka more and more, fainter Alnilam stars. Alnitak Scientifically, we quantify this apparent brightness Hatsya based of the measured flux Rigel of energy per unit area at Saiph our location. Hipparchus The most noted astronomical observer among the Greeks was Hipparchus (~190-120 BC). Provided accurate measure of Tropical year. Discovered precession of the equinox. Provided more accurate Earth-Moon distance (~70 REarth). Produced accurate stellar catalog with the positions of more than 850 stars. Established the magnitude system for stellar brightness, which is still used today. Farnes Atlas Apparent Magnitude In Hipparchus system, stars were divided into 6 magnitudes, with 1st magnitude the brightest and 6th magnitude the faintest visible to unaided eye. With the invention of the telescope, the system was extended to larger magnitudes. Note, dimmer stars are larger magnitudes! In 1856, Norman Pogson observed that a difference of 5 magnitudes corresponds to a change of a factor of 100 in apparent brightness. This led to a standardized system, including the planets and Sun. Ordinal vs. Cardinal Hipparchus system of magnitudes assigned the lowest numbers to the most prominent stars. Linguistically, these are ordinal numbers (first, second, third, fourth, …) indicating a ranking. They are equivalent to assigning precedence (primary, secondary, tertiary, quaternary, …) Linguistically, the ordinal numbers are distinct from the cardinal numbers (1, 2, 3, 4 …), the “counting” numbers. When the modern system was standardized, these magnitudes became cardinal numbers, including decimals. However, they retained the ordering of the ordinal numbers, making the smallest numbers correspond to the largest fluxes. Measuring Magnitude A difference of 5 magnitudes corresponds to a factor of 100 in brightness or flux. Thus, if m2 − m1 = 5, For an arbitrary difference in magnitude, or To establish a standardized scale, a “zero” is also needed. The star Vega was chosen, because it is bright and has little variability in brightness. Thus m = C − 2.5 log F where C = 2.5 log FVega . Absolute Brightness The ancients had long known that small distance could make a dim object appear brighter. Two objects that appear equally bright might be a closer, dimmer object and a farther, brighter one. In the heliocentric cosmology developed by Eudoxus, Aristotle and Ptolemy, all stars were at the same distance. Thus differences in apparent brightness were due to differences in absolute brightness. In fact, the inability to observe a noticeable parallax for the stars was taken as proof that the Earth was stationary and the center of the universe. Solar System Distances The first reasonably accurate measurement of solar system distances was Hipparchus’ determination that the Earth-Moon distance was ~ 70 × R⨁. Together with Eratosthenes’ determination of the Earth’s diameter, this provided a value good to within 35%. Ancient astronomers also estimated the Earth-Sun distance, but the first reasonable values (±10%) were measured in 1672 by Cassini & Richer using measuring parallax of Mars between Paris and Guiana (or perhaps by Huygens in 1658). From the measured astronomical unit, Kepler’s laws provided estimates of the distances to the other planets. Today, radar time of flight measurements provide very accurate distances, e.g. 1 AU = 149,597,870.7 km. Diurnal Parallax Hipparchus made his measurement of the Earth-Moon distance using diurnal parallax. The Earth’s rotation provides an opportunity for parallax with a baseline of 12,700 km. Hipparchus was able to measure angles to about 1/6° (10’), making his distance measuring limit Distance = R⨁ / tan (0.08°) 6 = 688 R⨁ = 4.4 × 10 km = 0.03 AU Annual Parallax The Earth’s orbit provides another opportunity for distance measurement, with a base line of 2 AU = 3.0 × 108 km. But even with the best observations of the pre-telescope era, Tycho Brahe’s limit of 0.03°, this has very limited range. Distance = 1 AU / tan (0.015°) = 3820 AU = 0.06 lightyear Even early telescopes could not resolve stellar parallax, until Friedrich Wilhelm Bessel measured the parallax to the star 61 Cygni in 1838 as 0.31” or 0.00009°, giving a distance of 3.2 parsec (10.4 ly). The Solar Neighborhood Following quickly on the heels of Bessel’s discovery, distance measurements were made for many stars in the solar neighborhood. For example, this figure shows the 30 closest stars to the Sun. Ground-based telescopes are limited by the atmosphere to resolving parallax angles of 0.01”, providing distances of 10 parsecs for 10% errors. Only a few hundred stars had well measured distances. Proper Motion Searches for stellar parallax also reveal proper motion, shifts of the star in the sky after correcting for parallax. These were first detected by Edmund Halley in 1718 for Sirius, Arcturus and Aldebaran. The poster child for proper motion is Barnard’s star, with a proper motion of 1°/350 yrs. 22 years later The large proper motion results from its close proximity, it is our second nearest neighbor at 1.83 parsecs (5.96 ly), and a fairly high relative velocity of 140 km s−1. Leaving a Trail Mira, a red Giant in the constellation Cetus, 92 pc (300 ly) from Earth, is moving across the sky at 130 km/s (291,000 miles/hr). Mira is losing gas, leaving a visible tail 13 13 LY lightyears long behind it. 2° It takes the ejected gas more than 30,000 years to cool to invisibility. Hipparcos In 1989, the European Space Agency launched the Hipparcos satellite. Hipparcos measured precision parallaxes to an accuracy of about 0.001”, extending the range of parallax measurements tenfold. This increased the number of stars with accurate distances to ~105 (118,218 entries, stars or multiple stars), greatly improving our sample of well studied stars. Gaia The Gaia mission, launched by ESA in December 2013, increases angular sensitivity another ~50 fold (.000024 ”). After traveling to Earth’s L2 Lagrangian point and commissioning, it has been taking data since July 2014. Over 5 years, it was planned to observe each star ~70 times to refine the parallax and distance. The current schedule extends through 2020. Gaia Data The first data release, in September 2016, pinpointed 1.143 billion stars, but provided parallaxes (distances) to only 2 million stars. The second data release expands the set of observed stars to 1.692 billion. Parallaxes are provided to 1.332 billion stars, as well as proper motions. Radial velocities for 7 million stars have also been measured. The third data release is scheduled for 2021 with a final release once the mission is complete. Solar System Perspective Taken together, the distance measurements paint an interesting picture of the solar neighborhood. Nearest star to the Sun is Proxima Centauri at 1.30 pc. It orbits the binary star α Centauri at 13,000 AU (0.63 pc), with a period > 500,000 years. α Centauri A & B are in close orbit with a period of 80 years. In about 27,000 years, Proxima Centauri will make its closest approach to Earth at ~ 1 pc. If Sun is a marble, Earth is a grain of sand orbiting 1 m away, the Solar System extends about 50 m from Sun. Proxima Centauri is a smaller ball, 2 mm in diameter, 270 km away. Solar Luminosity A measurement of the apparent brightness or flux of the Sun (1372 W m−2) allows calculation of the Sun’s apparent magnitude. m☉ = C − 2.5 log F = −18.91 − 2.5 log (1372) = − 26.75 where C = 2.5 log FVega = −18.91 With a measurement of the Earth- Sun distance (1AU = 1.5 × 108 km), and one can calculate the Sun’s luminosity or intrinsic brightness. 26 For the Sun, L☉= 3.86 × 10 W Absolute Magnitude To compare the intrinsic brightness of stars, one can adjust the apparent magnitude to account for the measured distance. m = C − 2.5 log F = C − 2.5 log L + 2.5 log(4π) + 5 log d The absolute magnitude is the apparent magnitude a star would have at a standard distance of 10 parsecs. 5 M = C − 2.5 log L + 2.5 log(4π) + 5 log(10 pc) The absolute magnitude of a star can be calculated from its measured apparent magnitude and distance. M = m − 5 log (d/1 pc) + 5 = m − 5 log (d/10 pc) For example, −6 M☉ = −26.75 − 5 log(5 × 10 pc/ 10 pc) = 4.83 Compared to the Sun Since we often refer to a star’s luminosity in solar units, it’s convenient to derive the relationship between their absolute magnitudes. M = C − 2.5 log L + 2.5 log(4π) + 5 M☉ = C − 2.5 log L☉ + 2.5 log(4π) + 5 Taking the difference removes common terms, M − M☉ = − 2.5 log L + 2.5 log L☉ or A reduction of 5 in magnitude corresponds to an increase in a factor of 100 in luminosity, due to the logarithmic response of the human eye. Distance Modulus The difference between the apparent and absolute magnitudes is the result of distance. M = m − 5 log(d/10 pc) We use this difference, called the distance modulus, to provide a logarithmic measure of the distance. M - m = DM = 5 log(d/10 pc) The distance modulus shows an even larger range than the absolute magnitude. DM☉ = −26.75 − 4.83 = −31.58 DM(Proxima Centari) = − 4.4 DM(Galactic Center) = 14.6 DM(Andromeda) = 29.5 Close or Bright The wide ranges in intrinsic brightness and distance mean that the stars we can see are include a range from bright, but distant to dim but close.
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