Stars/Sun/Heliometry

Heliometry is the science of measuring the properties of the Sun.

On the right is a coelostat which rotates a solar telescope so as to keep its orientation constant with relation to the Sun.

Astronomy
The color of a star, as determined by the peak frequency of the visible light, depends on the temperature of the star's outer layers, including its photosphere. The effective temperature of the surface of the Sun's photosphere is 5,778 K. The temperature at the bottom of the Sun's photosphere is 6600 K, while the temperature at the top of the photosphere is 4400 K. The photosphere is ~400 km in thickness.

The peak emittance wavelength of 501.5 nm (~0.5 eV) makes the photosphere a primarily green radiation source. The figure at the right shows the extraterrestrial solar spectral irradiance as compared with a blackbody spectrum. The sharper than black-body cutoff at the shorter wavelength end indicates an even lower likelihood that X-rays are emitted from the photosphere.

Radiation
Direct irradiance measurements have only been available during the last three cycles and are based on a composite of many different observing satellites. However, the correlation between irradiance measurements and other proxies of solar activity make it reasonable to estimate past solar activity. Most important among these proxies is the record of sunspot observations that has been recorded since ~1610. Since sunspots and associated faculae are directly responsible for small changes in the brightness of the sun, they are closely correlated to changes in solar output. Direct measurements of radio emissions from the Sun at 10.7 cm also provide a proxy of solar activity that can be measured from the ground since the Earth's atmosphere is transparent at this wavelength. Lastly, solar flares are a type of solar activity that can impact human life on Earth by affecting electrical systems, especially satellites. Flares usually occur in the presence of sunspots, and hence the two are correlated, but flares themselves make only tiny perturbations of the solar luminosity.

Solar irradiance and insolation are measures of the amount of sunlight that reaches the Earth. The equipment used might measure optical brightness, total radiation, or radiation in various frequencies.

"High precision radiometric observations of the Sun carried out by several satellites since the late-1970s have shown that the Sun undergoes small cyclic variations in brightness. These brightness changes are closely related to the ~ 11 year sunspot cycle. Over the last three solar sunspot cycles (Cycles 21, 22, & 23), the observed bolometric brightness of the Sun varied by 0.12 percent (see, e.g., Lean 1997). However, surprisingly (at least at first) the Sun is brightest during the times of maximum sunspot number and faintest during the sunspot minima. This has been explained (and modeled) by the larger changes in the areal coverage and intensity of magnetic white light facular regions that peak near sunspot maximum. Thus, the observed light variations of the Sun arise from the blocking effect of sunspots and increased facular contribution to brightness in which the facular contributions slightly offset the light blocking effects of sunspots."

"Even though the observed light (bolometric) variations are small over its activity cycle, variations over the sunspot cycle are much larger at shorter wavelengths (Lean 1997; Rottman, this volume). [The] typical variations of solar coronal X-ray emissions from the minimum to the maximum of the ~ 11 year activity cycle are nearly 500 percent. The cyclic changes arising from variations in the chromospheric and transition region emission range from 10 to 200 percent at NUV, FUV and EUV wavelengths. Also, the frequencies and intensities of flaring events and coronal mass ejections (CME) are strongly correlated with the Sun's activity cycle. For example, the rate of CME occurrences is larger during the sunspot maxima than during the [...] sunspot minima (Webb & Howard 1994). [The] changes in the XUV flux of the Sun over a typical activity cycle are significantly larger and these high energy solar emissions are absorbed and heat the Earth's stratosphere and thermosphere. Although the deposited energy is small, non-linear feedback mechanisms could amplify the effect on climate by, for example, altering the tropospheric heat exchange between the equator and polar regions."

Theoretical heliometry
Def. the "measurement of the diameters of heavenly bodies, their relative distances, etc." is called heliometry.

Electromagnetics
Notation: let the symbol $$ Q_{\odot} $$ represent the net solar charge.

"[A] variety of geophysical and astrophysical phenomena can be explained by a net charge on the Sun of -1.5 x 1028 e.s.u." This figure was later reduced by a factor of five.


 * $$ Q_{\odot} = -0.3\times 10^{28}\,\hbox{esu}.$$

Visuals
The luminosity of stars is measured in two forms: apparent (visible light only) and bolometric (total radiant energy). (A bolometer is an instrument that measures radiant energy over a wide band by absorption and measurement of heating.) When not qualified, "luminosity" means bolometric luminosity, which is measured either in the SI units, watts; or in terms of solar luminosities, $$ L_{\odot} $$, that is, how many times as much energy the object radiates as the Sun.

Notation: let the symbol $$ L_{\odot} $$ represent the solar bolometric luminosity.

The solar luminosity, [$$ L_{\odot} $$], is a unit of radiant flux (power emitted in the form of photons) conventionally used to measure the luminosity of stars. One solar luminosity is equal to the current accepted luminosity of the Sun, which is $3.839 W$, or $3.839 erg/s$. The value is slightly higher, $3.939 W$ (equivalent to $4.382 kg/s$ or $1.9 M_{☉}/d$) if the solar neutrino radiation is included as well as electromagnetic radiation. The Sun is a weakly variable star and its luminosity therefore fluctuates. The major fluctuation is the eleven-year solar cycle (sunspot cycle), which causes a periodic variation of about ±0.1%. Any other variation over the last 200–300 years is thought to be much smaller than this.

The solar luminosity is related to the solar irradiance measured at the Earth or by satellites in Earth orbit. The mean irradiance at the top of the Earth's atmosphere is sometimes known as the solar constant, [$$ I_{\odot} $$]. Irradiance is defined as power per unit area, so the solar luminosity (total power emitted by the Sun) is the irradiance received at the Earth (solar constant) multiplied by the area of the sphere whose radius is the mean distance between the Earth and the Sun:
 * $$L_\odot = 4\pi kI_\odot A^2\,$$

where A is the unit distance (the value of the astronomical unit in metres) and k is a constant (whose value is very close to one) that reflects the fact that the mean distance from the Earth to the Sun is not exactly one astronomical unit.

Solar radius
Notation: let the symbol $$R_{\odot}$$ indicate the solar radius.

The solar radius is a unit of distance used to express the size of stars in astronomy equal to the current radius of the Sun:


 * $$R_{\odot} = 6.955\times 10^8\,\hbox{m} = 0.0046491\,\hbox{AU} = 2.254\times 10^{-8}\,\hbox{pc}$$

The solar radius is approximately 695,500 kilometres (432,450 miles) or about 110 times the radius of the Earth ($$R_\oplus$$), or 10 times the average radius of Jupiter. It varies slightly from pole to equator due to its rotation, which induces an oblateness of order 10 parts per million.

The solar radius varies slightly from pole to equator due to its rotation, which induces an oblateness in the order of 10 parts per million.

The unmanned Solar and Heliospheric Observatory (SOHO) spacecraft was used to measure the radius of the Sun by timing transits of Mercury across the surface during 2003 and 2006: a measured radius of 696342 +/-.

A "determination of the solar diameter derived from the total solar eclipse observation in Turkey and Egypt on 29 March 2006 [indicated] the solar radius carried back to 1 AU was 959.22 ± 0.04 arc sec at the time of the observations."

"The inspection of the compiled 19 modern eclipses data, with solar activity, shows that the radius changes are nonhomologous, an effect which may explain the discrepancies found in ground-based measurements and implies the role of the shallow sub-surface layers (leptocline) of the Sun."

Precise "solar radius measurements are at the cutting edge of the techniques used: a difference of 20 mas (milli arc sec) between the greatest solar radius (at the equator) and the lowest one (around 45-50° of latitude) has been detected, and the mean radius does not change more than 15 mas, over time scales of eleven years or so (Kuhn et al., 2004; Lefebvre, Kosovichev, and Rozelot, 2007)."

Haberreiter, Schmutz & Kosovichev (2008) determined the radius corresponding to the solar photosphere to be 695660 +/-. Previous estimates using inflection point methods had been overestimated by approximately 300 km.

In 2015, the International Astronomical Union passed Resolution B3, which defined a set of nominal conversion constants for stellar and planetary astronomy: the nominal solar radius (symbol $$R^{N}_{\odot}$$) is equal to exactly 695,700 km.

The image on the right graphs the evolution "of the solar radius since 1567, before the Maunder Minimum. The T+WD data are after Toulmonde (1997) and Wittmann and Débarbat (1990), whose error bars were computed by one of us from the records kept at the Paris Observatory. The triangles are the data of Toulmonde (T corr series) with his correction for diffraction and refraction. Historical eclipses represented by circles are after Fiala (1994) and Sigismondi (2008). Their error bars are smaller than the dimension of the circle. Toulmonde’s data are averaged over many observations made by respective authors, each lasting for several years; therefore they seemingly distribute around an average value of 960 arc sec."

In the second image down on the right are solar "radius variations from recent eclipse observations, with the data of SDS Egidi (2006) superimposed. Eclipse data are from Parkinson (1980) for the 1966 hybrid eclipse; Kubo (1993) for 1970, 1973, 1980 and 1991 total eclipses from centerline; Fiala (1994) and Dunham (2005) for the majority of eclipses; Sigismondi (2008) for 2006 annular eclipse of September 22; and for the 29 March 2006 total eclipse and the 22 September 2006 annular eclipse [...]. The error bars are statistical, based on the number of Baily beads events observed during each eclipse."

Solar radius measurement techniques
Visual "solar radius measurement techniques are divided into two classes; one is time observation, such as meridian drift transits, planetary transits (Mercury or Venus), and the other is angular observation, such as with Danjon modified astrolabes, heliometers and spectrographs. Eclipse observations fall in the first category. If we compare these techniques in terms of the number of datasets attainable, meridian transit observations give one solar radius data per day, Mercury or Venus transits, a very few per century and eclipse observations no more than three per year. Angular measurements and drift- scan methods out of the meridian permit to increase significantly the number of observations in one day, and thus, in principle, would permit to increase the statistics and the accuracy."

The "long database obtained by means of solar astrolabes since 1975, located at different places (France, Brazil, Chile, Turkey and now Spain and Algeria) yields incompatible results. Such discrepancies cast doubt on the measurements obtained through the astrolabes. If the large amplitude variations in the solar diameter found over a solar cycle (more than 150 mas in certain case), are real, the only way to interpret them seems to be through a magnification of the intensity at the limb due to feedback mechanisms in the UTLS (Upper Troposphere Lower Stratosphere) region of our atmosphere (Badache-Damiani and Rozelot, 2006; Badache-Damiani et al., 2007). If they are from solar origin, then, they will be due to a magnification of the intensity at the limb due to the magnetic field (Li et al., 2006). However, in the latter case, the solar variations of the diameter of 150 mas or more lead to limb radius variations irreconcilable with the results obtained from the f-mode observations (Lefebvre, Kosovichev, and Rozelot, 2007) and to incoherent astrophysical consequences (Rozelot, 2009)."

"Observations through other techniques (spectrograph, transit drift time or heliometer) are more reliable; heliometer measurements at Pic du Midi Observatory (France) and spectrograph measurements at Mount Wilson (USA) are in good agreement (Lefebvre, Kosovichev, and Rozelot, 2007; Lefebvre, Nghiem, and Turck-Chièze, 2009)."

"The main limitation of [solar eclipse] observations arises from the number of obtained data because of poor number of eclipses per year. However, due to the short eclipse duration and the fast-moving trajectory in space, total and annular solar eclipse observations can provide more accurate calibration points by comparison to other data bases (Fiala, Dunham, and Sofia, 1994; Dunham et al., 2005)."

Saros cycles
A sar is one half of a saros.

The earliest discovered historical record of what is known as the saros is by Chaldean (neo-Babylonian) astronomers in the last several centuries BC. It was later known to Hipparchus, Pliny the Elder and Ptolemy.

The Suda says, "[The saros is] a measure and a number among Chaldeans. For 120 saroi make 2220 years (years of 12 lunar months) according to the Chaldeans' reckoning, if indeed the saros makes 222 lunar months, which are 18 years and 6 months (i.e. years of 12 lunar months)."

The Greek word apparently comes from the Babylonian word "sāru" meaning the number 3600.

Mechanical calculation of the saros cycle is built into the Antikythera mechanism.

After one saros, the Moon will have completed roughly an integer number of synodic, draconic, and anomalistic periods (223, 242, and 239) and the Earth-Sun-Moon geometry will be nearly identical: the Moon will have the same phase and be at the same node and the same distance from the Earth; because the saros is close to 18 years in length (about 11 days longer), the earth will be nearly the same distance from the sun, and tilted to it in nearly the same orientation (same season).

For solar eclipses the statistics for the complete saros series within the era between 2000 BCE and 3000 CE are given in the references.

Saros series, as mentioned, are numbered according to the type of eclipse (lunar or solar).


 * 1) In odd numbered series (of either kind) the sun is near the ascending node, whereas in even numbered series it is near the descending node;
 * 2) generally, the ordering of these series determines the time at which each series peaks, which corresponds to when an eclipse is closest to one of the lunar nodes;
 * 3) for solar eclipses, the 40 series numbered between 117 and 156 are active (series 117 will end in 2054), whereas for lunar eclipses, there are now 41 active saros series (these numbers can be derived by counting the number of eclipses listed over an 18-year (saros) period from the eclipse catalog sites.

After a given lunar or solar eclipse, after 9 years and 5.5 days (a half saros) an eclipse will occur that is lunar instead of solar, or vice versa, with similar properties.

For eclipses "separated by a whole Saros cycle [...] the solar radius changes over such timescales."


 * 1) Jean Meeus and Hermann Mucke (1983) Canon of Lunar Eclipses.  Astronomisches Büro, Vienna
 * 2) Theodor von Oppolzer (1887). Canon der Finsternisse.  Vienna
 * 3) Mathematical Astronomy Morsels, Jean Meeus, Willmann-Bell, Inc., 1997 (Chapter 9, p. 51, Table 9. A Some eclipse Periodicities)

Solar cycles
In the diagram on the right, solar "radius variations as deduced from eclipse observations [are] plotted versus sunspot activity. The international sunspot numbers (ISN) are taken from NOAA data. The deviations from the standard radius are larger in amplitude during low activity periods and smaller when high activity occurs. SOHO data during the 2003 Mercury transit is represented by a square. The external curves are ±0.7 exp(ISN/50)."

"The eclipse data show significant variations of the solar radius observed over timescales shorter than one year."

"The analysis of the available data in terms of the solar activity shows that the Sun does not expand or contract homologously with a change in solar activity (Sofia, Basu, and Demarque, 2005; Lefebvre and Kosovichev, 2005). We believe that there is here a key to understand results obtained from ground-based measurements of the solar radius. We would want also to emphasize the role of the shallow photospheric layers (the leptocline, see (Bedding et al., 2007; Lefebvre, Rozelot, and Kosovichev, 2007)), which are the seat of many physical phenomena which couple the internal dynamics to the dynamics of the solar atmosphere."

Physics
The center of mass plays an important role in astronomy and astrophysics, where it is commonly referred to as the barycenter. The barycenter is the point between two objects where they balance each other; it is the center of mass where two or more celestial bodies orbit each other. When a moon orbits a planet, or a planet orbits a star, both bodies are actually orbiting around a point that lies away from the center of the primary (larger) body.

The Sun's motion about the center of mass of the Solar System is complicated by perturbations from the planets. Every few hundred years this motion switches between prograde and retrograde.

Def. "the mass of the Sun" is called the astronomical unit of mass.

Notation: let the symbol $$M_{\odot}$$ indicate the solar mass.

The solar mass ($$\begin{smallmatrix}M_\odot\end{smallmatrix}$$) is a standard unit of mass in astronomy, used to indicate the masses of other stars, as well as clusters, nebulae and galaxies. It is equal to the mass of the Sun, about two nonillion kilograms:


 * $$M_{\odot}=( 1.98855\ \pm\ 0.00025 )\ \times10^{30}\hbox{ kg}$$

This is about 332,946 times the mass of the Earth or 1,048 times the mass of Jupiter.

Because the Earth follows an elliptical orbit around the Sun, the solar mass can be computed from the equation for the orbital period of a small body orbiting a central mass. Based upon the length of the year, the distance from the Earth to the Sun (an astronomical unit or AU), and the gravitational constant (G), the mass of the Sun is given by:


 * $$M_\odot=\frac{4 \pi^2 \times (1\ {\rm AU})^3}{G\times(1\ {\rm year})^2}$$.

The value of the gravitational constant was derived from 1798 measurements by Henry Cavendish using a torsion balance. The value obtained differed only by about 1% from the modern value. The diurnal parallax of the Sun was accurately measured during the transits of Venus in 1761 and 1769, yielding a value of 9&Prime; (compared to the present 1976 value of 8.794148&Prime;). When the value of the diurnal parallax is known, the distance to the Sun can be determined from the geometry of the Earth.

Technology
Def. a "device used to measure the heating power of electromagnetic radiation, especially that of solar radiation" is called an actinometer.

Def. an "actinometer used to measure solar radiation incident on a surface" is called a pyranometer.

At right is an SR20 solar radiation sensor. It complies with the "secondary standard" specifications within the latest ISO and WMO standards.

Def. the "total solar radiation from sun and sky on a horizontal surface" is called the global radiation.

The Radiation Observatory, University of Bergen, Bergen, Norway, latitude 60° 24' N and longitude 5° 19' E, at 45 m elevation above sea level, uses one or more pyranometers to measure the Global Radiation.

A sensitivity check is made of each pyranometer against a standard using the sun/shade method on a cloudless day.

A sensitivity may be similar to 4.818 V/Wm-2, which should be a small factor such as 1.0165 times the original sensitivity when first manufactured.

The diffuse (sky) radiation is measured by a pyranometer. "When measuring the sky radiation, the direct solar radiation is constantly shadowed off by means of a 6 cm diameter circular disc mounted on a 30 cm long rotating arm."

Def. a "device that measures the intensity of solar radiation received on the surface of the earth" is called a pyrheliometer.

The normal incidence beam radiation is measured by a normal incidence pyrheliometer with a known and calibrated sensitivity, e.g. 8.15 V/Wm-2.

The pyrheliometer is mounted on an automatic solar tracker.

Def. a "device that measures radiant energy" is called a radiometer.

Ultraviolet radiation is measured by means of a total ultraviolet radiometer with a specific wavelength response such as 290 - 385 nm.

"For the measurement of long-wave radiation, a ventilated [...] pyrgeometer [...] with coated silicon hemisphere [is] used. This makes it possible to compute the [Downward Atmospheric Radiation], since the temperature of the instrument is also recorded."

"The [Duration of Sunshine] is measured by a Campbell-Stoke sunshine recorder with blue paper strips. The strips are read according to the rules of [the World Meteorological Organization] WMO [3]. Maximum possible duration gives the number of hours the sun is above the natural horizon, as found from the records on days with clear skies at sunrise or sunset. The [Duration of Sunshine] is also given as the number of minutes during which the [NIP records] irradiance above 120 Wm-2 (with one instantaneous recording counted as 20 seconds)."

"The necessary routine calibrations of the pyranometers and the NIP pyrheliometer are carried out by means of the absolute self-calibrating cavity pyrheliometer [which in turn is] compared to the World Radiation Reference Scale (WRR)".

Hypotheses

 * 1) The Sun may not be gaseous all the way through.