Physics & Astronomy Modules

# Observables in Astrophysics

## Our passive interaction with the universe

What can we measure that is quantitatively useful in "Observational" astronomy? Limiting the focus to optical and infrared with light, the observables are information carried by the light may depend on time and are

• Position (direction to where the light originates)
• Flux (in energy or photons)
• Spectral properties (what frequencies or wavelengths are present in what amounts)
• Polarization

That's the sum of it. The photons we detect have energy, momentum, and spin, and they arrive a rate which can be determined by observing for a time interval and finding how many come within it. Photon energy is frequency, momentum includes direction, and spin is the polarization. Whether we use the language of quanta or of electricity and magnetism it still comes down to these observables. They are as an ensemble affected by what is happening at the light source. The task of observational astronomy is not simply to count and catalog photons, but to make the connection to an understanding of the unverse derived from the signals they convey.

## Position

Objects in the sky are located by celestial coordinates, Right Ascension and Declination, measured on the celestial sphere. Begin with a sphere surrounding the Earth, its equator being an extension of Earth's equator, and its poles extensions of Earth's axis.

Declination is measured in degrees north (+) or south (-) from the equator.

Right Ascension is measured like a clock on the sky in hours toward the east from a reference meridian. This meridian is set by convention at a point where the plane of Earth's orbit around the Sun intersects with the plane of its equator projected on the sky. There are two such points, and the one we use for the zero point of Right Ascension is where the Sun passes from below the equator to above the equator as seen from the center of the Earth. Since this happens in a northern hemisphere spring when days and nights have equal duration, it is the point that marks the vernal equinox.

Earth's axis slowly precesses in space, approximately rolling around a perpendicular to the plane of its orbit, creating a virtual cone in space. The sense of rotation seen from above looking down on it is clockwise, opposite the sense of Earth's daily rotation. The precession period is 26,000 years, over which the point of the vernal equinox moves around the celestial equator one full turn, or it makes 1/26,000 of a turn per year. That's more than 1 degree per century. The places in the sky marking the north and south celestial poles will trace out a full circle over 26,000 years, centered on the normal to the plane of the solar system and with an angle of about 23.5 degrees to it that is the tip of Earth's axis from being perpendicular to its orbital plane.

If we look up at the pole, the circle over this 26,000 years is progressed in a counter clockwise sense. The star we might call the "north" or "pole" marking this point will change and then repeats every 26,000 years.

Consequently the celestial coordinates of any star will depend on the year the reference for the equinox is set. It may be convenient to set a specific year close to the one for which measurements are made for high precision data, or to refer positions to a standard epoch. Currently this is year 2000, which replaced 1950. When giving coordinates, it is essential to state the epoch.

## Orbits, Ephemerides, and Proper Motion

Everything is in motion, and the Sun's motion through space as well as the motion of stars or any object we look at alters where it will appear in the sky.

The objects in our solar system, Earth included, move on paths determined by gravity and the laws of motion. If we know where they are now and how they are moving at this moment, then from their mutual gravitational interactions we can compute where they will be and how they will be moving at any time past or future. The parameters that enable that modeling are called the orbital elements, and they are known for all identified bodies in the solar system, naturala and man-made. Given the orbital elements we can predict events, and timing is termed an "ephemeris". From these we know where the Earth will be at any time, how it will be oriented in space, and where any object of interest will be too. The calculation of the direction to look that is of the instantaneous celestial coordinates, is a matter of vector algebra. It is complex for objects which are nearby and have a fast apparent motion on the sky, while for distant objects it is straightforward because the motion on the sky is nearly linear in space and time. If we take the Earth's motion out of the problem of where to look at a distant object typically beyond the solar system, then we would need to know the intrinsic motion of that object due to its velocity in space, and how far it is from us. The resulting motion on the sky each year is its "proper motion".

The proper motion of our nearest stellar neighbor, Proxima Centauri, is -3781 mas/year in right ascension, and +770 mas/year in declination.

## Parallax

Even if it had no space velocity relative to the Sun, because Earth moves in orbit around the Sun the direction we must look to see something depends on where we are, and varies annually. When the object of interest is directly above the plane of our solar system Earth's motion causes the object to trace out a minature copy of Earth's orbit on the sky. That makes a nearly circular path with a radius of 1 second of arc when the object is at a distance such that Earth's orbit would appear to be 1 second of arc in average radius seen from it. That distance is called a parsec (pc) and is given by "d" where $\tan(1 \; arcsecond) = a / d$ and "a" is representative of an average of Earth's orbit that we call an astronomical unit (au). Since an arcsecond is 1/3600 of a degree, this relation gives $d = 206264.81 \; a$ and a parsec is 206264.81 astronomical units. The astronomical unit was originally defined as the average of Earth's perihelion and aphelion distances, but is now more simply set exactly to be $1.495978707 \times a10^{11}$ meters (roughly 150 million km). It is nearly, just not quite, the semimajor axis of Earth's orbit which is 1.000001018 au.

The speed of light is 299792458 m/s. Consequently it takes light 499.005 seconds or 8.32 minutes to traverse 1 AU. To go 1 parsec, light would take $206264.81 \times 499.005$ seconds. Since an astronomical year of 365.25 days (a Julian year) has 3155760 seconds, the parsec is $d = 206264.8 499.00478 /31557600 = 3.262 \;light\, years$ We catalog the apparent annual shift of a star's angle on the sky as the parallax, typically measured in milliarcseconds (mas), for the angle it would move if observed from two locations separated by 1 au. In practice, if we observe from the moving Earth when the object is at the ecliptical pole its apparent shift is a cicle with an angular diameter that is inversely proportional to its distance from us. When the object is on the ecliptic equator it shifts along a line instead. The extremes of these motions result from the Earth translating 2 au the shift observed on the sky annually is twice the cataloged parallax.

A measurement of parallax provides the distance to the object. For a parallax $\pi$ (unforunately the standard symbol for it astronomy), the distance to the object in parsecs is $tan(\pi) = a / d$ since parallax is defined for a shift of 1 au in position and a parsec is how far it would be for a shift of an arcsecond. For small angles the tangent and the angle are the same in radians and simply $\pi = a/d$ Thus measuring d in parsecs and $\pi$ in milliarcseoncds we have $d [pc] = 1 / \pi [arcsec]\; or\; d [pc] = 1000 / \pi [mas]$ To take a simple but profound example, the parallax of our nearest stellar neighbor Proxima Centauri is 768.5 mas. It is $d = 1000/768.5 \; pc = 1.30\; pc \; or \; 4.24 \; ly$ away.

At a distance of 1.30 pc from us to Proxima Centauri, its proper motion in declination of +770 mas/year on the sky requires a $770/1000 \times 1.30 \; au$ traversal through space. That would be a space velocity of 1.001 au/year. The astronomical Julian year is 365.25 days of 86400 seconds each. Proxima Cen is moving at $v = 1.0 \times 1.496\times10^{11} / (365.24 \times 86400) = 4740 \; m/s \; or \; 4.74 \; km/s$ in the direction parallel to "declination" in our sky. Earth's orbital velocity, for comparison is 29 km/s. Space velocities of hundreds of km/s are common within our galaxy.

## Flux

We use the term flux because it evokes flow. More precisely there are specific terms describing the emission of light as seen by a distant observer. Ultimately we measure either energy/time or photons/time depending on the detector techonology. If it is counting photons, then multiplying the number of photons by their energy per photon $h\nu$ will give their energy, and the energy/time is power. Our instruments gather photons over an area, and thus from a single point source of light that is spatially unresolved such as a star we can have a measurement of power/area-time in specfic range of frequencies or wavelengths. Astronomers often use the generic term flux for this quantity but it should be reserved for total energy/time in units of watts.

• Luminosity The rate of change of energy, often denoted by L, and in SI units given in watts (W). It is synonomous with flux.
• Flux The rate of change of energy denoted by $\Phi$ or F and in SI units given in watts (W).
• Irradiance The flux per unit area on a surface, denoted by E and given in units of watts per square meter $(W/m^2)$.
• Intensity Total flux per unit solid angle from a source, denoted by I and given in units of watts per steradian $(W/sr)$.
• Specific Intensity Total flux per area per frequency band width flowing thorugh space in units of watts per square meter per steradian per hertz $(W/sr - m^2 - Hz)$. The symbol I is also used for this.
• Radiance Flux emitted per unit solid angle per area given in units of watts per square meter per steradian per hertz $(W/sr - m^2 - Hz)$. Sometimes this called brightness, but that's a carryover from the days of visual astronomy. The symbol L is commonly used.

What we observe is affected by the source itself, the distance the light travels, intervening absorption and scattering, the collection area of the telescope or sensor, the transmission efficiency of the optics, the quantum efficiency of the detector, and the scaling of the detected photons over a time interval into a digital record. The detection is always within a frequency range limited intentionally by filters, or as a consequence of the detector's response and the transmission of the Earth's atmosphere and the optics. Consequently, we measure point sources on a scale set by comparisons of one to another and assign a magnitude to represent the flux from the source. Magnitudes are related to radiance by $m_1 - m_2 = 5/2 \log_{10} (L_2 / L_1)$ which means that two objects differ in magnitude by 5 if one is 100 times more luminous than the other. A magnitude is larger and more positive if the object is fainter.

The reference star was originally $\alpha$ Lyrae, Vega, which has a nominal magnitude of 0 in each of the photometric bands commonly used for stellar measurements. With improvements in calibration and reference to absolute standards, Vega's magnitudes are now slightly off from from the zeropoint. We set a bolometric magnitude at 0 for a luminosity of $L_0 = 3.0138 \times 10^{28} W$ which is to say that an object at 10 parsecs, the standard distance for making an absolute measurement, will appear to have a magnitude of 0 integrated over all frequencies emitted if it is emitting $L_0$ watts into all directions in space. The zero point was selected so that the nominal solar luminosity $L_{sun} = 3.828 \times 10^{26} W$ would correspnd to the absolute bolometric magnitude $M_{sun} = 4.74$ that was the accepted value when this standard was adopted in 2015. The zero point for irradiance is $E_0 = 2.518021 \times 10^{-8} W/m^2$

We make the distinction between apparent magnitude m that we observe, and absolute magnitude M that would be observed from 10 parsecs. Because of the inverse square law for flux, $m - M = 5 log_{10} d - 5$ when d is measured in parsecs.

We also distinguish flux meaasured bolometrically, that is in total emitted energy, and in specfic bands or through specific filters that limit the spectral range of the detected light. There are many filter systems in use.

## Spectral Properties and Filters

Two common systems are known as Johnson-Cousins-Bessell and Sloan filters. Letter designations are used somewhat representative of the color seen when looking through the filter. These are summarized here.

Johnson-Cousins or UBVRI systems based on filter specifications refined by Bessell in 2005 are

The filter properties may vary slightly depending on manufacturer and may be matched to the sensor. These values are typical.

### Johnson-Cousins-Bessell

 U Ultraviolet Peak transmission wavelength 370 nm FWHM 70 nm B Blue Peak transmission wavelength 430 nm FWHM 100 nm V Visible Peak transmission wavelength 530 nm FWHM 80 nm R Red Peak transmission wavelength 660 nm FWHM 160 nm I Green Peak transmission wavelength 800 nm FWHM 160 nm

### Sloan or SDSS

 u' Ultraviolet Center transmission wavelength 360 nm, FWHM 80 nm g' Blue-green Center transmission wavelength 460 nm, FWHM 120 nm r' Red Center transmission wavelength 620 nm FWHM 120 nm i' Near-infrared Center transmission wavelength 750 nm, FWHM 140 nm z' Infrared Center transmission wavelength 820 nm, FWHM 180 nm

A star which is brigher in the R or I band than it is in the B band would be "red", and the ratios of fluxes or the differences in magnitudes for different filters is a color measurement. Commonly these would be

• U-B
• B-V
• V-R
• R-I
such that a positive difference means the redder filter lower (e.g. brighter) magnitude. The magnitude scale for Vega was set such that its colors were all nominally zero.

A blackbody source emits light with a maximum in its spectrum at $\lambda_{max}\; [\mu m] = 2897/T[K]$ which tells us that at 3000 K a perfect absorber emits the peak of its spectrum at 0.96 microns or 960 nm in the near-infrared. A star such the Sun at a mean surface temperature of 5772 K emits its peak radiation at 500 nm. Its B-V color is 0.63.

Click the image to try the simulation.

The precise measurement of flux resolved finely with frequency or wavelength is spectroscopy, as contrasted with spectrophotometry which is broaded banded. High resolution spectroscopy reveals the atomic and molecular species present in the source, and by virtue of measurement of their apparent wavelengths, the relative velocity of the source and the observer.

Typically astronomical measurements of spectra may be made at resolutions of 1000 (a spectral bandwith of 1/1000 of the wavelength) for fain sources where light is at premium, up to 1,000,000 for sources where there is enough light to disperse it into small spectral elements. The radial velocities of stars hosting extrasolar planets may be measured with spectroscopic resolutions of 100,000 sufficient to resolve turbulence in their atmospheres and the effects of stellar resolution. The features identified in such spectra can yield Doppler shifts with sufficient precision to establish relative velocities of less tha 1 m/s, essentially walking speed. This is sufficient to detect the reflex motion of the star from the tug of its planets, and use that to measure the mass of the planets it hosts.

## Polarization

It is possible to measure the polarization of light from astronomical sources which arises from scattering off of dust in reflection nebula, surface properties of planets and satellites, and alignment of atoms and molecules in magnetic fields. The latter depends on a technology combining polarimetry with spectroscopy to reveal stellar magnetic fields, star spots, and stellar activity cycles. Optical polarization is characerized by measuring either linear polarization, the classical direction of the electric field, or circular polarization, the rotation of the field in a clockwise or counter-clockwise sense as the light propagates, or both. The polarization state is represented by four Stokes parameters that can be thought of as a 4-dimensional Stokes vector. Since the first parameter is the actual flux from the object, the others relative to the first give the polarization state. $S_0 = 1$ $S_1 = p \, \cos(2\psi) \, \cos(2\chi)$ $S_2 = p \, \sin(2\psi) \, \cos(2\chi)$ $S_3 = p \, \sin(2\chi)$ where p is the fractional degree of polarization. Fully polarized light has p=1. In astronomy, much smaller values are found for stars. The three components of the vector $(S_1, S_2, S_3)$ can be used to identify the nature of the polarized light.

The angles $\chi$ and $\psi$ describe the ellipse which the electric vector traces out as seen looking into the light. If $\psi$ is zero, then the light is aligned on the x axis and a positive \chi indicates a rotation that is clockwise looking into the beam, while a negative chi is counter clockwise.

 (1 1 0 0) linear horizontal $(\psi = 0 \; \chi = 0)$ (1 -1 0 0) linear vertical $(\psi = 90 \; \chi = 0)$ (1 0 1 0) linear at +45 degrees $(\psi = 45 \; \chi = 0)$ (1 0 -1 0) linear at -45 degrees $(\psi = 135 \; \chi = 0)$ (1 0 0 1) righthand circular $(\psi = any \; \chi = +45)$ (1 0 0 -1) lefthand circular $(\psi = any \; \chi = -45)$ (1 0 0 0) unpolarized $(\psi = any \; \chi = any)$

Right-handed circular polarization is the case where the electric field appears to rotate counter-clockwise when the observer is looking into the beam of light. This is the sense the fingers of your right hand would wrap around the beam if your thumb is pointed in the direction it is going. The handedness reverses depending on whether it is with respect to the source or to the observer.

## Conclusions

Our information about the universe comes from what we can detect and infer from those signals. Light, electromagnetic radiation, is the most informative of the interactions we have with our environment and we have learned how to tease an understanding of the universe from what it sends our way in this form. While the new science of gravitational sensing offers complementary data, most of what we know now has been derived from light in one form or another. The observable data are being analyzed and catalogued, and are available to augment any new measurements we may make.