sign in sign up
<<
Home , Astronomical unit, Hertz, Solar mass, Unit of length

and the new SI

Prasenjit Saha1

1Physik-Institut, University of Zurich, Winterthurerstrasse 190, 8057 Zurich, Switzerland

Abstract In 2019 the International System of units (SI) conceptually re-invented itself. This was necessary because quantum-electronic devices had become so precise that the old SI could no longer calibrate them. The new system defines values of fundamental constants (including c, h, k, e but not G) and allows units to be realized from the defined constants through any applicable equation of physics. In this new and more abstract SI, units can take on new guises — for example, the is at present best implemented as a derived electrical unit. Relevant to astronomy, however, is that several formerly non-SI units, such as -, -, and what we may call “ seconds” GM/c3, can now be interpreted not as themselves units, but as shorthand for volts and seconds being used with particular equations of physics. Moreover, the classical astronomical units have exact and rather convenient equivalents in the new SI: zero AB amounts to ' 5 × 1010 m−2 s−1 per logarithmic or interval, 1 au ' 500 light- seconds, 1 pc ' 108 light-seconds, while a solar ' 5 gravity-seconds. As a result, the unit conversions ubiquitous in astrophysics can now be eliminated, without introducing other problems, as the old-style SI would have done. We review a variety of astrophysical processes illustrating the simplifications possible with the new-style SI, with special attention to gravitational dynamics, where care is needed to avoid propagating the uncertainty in G. Well-known systems (GPS , GW170817, and the M87 ) are used as examples wherever possible. Keywords: methods: miscellaneous, , gravitational lensing: strong, gravitational , : kinematics and dynamics, : Local Group, galaxies: supermassive black holes, cosmology: distance scale

1 Introduction than the SI-prescribed standards involving be- tween current-carrying wires. Not only that, the de- rived electrical unit While the whole point of units is to stay at fixed val- ues, definitions of units do in practice change from 2 −1 −2 3 V90 Ω90 m s (1) time to time, in response to scientific developments.

arXiv:1911.10204v1 [astro-ph.IM] 22 Nov 2019 From 1990 the SI (Syst`emeInternational d’unit´es) was equal to a kilogram but more precise than the faced a rebellion of electrical units. The development SI kilogram. The presence of an alternative standard of Josephson junctions beginning in the 1960s pro- that was more precise threatened to make the SI irrel- vided voltage as precisely h/(2e) times a frequency. evant. Faced with this crisis, the SI began a long pro- The discovery of the quantum Hall effect in 1980 pro- cess of reinventing itself (see Taylor & Mohr, 2001) vided a standard resistance e2/h. From these two leading to the reforms of 2018/2019. In the new SI processes a new conventional V90 and conven- (BIPM, 2019) only the is specified by a specific tional Ω90 emerged, which were more precise physical process (a spectral in Cs). Other units

1 are defined implicitly, as whatever makes the in Newtonian dynamics. Two more significant of light come out as 299792458 m s−1, Planck’s con- digits are available if general-relativistic time di- stant come out as 6.626070040 × 10−34 J s, and so on. lation is included, which we will discuss later (Table 1 in the Appendix summarizes.) Any equa- (Section 3.4). tion of physics may be used for realisation of units. Thus, if one wishes to interpret the kilogram as a • The AB-magnitude scale (Oke & Gunn, 1983) derived electrical unit, there is no longer a conflict sets a spectral flux density of a (that with the SI. A nice toy example where this happens is, 10−26 W m−2 Hz−1) as AB = 8.90. This is the LEGO balance (Chao et al., 2015) which style is not unique to astronomy — in the measures mass as electrical times s3 m−2. moment-magnitude scale in seismology (Hanks Astronomers cannot be said to have rebelled & Kanamori, 1979) a zero-magnitude earthquake against the SI, because they never really joined. In corresponds to a gigajoule (or rather 109.05 J), the 19th century astronomers set up units standard- and two magnitudes correspond to a factor of a ized from the sky rather than in the laboratory. An- thousand in . cient stellar magnitudes were formalized as a loga- rithmic scale for brightness (see Jones, 1968; Schae- Along with these SI-calibrated classical units, as- fer, 2013, for this history). The became the pro- tronomers commonly use several named units that totype mass. Most interesting was the astronomical are SI units times a simple multiplicative factor. unit of (for the history, see Sagitov, 1970). Some of them are universally understood (, min- Anticipating the explicit-constants style of the new utes, also degrees, arc- and arc-seconds) and SI, the au was defined such that the formal angular recognized in Table 8 of the SI brochure as useful p 3 ˚ GM / au equals exactly 0.01720209895 alongside SI units. Others (ergs, gauss, and A) are per (which is close to 2π per sidereal legacies of pre-SI conventions. year). But there remains one important unsolved problem Modern practice has continued with the classical in calibrating conventional astronomical units against units of length, mass, and brightness, but calibrated the SI, and again it has to do with the kilogram. In them against the SI. the laboratory, mass is a measure of inertia, but in as- trophysics, mass is observable from the gravitational • The astronomical is now defined field it produces. Whether the mass is the Sun, or as 1 au = 149 597 870 700 m. A is the −12 an less than 100 km across and ∼ 10 M distance at which 1 au subtends one arc-second, (e.g., Goffin, 2014), the astrophysical observable is hence now also defined as a fixed (albeit irra- not M but GM. Unfortunately, G is known only tional) number of . This calibration of as- to 10−4 ( et al., 2018), and while there are some tronomical distances to the SI, retiring the 19th- creative new ideas for measuring G (Christensen- century definition of the au, was adopted quite Dalsgaard et al., 2005; Rosi et al., 2014), there is 1 recently, in 2012. no near-term prospect of more than four significant digits. The error propagates to any astrophysical • The Sun’s mass times the mass expressed in , and in particular to (known as the parameter) is nowadays M = 1.988(2) × 1030 kg. This much uncertainty stated in SI units.2 It is measured as in the mass of the Sun would be fatal for precision 20 3 −2 applications like dynamics. Hence, kilo- GM = 1.3271244 × 10 m s (2) grams cannot be used to express gravitating 1Resolution B2 on the re-definition of the in precision applications. With the kilogram unus- of length, adopted at the IAU General Assembly (2012). able, it is not surprising that astronomers have little 2Resolution B3 on recommended nominal conversion con- stants for selected solar and planetary properties, adopted at appetite for changing any of their conventions to SI the IAU General Assembly (2015). units.

2 The new SI, however, changes the situation. There Table 2 in the Appendix gives precise conversions. are no new units in the new SI, nor do any values To use light-seconds more than informally (that change significantly, instead the reform is conceptual. is, in formulas and programs), we need In particular, the kilogram is now defined implic- to decide whether a light second is a length or a itly. This suggests leaving astrophysical masses in time. That is, a light-second could be a synonym kilograms implicit (pending future developments in for 299792458 m, or it could be this length divided the measurement of G) and working with the mass by c. In this paper, the latter interpretation will be parameter in SI units. Of course, in solar-system used. That is, a light-second will be taken as a nor- dynamics, this is already done, only m3 s−2 seems mal second measurable by a , just being used to too un-memorable for general use. But that prob- measure a length divided by c. lem is easily remedied, by using GM/c2 in metres, or An analogous situation applies to the . GM/c3 in seconds. Basically, one could with The list of useful non-SI units in Table 8 of the the formal in place of the mass. SI brochure gives the electronvolt as 1.602176634 × This paper will develop some ideas like the above. 10−19 J. But the electronvolt is also used to mea- The aim is not some kind of formal compliance with sure mass or even frequency. Hence, it is more useful the SI, but to suggest some formulations that (a) have to understand the electronvolt as just a volt, with a precise meaning in the new SI, (b) simplify formulas the ‘electron’ label denoting that, according to con- and help understand astrophysical processes better, text, one is measuring energy divided by the elec- and (c) would be reasonably easy to convert to. This tron charge e, or mass times c2/e, or frequency times paper will focus on the au, pc, M , and optical mag- h/e. Formerly, such a multiplication depended on the nitudes. If those get converted, units like the A˚ and experimentally-determined value of e, but not any gauss can be trivially replaced, and do not need dis- more, because in the new SI e is a defined constant. cussion here. Section 2 discusses SI formulations that Later in this paper, we will write several distances are equivalent to the classical astronomical units, but in light seconds. For this, let us fix the notation have somewhat different interpretations. Section 3 is devoted to example applications, where we see that  a¯   a/c  replacing the au, pc, solar mass, optical magnitudes R¯ ≡ R/c (4) with SI equivalents is quite convenient and can pro-     ¯ vide insight into diverse astrophysical phenomena. D D/c to express as times. 2 Length, mass, and brightness units

Astronomical distances are often stated in light sec- 2.2 Gravity seconds onds. In topics where plays a role, We could choose GM/c2 in metres as an easier vari- GM/c3 in seconds may appear as a surrogate for ant of the mass parameter, but GM/c3 in seconds mass. AB magnitudes are equivalent to flux blends somewhat better with light-seconds. There is per logarithmic wavelength interval (see Eq. 1 from no standard term for measuring Tonry et al., 2012). Let us discuss these in . GM M ≡ (5) 2.1 Light-seconds c3 The light-second is a common and useful informal in seconds, but it would be useful to have one. Let unit, and has conveniently round conversions: us use ‘gravity-second’ to denote a second being used in this way. The solar mass in gravity-seconds is 1 au −→ 5.0 × 102 s 8 (3) −6 1 pc −→ 1.0 × 10 s . M ' 4.9 × 10 s (6)

3 while for the , the value is 15 ps. Again, Table 2 unit of W m−2 Hz−1 is extremely large, because 1 Hz gives precise values. The values of 2M c (3 km for the is a very small spectral range to pack 1 W m−2, and Sun, 1 cm for the Earth) are more familiar, but the the convention in astronomy is to use a jansky, de- gravity-second values are not difficult to remember. fined as 10−26 of the SI unit. At optical , Because of the uncertainty in G, we do not know detectors in use nowadays measure the photon flux precisely how many kilograms a gravity-second is, but in some band Z since the constancy of G has been much better tested fν Wν dν (11) than the value has been measured, we do know that a hν gravity-second is precisely some number of kilograms. where Wν is the throughput of the filter being used. One seemingly weird consequence of light-seconds Many filters with calibrated transmissions are in use and gravity-seconds is that density will come out in (Bessell, 2005). gravity-seconds per cubic light-second, which is fre- Since h is a defined constant in the new SI, let us quency squared. To get kg m−3 we need to divide by define f G, as in φ ≡ ν (12) M 1 ν h × . (7) (4π/3)R¯3 G which has units of counts m−2 s−1. If we then change In contexts where particle interactions are of interest, the frequency scale to logarithmic, the photon flux density in eV m−3 may be more useful. That is easily (11) becomes obtained by a further factor, as in Z W φ d(ln ν) . (13) M c2 ν ν × . (8) (4π/3)R¯3 Ge Applying the condition from Oke & Gunn (1983) that −1 −2 −1 23 For gravitational phenomena, however, density as fre- hφν = 1 erg s cm Hz = 10 Jy corresponds to quency squared is ideal — recall that the crossing AB = −48.60 gives time in a gravitating system depends only on the −22.44−AB/2.5 −2 enclosed density, as manifest in, for example, the hφν = 10 J m (14) destination-independent travel time of 42 minutes on the “gravity-train” through the Earth (Cooper, relating φν to AB. Dividing by h we have 1966). We can also write the gravitational constant φ = 10−AB/2.5 × 5 × 1010 m−2 s−1 (15) as ν 2 −3 1/G = (1.075 hr) g cm (9) and the precise coefficient is given in Table 2. Colours explicitly relating density and time squared. are simply the ratio of photon fluxes 10 ∆AB/2.5. One thing one musn’t do with mass in gravity sec- To get the photon flux over a bandpass, we need onds is to take M c2 to get energy! Instead, to turn to take the integral (13). If the is gravity-seconds into , we have to multiply by fairly flat and the throughput is ≈ 1, the result will be ∼ φν ln(ν2/ν1) where ν2 and ν1 are the edges of c5 = 3.628 × 1052 W . (10) the band. If ratio of those is ' 1.2 (which is typical G of optical bands) the photon flux over comes This constant, sometimes called the Planck , is to roughly 10−AB/2.5 × 1 × 1010 m−2 s−1. In other the scale of merging black holes. words, zero AB magnitude in a typical optical band gives about 1010 photons m−2 s−1. 2.3 Photon flux vs magnitude The photon flux φν is clearly much more intuitive that the magnitude scale. A logarithmic frequency The usual physical measure of brightness in astron- scale also brings some advantages. First, it does not omy is the spectral flux density fν . The relevant SI matter whether the scale is frequency or wavelength

4 or energy: φν = φλ = φE. Secondly, redshift is systems inside a derivation or a computation. Funda- simply a shift along the spectral scale and does not mental constants relating different SI units (see Ta- change the shape of φν . ble 1) will be used whenever needed or useful. The We remark in passing that the SI has three spe- exception is G, which we will use explicitly only where cial units relating to brightness: the , the lu- it is unavoidable, because of its large uncertainty. men, and the . These units, however, are designed For the purposes of this paper, it is not necessary to quantify the physiological effect of light. The SI to consider the most general or most sophisticated specifies that a watt of monochromatic light form of each process. The approximate conversions at 540 THz is worth 683 lumens, thus defining a lu- Eqs. (3), (6), and (15) will be useful, as will some men. (A lux is a per square , while a common spherical-cow idealizations. candela is a lumen per .) Light at other vis- ible has fewer lumens per watt, the exact 3.1 The Sun and the CMB number being given by a model throughput function for human vision, known as the luminous efficacy. Let us compare the nearest and furthest sources of Household LED deliver about 100 lumens of light in astronomy. For this purpose, we approximate light per watt of electrical power. the solar surface by a blackbody at T = 5800 K, and the cosmic background by an ideal Planckian with Tcmb = 2.725 K. The for- 3 Examples mer approximation is comparatively drastic, but that does not matter for this example. The preceding sections drew attention to the char- The photon flux per logarithmic spectral interval acteristic style of the new SI, wherein constants are has a simple interpretation for thermal sources. A defined explicitly and units are defined implicitly blackbody has thereby, and suggested that the classical astronom- c 2π ical units for length, mass, and brightness could be φ = × (16) λ 3 hc/λkT replaced by SI units while leaving metres, kilograms, λ e − 1 and watts per implicit. We will now discuss ex- at its surface. For a spherical blackbody of ra- amples from different topics in astrophysics, illustrat- dius r1 observed from distance r2, φλ is reduced 2 ing how expressing distance in light-seconds, mass in by a factor (r1/r2) . The φλ spectrum peaks at −1 gravity-seconds, and brightness as φν or φλ can be 5.100 mm×T K. At this peak value, the second fac- useful in describing diverse astrophysical processes. tor in Eq. (16) is of order unity (actually 0.40). This The basic strategy is to work in the SI, while using makes it useful to imagine the c/λ3 as a packed col- the freedom given by the new SI to use any equa- umn of ball-like photons whose diameter is the peak tion of physics to change variables or introduce new wavelength, moving at the . quantities. Light-seconds, gravity-seconds, and loga- From the photon spectrum (16) it is clear that rithmic spectral scales are particular instances of this the photon flux will be ∝ T 3. Thus, at the solar 3 strategy — a time variable may be used to repre- surface, the Sun has a photon flux (T /Tcmb) ' sent a length divided by c, and so on. There is no 1010 times that of the CMB. From a distance of 3/2 requirement to use light-seconds for all lengths, or (T /Tcmb) R ' 450 au the photon fluxes become gravity-seconds for all masses — we can use what- equal. Figure 1 illustrates. The corresponding peaks ever formulation is most convenient, while keeping to are at 1.871 mm (or 160.2 GHz) for the CMB and the principle that all dimensional quantities are un- 880 nm for the Sun. The latter may seem wrong, be- ambiguously in SI units. Classical astronomical units cause the solar spectrum is well-known to peak in the will be converted to SI units at the input stage, and middle of the visible range — but that is only because sometimes classical units will be re-introduced at the the quantity usually plotted for the solar spectrum is −2 output stage. What we want to avoid is mixing unit the energy against wavelength, or hcλ φλ. The AB

5 The classical Oort parameters A and B, defined as v ∂v 2A = φ − φ r ∂r (19) v ∂v −2B = φ + φ r ∂r in cylindrical coordinates, describe the differential rotation of the Galaxy. Solid-body rotation would give A = 0, whereas a flat rotation curve gives A + B = 0. Recent measurements (e.g., Li et al., 2019) give A − B ' 28 m s−1 pc−1 and all the other components are an order of magnitude smaller. Ex- Figure 1: Approximate photon spectrum of the Sun, pressing this in SI units as A − B ' 9 × 10−16 s−1 if seen from about 450 au, and the all-sky spectrum does not seem much of an improvement. If we note, from the CMB, shown as photons m−2 s−1 and AB however, that A − B is an , then magnitudes. A − B = vφ/r ' 2π/(220 Myr) is simple. In the case of the Hubble parameter, it is easy to work with the Hubble time magnitude of the Sun is indeed brightest in filters near 900 nm (see e.g., Willmer, 2018). −1 17 H0 = (4.4 ± 0.2) × 10 s = (14 ± 0.5) Gyr (20) which, as well as being the reciprocal expansion , 3.2 The Oort and Hubble parameters is also approximately the time since the Big Bang. This already applied in the old SI.3 The new SI, how- Classical astronomical units often appear mixed with ever, encourages further useful formulations. Recall 2 SI units. Thus, we may have an expression of the that 3/(8πG) × H0 is the critical density of the Uni- type verse. The result in kg m−3 is not very intuitive, but ∂v with the help of the SI constants we can also write i (17) ∂x 3 H2 c2 j 0 × ' 5 GeV m−3 (21) 8π G e where vi is a velocity field (mean velocity at location −1 or roughly 5 atomic mass units per cubic metre. (Of xi), with vi measured in km s and xi in pc. Di- mensionally, such expressions are inverse times, and course, only a fraction Ωb ' 0.05 of this will be in it is useful to remember baryonic matter.) If the Hubble time is expressed in seconds, no astro-specific unit-conversion factors are −1 −1 −1 −1 −1 needed, just constants in SI units. Furthermore, the 1 m s pc = 1 km s kpc ≈ (1 Gyr) (18) 2 density expression 3/(8π)×H0 in gravity-seconds per cubic light-seconds can also be of use. Dividing by while more digits are given in Table 2. The divergence M gives the density in solar masses per cubic light (trace of Eq. 17) of the velocity field of galaxies is the second. We can then calculate a notional distance Hubble constant. The (generalized) Oort constants are analogous quantities for the stellar in  3 Ω H2 −1/3 b 0 ∼ c × 5 × 1010 s (22) the solar neighbourhood. Of course, none of these 8π M are really constants, just values at our location or 3 time, so ‘parameter’ may also be used. Conversions of For example, Sandage (1962) in his prediction of redshift −1 drift (which has yet to be observed, but see Lazkoz et al., the type 100 km s−1 Mpc ' (9.78 Gyr)−1 are often −1 2018) gives H0 = 13 Gyr as an illustrative value, and does used in both contexts. not bother with km s−1 Mpc−1 at all.

6 meaning that if all the baryonic matter in the Uni- to define a dimensionless constant β. For a circular verse were in Sun-like , these would be on aver- β is clearly the in light units. For age roughly 500 pc apart. general bound , β can be worked out using the virial theorem as the orbit-averaged rms speed 3.3 The and Andromeda phv2i β = (26) We saw above that classical astronomical units can c leave us having to disentangle a mixture of pc, −1 in light units. The can be expressed as km s , and M , with G in SI or cgs units thrown in for good measure. Another example where this hap- M P = 2π (27) pens is the Local Group timing argument, in which β3 the Galaxy and M31 appear as an unusual but very interesting binary system. thus relating mass in gravity-seconds to an observable The basic observational facts are that the two time. Expressinga ¯ in Eq. (25) as a time is not simply galaxies are some 800 kpc apart and are approaching formal either — in pulsar binaries (see e.g., Lorimer each other at roughly 120 km s−1. Using this infor- & Kramer, 2012) the light crossing time is the ob- mation, one can estimate the combined mass of these servable size of the orbit (because no resolved image two galaxies by computing how much mass would be of the system is observed), and is known as the Roe- needed to have countered the expansion of the Uni- mer time delay after the 17th-century measurement verse in the Local Group in a Hubble time. The idea of the light-travel time across the . The goes back to Kahn & Woltjer (1959) and has been Earth’s orbit provides a nice illustration of Eqs. (25) −6 developed further by many researchers (e.g., Banik and (27). Since M = 5 × 10 s anda ¯ = 500 s we −4 −1 & Zhao, 2016). The inferred mass provides a simple have β = 10 (or 30 km s ). For the orbital period and robust estimate of the dark matter in the Local we recover the well-known mnemonic that a year is 7 Group. π × 10 s. Converting the distance to light seconds, we have Turning now to the Galaxy and Andromeda, let us consider these as being on a radial two-body orbit. ¯ 13 ¯ −4 RM31 ' 8 × 10 s dRM31/dt ' −4 × 10 (23) Such a system has a well-known solution, which in using the notation from Eq. (4) to denote distance our notation is in light-seconds. Dividing distance by speed gives a t = M β−3(ψ − sin ψ) formal time of 2×1017 s, which is of the same order as R¯ = β−2(1 − cos ψ) (28) the Hubble time. This suggests modelling the system M31 M ¯ as a radial two-body orbit that started to move out at dRM31/dt = β sin ψ/(1 − cos ψ) the Big Bang, and has since turned around and is now with a formal independent variable ψ serving to give approaching a collision. To do so, let us introduce the time dependence implicitly. To find the current some notation for binaries in general. value of ψ, we consider the dimensionless product Consider two masses, M1, M2 in gravity seconds, in a two-body orbit, with t dR¯ sin ψ (ψ − sin ψ) × M31 = (29) ¯ 2 M1M2 RM31 dt (1 − cos ψ) M ≡ M1 + M2 η ≡ (24) M 2 and in it we put t = 4 × 1017 s (about a Hubble time) being the total mass and the symmetric mass ratio. and the distance and velocity values from Eq. (23) Leta ¯ be the orbital semi-major axis in light-seconds, The value of the expression (29) comes to −2. A and let us use the expression numerical solution for ψ yields ψ ' 4.2. With ψ M determined, it is easy to solve for β and M . The a¯ = (25) 7 β2 result is M = 2.5 × 10 s. Recalling the conversion

7 12 (6) gives a mass of 5 × 10 M , which is more than gives M = 15 ps [Table 2 gives the precise value.] an order of magnitude above the in the (iii) equating 2πM /β3 to the orbital period of 12 hr Local Group. for GPS satellites gives β, (iv) using β2 = M /a¯ In conventional astronomical units, the above steps givesa ¯. Doing the arithmetic, we find a delay of would be basically the same. But because we con- 38.5 microsec per day. verted the input to SI at the start, and converted GPS are also in orbit around the Sun. the inferred mass from gravity-seconds to M at the Hence, though they tick faster than terrestrial clocks, very end, the calculation was much easier. A com- clocks run slower than a notional TCB clock. puter is useful when solving for ψ, but the rest of the To find the time delay over one orbit (that is, one arithmetic is trivial. year), we use the first term in Eq. (30) and substi- tute the solar mass and the Earth’s orbital speed. 3.4 Time dilation in orbiting clocks The latter, we have already seen, is yet another con- veniently round number β = 10−4. The result is When the SI was first instituted in 1960, the second 0.5 s per year. The solar mass parameter (cf. Eq. 2) 20 3 −2 was defined from astronomy, as a fraction of a mean is GM = 1.32712442099(10) × 10 m s in TCB solar day. The -clock standard changed that but measurably different with respect to terrestrial a few years later. Still, astronomy has not entirely time (Luzum et al., 2011). ceded time measurement to atomic physics, because For eccentric orbits, the time dilation is more com- some applications require time dilation to be taken plicated in detail, but of the same order. A good into account. The TCB (temps coordonn´eebarycen- example is the /S0-2, which orbits the black trique) is defined as the time kept by clocks moving hole at the center of the Milky Way in a highly eccen- with the solar-system but outside all grav- tric orbit. The spectral features of the star amount to itational fields (Brumberg & Groten, 2001). a natural clock, and its orbital time dilation has re- It is well known that global navigation satellites cently been measured (Abuter et al., 2018; Do et al., have to correct for relativistic time dilation. A de- 2019). Further observable times of the form M βn tailed treatment can be found in Ashby (2003) but a (with positive n) can emerge from higher-order rel- simple estimate makes a nice illustration of gravity- ativistic effects (Ang´elil& Saha, 2014) but have not seconds, as follows. Let us consider a two-body or- been measured yet. bital system as in subsection 3.3 above, except that η → 0 since the satellite mass is negligible, and M is simply the Earth’s mass. Let us write R¯ for the 3.5 Shapiro and Refsdal delays radius of the Earth in light seconds. A clock at the As well as the various M β−n in seconds, there is a surface of the Earth runs slower than TCB by a frac- ¯ measurable time that is simply M times a numerical tion M /R, neglecting the spin of the Earth. From factor. As one might guess from the absence of β, it weak-field relativity, a clock in a circular orbit of ra- involves light. diusa ¯ (in light-seconds) runs slower than TCB by A light ray, that on its way between source and 3 M /a¯. The difference of these two time dilations is 2 observer has flown past a mass M , experiences a time observable. Multiplying by the orbital period (27) delay and rearranging, we have a delay of − 2M ln(1 − cos θ) (31) 3 a¯  M 2π − (30) where θ is the angle on the observer’s sky between 2 R¯ β the mass and the incoming ray. The in per orbit. We can estimate its value from easily- (31) will be negative, assuming θ is not too large, and remembered quantities, as follows: (i) the circum- hence the whole expression will be positive. This de- ference of the Earth is 40 000 km, which gives R¯, lay is well known in ranging experiments in the solar (ii) using g/c = M /R¯2 and putting g ' 9.8 m s−2 system, and in pulsar timing, and is known as the

8 Shapiro delay after the prediction by Shapiro (1964). M , η, and β. For this system let us write Another manifestation of the same phenomenon, ω = β3/M (33) predicted by Refsdal (1964) by very different argu- ments, appears in gravitational lensing. In the regime for the angular orbital frequency, and of lensing, θ is small. Taking the small-θ limit of (31) = − 1 η β2 (34) gives −2M (2 ln θ − ln 2). But at small θ, the deflec- E 2 M tion ∆θ of the light ray is important, as it contributes for the orbital energy in gravity-seconds. a further time delay. The time delay depends on θ Such a binary will produce gravitational waves of and ∆θ as 2ω, and the strain components at distance D¯ will be t(θ) = D¯ ∆θ2 − 4M ln θ (32) E h ∼ (35) GW D¯ where D¯ is an effective distance (in light-seconds) de- pending on the distances to the mass and the light with numerical coefficients of order unity, depend- source. Light then follows Fermat’s principle and ing on orientation. It may be convenient to think of chooses θ and ∆θ so as to make the total light travel the strain as a kind of gravitational potential whose time extremal (e.g., Blandford & Narayan, 1986). source is not the mass but the orbital energy. If we There can be more than one extremal light path, giv- further associate the with an energy density 2 2 ing multiple images with different light travel times. ∝ ω hGW (cf. Mathur et al., 2017) and moving with A good example is Refsdal observed to a speed of light, it follows that energy will be emit- 2 2 appear at displaced locations with time delays (Kelly ted at a rate ∝ ω E . Writing the proportionality et al., 2016). In general, a will not constant as κ, we have be a single mass but an extended mass distribution. dE = −κ ω2E 2 . (36) Accordingly, the last term in Eq. (32) has to be re- dt placed by an integral over the mass density. The ob- This paragraph is not a derivation, just a plausi- servable time delay between different lensed images bility argument. Nevertheless, the expression (36) then depends on details of the how the mass is dis- with κ = 128/5 turns out to be the correct leading- tributed (see e.g., Mohammed et al., 2015). order relativistic result for circular binaries (Peters If a mass is at a cosmological distance (say at - & Mathews, 1963). Eccentric orbits and higher order shift z), the Shapiro or Refsdal delay as measured modify the numerical factor, but do not change the by an observer will be time-dilated by (1 + z), just units needed. Notice that the power output (36) is like any other time interval at that redshift. That is, in gravity-seconds per second. To convert to watts, mass in gravity-seconds gets redshifted. we need to multiply by c5/G (as noted at Eq. 10). Low-mass gravitationally radiating binaries can be 3.6 Gravitational-wave inspiral just as luminous as supermassive binaries, but the latter take longer. To see this, we eliminate E and ω In general relativity it is common to put G = from the formula (36) and rearrange to get c = 1, leaving units and dimensions implicit (ge- dβ β9 ometrized units — see Appendix F in Wald, 1984). = 1 κη × (37) dt 4 Light seconds and gravity seconds are similar, but M with explicit variable changes to keep track of units giving the increasing speed of the inspiralling system. and dimensions, and using these we can express Integrating to get the time left before β = 1 (when gravitational-wave inspiral very concisely, while also the system would merge), we get keeping the comparison with observations simple. 1 M T = (38) Let us again consider a binary with parameters insp 2κη β8

9 for the inspiral time. Here we have yet another time ' 20 Hz (wave frequency of ' 40 Hz), while Eq. (38) n scale of the form M /β . implies Tinsp ' 30 s, reproducing the observed values. The parameters M , η, and β are, in general, not Galactic binary pulsars are younger systems like directly observable for inspiralling binaries. The ob- GW170817. The gravitational-wave inspiral of these servables are the two components (polarizations) of systems has been known since the first measure- the strain, the wave frequency 2ω, and its derivative, ment by Taylor et al. (1979), but detecting the known as the chirp. In terms of there we can write gravitational-wave strain from these low-frequency sources is not feasible yet. Interestingly, the hGW 1 dω 3 −1 3 2 from Galactic binary pulsars is comparable to that = 8 Tinsp = 2 κω E . (39) ω dt from GW170817. To see this, we can simply scale Since the left expression is observable, the other two −2 ¯ −4 ¯ −1 β → 10 β, D → 10 D expressions become measurable. That Tinsp can be inferred from the frequency and chirp is not surpris- which leaves hGW the same. The distance changes ing. That E is measurable is remarkable, and has im- to 4 kpc, a 20 Hz orbital frequency (∝ β3) changes 1 5/3 2/3 portant consequences. Writing E as 2 ηM ω , we to a half-day orbital period, and a 30 s inspiral time see that the combination η3/5M is also measurable. (∝ β−8) changes to 1010 yr. These values are typical It is known as the chirp mass. Still more interest- of Galactic binary pulsars (see Lorimer, 2008). ing is that gravitational-wave binaries can serve as “standard sirens” enabling distance measurements, 3.7 Eddington luminosity and the M87 black as first noted by Schutz (1986). To see how, recall hole from Eq. (35) that E relates strain and distance. The coefficients in the equation can be determined In the preceding examples, matter only contributed a from the measurable polarization of the gravitational gravitational field, and hence mass always appeared ¯ wave. This makes D measurable too, and from it H0 multiplied by the gravitational constant, which we as well if the redshift is measured. were able to absorb inside M , thus eliminating the Regarding redshifts, it is important to note that a uncertainty in G. If matter contributes in other ways redshift z dilates t in the observer frame by 1 + z. As too (such as producing gas ), mass will ap- a result, observed frequencies are slowed down, and pear as both M and GM. This will make the un- the inferred chirp mass is dilated, by the same factor. certainty in G unavoidable — and also offer a way to The speed β is not affected. To make the strain in measure G, as in Christensen-Dalsgaard et al. (2005). Eq. (39) come out right, we need to dilate D¯ by 1 + z An exceptional but important situation, in that as well (i.e., use the luminosity distance). Again, mass appears only as GM even though non- as was the case in lensing, we see mass in gravity- gravitational processes are involved, is Eddington lu- seconds at cosmological distances getting redshifted. minosity. In this one has a spherical mass M of ion- GW170817 (Abbott et al., 2017) was a particu- ized gas, and some energy source which gives it a lu- larly interesting gravitational-wave source, providing minosity L, and the from the latter a wealth of observations in addition to the inspiral, balances the self-gravity. At any radius r inside the and taken together these enabled a reconstruction of sphere, we have the system as two neutron stars at ' 40 Mpc, which  2 corresponds to GMmp L/c 2 αh 2 = 2 × (40) r 4πr 3π mec −5 1 ¯ 15 M ' 1.5 × 10 s η ' 4 D ' 4 × 10 s. where mp and me are the masses of the proton and The level of strain at merger would be M /D¯ ∼ 10−21. electron, and α is the fine-structure constant. On If we assume β = 0.12 at the start of the detected the left of this equation we have the weight of an ion, event, Eq. (33) gives an initial orbital frequency of and on the right we have the outward momentum flux

10 times the Thomson cross-section. Radiation pressure have a much lower effective , because it is acts on the , but the force is transmitted accreting only weakly now. The measured brightness electrostatically to the ions. Expressing the particle temperature at mm wavelengths is, however, much 2 masses as equivalent frequencies νe = mec /h, νp = higher. This tells us that the mm radiation cannot 2 mpc /h and rearranging, we get the Eddington lumi- be thermal and must be predominantly reprocessed. nosity L = 6π2 hν (ν /α)2 (41) p e M 4 Discussion written with mass in gravity-seconds. Although originally developed for massive stars, The recent reforms of the SI have made the formerly the Eddington luminosity is nowadays also often ap- unexciting subject of units scientifically novel. As- plied to accreting black holes to estimate the max- tronomers, however, have always been unwilling to imum possible luminosity. by black holes adopt SI units. is not a spherical process, so applying the formula The persistence of classical and other pre-SI units to black holes gives a rough estimate at best, but in astrophysics actually has an interesting scientific is nonetheless interesting. Let us accordingly ap- reason, namely the difficulty of calibrating astronom- proximate an accreting black hole as a blackbody ical observables against the laboratory standards on sphere at temperature T . It will radiate at 2π5c/15× which the SI and its predecessors are based. The (kT )4/(hc)3 per unit . For the radius of the calibration problems are mostly solved now, but one sphere, we take the radius of the innermost stable very important problem remains: the uncertainty in orbit, which is 6M (or 6M c in length units) for a G, which makes the kilogram unusable in some pre- non-spinning black hole. Equating the total lumi- cision applications. Without the kilogram, any pro- nosity to the Eddington luminosity we can define an posal to change to SI units becomes a non-starter. effective temperature At most, one sees arguments for a partial change to SI units (see Dodd, 2011). And so it is that every h 5ν2ν 1/4 T = e p (42) new research student in astronomy, having mastered 2πk α2M basic physics with SI units, is confronted with mag- nitudes, and solar masses, as well as pre-SI for the accreting system. The formula for this decimal like A,˚ ergs, and gauss. Ex- “Eddington temperature” seems strangely reminis- pressions mixing different unit systems are especially cent of the much much colder Hawking temperature painful, and make it difficult to catch errors. h/(4k ). M The new SI, by giving physical constants the cen- Akiyama et al. (2019) present an interferometric tral role, encourages reformulation of observables by image of the silhouette of the inserting factors of c, h, and so on. With this freedom, in M87 at a resolution of 20 µas or 10−10 radians. The the classical units of length, brightness, and mass can resolution is as expected for mm-wavelengths with a be usefully replaced by SI units having different di- baseline of ∼ 104 km. The distance to M87 being mensions. ' 20 Mpc, the resolved size comes to 2 × 10−3 pc, 5 which is like 2 × 10 s or two light days. The inferred • The au and pc are easily replaceable by light- mass is about an order of magnitude smaller than this seconds, and both conversions are close to round 9 4 scale: 6 × 10 M or 3 × 10 gravity-seconds. Plug- numbers (500 s and 108 s). ging the mass in the formula (42) gives T ' 8×104 K. Taken as an upper limit, this value is very reasonable, • AB magnitude is equivalent to the photon since a continuum peak around 100 nm, correspond- flux per logarithmic spectral interval, which ing to an effective temperature of T ≈ 3 × 104 K, is is much easier to understand and work with. typical of (Francis et al., 1991; Vanden Berk For typical optical bands, zero magnitude ' et al., 2001). The M87 black-hole system itself would 1010 photons m−2 s−1.

11 • Measuring astronomical mass in gravity-seconds has already been used in this paper.

may seem contrived at first, but it is really a 8 simple variant of the mass parameter used in • Rounded parsecs of exactly 10 light-seconds solar-system dynamics, and can help gain new (which is just over π light-years) would be 3% insight into diverse astrophysical processes. Es- smaller than parsecs. For σ8 in cosmology, the pecially nice is how some very different observ- power-spectrum would get averaged over a vol- able time scales in orbital systems are set by the ume about 10% smaller, and the change in that mass in gravity-seconds and the dimensionless average may be insignificant. A rounded au of 500 light-seconds would also be useful. speed: the classical Roemer delay is ∼ M /β2 3 and the orbital period is 2πM /β , in general • Zero magnitude could be conveniently rounded relativity the time dilation per orbit is ∼ M /β, to 1010 photons m−2 s−1, with reference to a 8 the gravitational-wave inspiral time is ∼ M /β broad band whose width is 20% of its median. and the Shapiro and Refsdal delays are ∼ M . Rounding the classical astronomical units in this way The light-second and gravity-second are not to be would be harmless for most applications. considered as new (and therefore non-SI) units with dimensions of length and mass. They are simply the Thanks to P.R. Capelo, J. Magorrian, A. Saha, second being used to measure a length times c−1, or R. Sch¨onrich, L.L.R. Williams, and the referee for measure a mass times G/c3. comments on earlier versions. An indirect benefit of the classical astronomical units is that working astronomers are quite used to converting between different unit systems. Theoret- References ical calculations of idealized systems may be done Abbott, B. P., Abbott, R., Abbott, T. D., et al. 2017, in geometrized units or even Planckian units (e.g., Phys. Rev. Lett., 119, 161101 Saha & Taylor, 2018), and compute-intensive work often favours internal units to improve numerical per- Abuter, R., Amorim, A., Anugu, N., et al. 2018, formance. All of these require unit conversion at A&A, 615, L15 input and output stages, but provided unit conver- sion is a minor overhead, it does not cause problems. Akiyama, K., Alberdi, A., Alef, W., et al. 2019, Hence, SI replacements for the classical astronomi- ApJL, 875, L1 cal units can simply be incrementally introduced by Ang´elil,R. & Saha, P. 2014, MNRAS, 444, 3780 early adopters, without requiring any formal policy changes. Ashby, N. 2003, Living Reviews in Relativity, 6, 1 All that said, which astronomer does not love their Banik, I. & Zhao, H. 2016, MNRAS, 459, 2237 parsecs and magnitudes? Moreover, there are param- eters that have a parsec inside their definitions: ab- Bessell, M. S. 2005, &A, 43, 293 solute magnitude and the conventional normalization BIPM. 2019, Le Syst`emeinternational d’unit´es/ The of the cosmological power spectrum σ8. Now, there is an interesting social phenomenon that sometimes International System of Units (‘The SI Brochure’), the word for an archaic unit survives, but changes ninth edn. (Bureau international des poids et its meaning to a round number of the new unit. For mesures) example, contemporary German usage has rounded Blandford, R. & Narayan, R. 1986, ApJ, 310, 568 up a ‘Pfund’ from a pound to 500 g, while in South Asia a ‘tola’ has been rounded down to 10 g. One can Brumberg, V. A. & Groten, E. 2001, A&A, 367, 1070 imagine the same for the classical astronomical units. Chao, L. S., Schlamminger, S., Newell, D. B., et al. −6 • A rounded solar-mass unit as M = 5 × 10 s 2015, American Journal of Physics, 83, 913

12 Christensen-Dalsgaard, J., Di Mauro, M. P., Schlattl, Peters, P. C. & Mathews, J. 1963, Physical Review, H., & Weiss, A. 2005, MNRAS, 356, 587 131, 435

Cooper, P. W. 1966, American Journal of Physics, Refsdal, S. 1964, MNRAS, 128, 295 34, 703 Rosi, G., Sorrentino, F., Cacciapuoti, L., Prevedelli, Do, T., Hees, A., Ghez, A., et al. 2019, Science, 365, M., & Tino, G. M. 2014, Nature, 510, 518 664 Sagitov, M. U. 1970, Sov. Astr., 13, 712 Dodd, R. 2011, Using SI Units in Astronomy (Cam- Saha, P. & Taylor, P. A. 2018, The Astronomers’ bridge University Press) Magic Envelope (Oxford University Press) Francis, P. J., Hewett, P. C., Foltz, C. B., et al. 1991, Sandage, A. 1962, ApJ, 136, 319 ApJ, 373, 465 Schaefer, B. E. 2013, Journal for the History of As- Goffin, E. 2014, A&A, 565, A56 tronomy, 44, 47 Hanks, T. C. & Kanamori, H. 1979, JGR, 84, 2348 Schutz, B. F. 1986, Nature, 323, 310 Jones, D. 1968, Leaflet of the Astronomical Society Shapiro, I. I. 1964, Physical Review Letters, 13, 789 of the Pacific, 10, 145 Taylor, B. N. & Mohr, P. J. 2001, IEEE Transactions Kahn, F. D. & Woltjer, L. 1959, ApJ, 130, 705 on Instrumentation and Measurement, 50, 563

Kelly, P. L., Rodney, S. A., Treu, T., et al. 2016, Taylor, J. H., Fowler, L. A., & McCulloch, P. M. ApJL, 819, L8 1979, Nature, 277, 437

Lazkoz, R., Leanizbarrutia, I., & Salzano, V. 2018, Tonry, J. L., Stubbs, C. W., Lykke, K. R., et al. 2012, European Physical Journal C, 78, 11 ApJ, 750, 99

Li, C., Zhao, G., & Yang, C. 2019, ApJ, 872, 205 Vanden Berk, D. E., Richards, G. T., Bauer, A., et al. 2001, AJ, 122, 549 Li, Q., Xue, C., Liu, J.-P., et al. 2018, Nature, 560, 582 Wald, R. M. 1984, General relativity (University of Chicago Press) Lorimer, D. R. 2008, Living Reviews in Relativity, 11 Willmer, C. N. A. 2018, ApJS, 236, 47 Lorimer, D. R. & Kramer, M. 2012, Handbook of Pulsar Astronomy (Cambridge University Press) A Constants and conversion factors Luzum, B., Capitaine, N., Fienga, A., et al. 2011, Ce- lestial Mechanics and Dynamical Astronomy, 110, This Appendix summarizes the various constants and 293 conversion factors used in this paper. Table 1 consists of physical constants relevant to astronomy, while Ta- Mathur, H., Brown, K., & Lowenstein, A. 2017, ble 2 has conversion factors from the classical astro- American Journal of Physics, 85, 676 nomical units. Mohammed, I., Saha, P., & Liesenborgs, J. 2015, The first four constants in Table 1 have defined PASJ, 67, 21 values in the new SI. The others are experimentally determined and hence have uncertainties. Neither is Oke, J. B. & Gunn, J. E. 1983, ApJ, 266, 713 a complete set, but simply the subset important in

13 astrophysics. Taking advantage of equivalences in the new SI, electric charge is written in joules per volt, and particle masses in e/c2 times volts. It is worth mentioning that the vacuum permeability µ0 is no longer a defined constant; instead, permeability and are given by Table 1: Physical constants in the new SI 1 h −1 c 299792458 m s cµ0 = = 2α 2 (43) c0 e h 6.62607015 × 10−34 J s −19 −1 the latter constant being the vacuum impedance ' e 1.602176634 × 10 JV 377 Ω. k 1.380649 × 10−23 JK−1 Table 2 expresses the classical astronomical units G 6.6743(2) × 10−11 kg−1 m3 s−2 in terms of SI units. Note that in each case, some simple change of variable is involved. The first four 1/α 137.03599908(2) numerical factors are actually exact numbers (that 2 me e/c × 0.5109989500(2) MeV is, derived from defined constants) but have been 2 mp e/c × 0.9382720882(3) GeV rounded to eight digits in the table. The mass values are measured quantities whose current uncertainties are smaller than the eight digits given here. As noted in the main text, all the numerical values are close to some round number and hence easy to remember ap- proximately.

Table 2: Classical astronomical units in SI terms AB = 0 5.4795384 × 1010 m−2 s−1 au c × 499.00478 s pc c × 1.0292713 × 108 s m s−1 pc−1 3.2407793 × 10−17 s−1 (0.9777922 Gyr)−1 3 −6 M c /G × 4.9254909 × 10 s 3 −11 M⊕ c /G × 1.4793661 × 10 s

14