Solar-Mass Geometric Conversions

Solar-mass geometric units turn a dimensionless relativistic calculation into lengths, times, and frequencies without first assigning the Sun a mass in kilograms. The safe input is the mass parameter $GM$: astronomical dynamics usually determines this product more directly than either factor separately. This card uses an exact nominal solar conversion in the denominator and keeps the chosen physical mass, frequency convention, and reference frame explicit.

The nominal solar mass parameter

The International Astronomical Union defines the nominal solar mass parameter

$$(GM)_\odot^{\mathrm N}=1.3271244\times10^{20}\ \mathrm{m^3\,s^{-2}}$$

as an exact conversion constant, not as a measurement of the present Sun IAU 2015, Resolution B3, Recommendations 1–4 and solar conversion table, printed pp. 1–2, PDF. For a chosen system mass parameter $GM$, define on this card

$$\mathcal M\equiv\frac{GM}{(GM)_\odot^{\mathrm N}}.$$

Thus $\mathcal M$ is a dimensionless scaling ratio. It inherits any uncertainty in the inferred numerator; the nominal denominator contributes none. Here $\mathcal M$ does not automatically mean the binary chirp mass, despite the common use of the same calligraphic letter for that different quantity.

Exact geometric units per nominal solar mass

The SI speed of light is exactly $c=299\,792\,458\ \mathrm{m\,s^{-1}}$ BIPM 2026, SI Brochure, §2.2, Table 1, printed p. 124, PDF. Combining two exact conversion constants gives

$$\begin{aligned} r_{g,\odot}^{\mathrm N} &\equiv\frac{(GM)_\odot^{\mathrm N}}{c^2}, &t_{g,\odot}^{\mathrm N} &\equiv\frac{(GM)_\odot^{\mathrm N}}{c^3},\\ r_{s,\odot}^{\mathrm N} &\equiv 2r_{g,\odot}^{\mathrm N}. \end{aligned}$$

Conversion	Rounded decimal display
$r_{g,\odot}^{\mathrm N}$	$1.476625038\ldots\ \mathrm{km}$
$t_{g,\odot}^{\mathrm N}$	$4.925490948\ldots\ \mathrm{\mu s}$
$r_{s,\odot}^{\mathrm N}$	$2.953250076\ldots\ \mathrm{km}$

The formulas, rather than the displayed decimals, are exact. The ellipses mark rounding of nonterminating decimal expansions; they are not experimental error bars. The identities $ct_{g,\odot}^{\mathrm N}=r_{g,\odot}^{\mathrm N}$ and $r_{s,\odot}^{\mathrm N}=2r_{g,\odot}^{\mathrm N}$ provide immediate unit and normalization checks.

Rescaling length, time, and frequency

Suppose a calculation uses $GM/c^2$ as its length unit and $GM/c^3$ as its time unit. If $\widehat L$, $\widehat t$, and $\widehat\omega$ are defined by

$$\widehat L=\frac{Lc^2}{GM},\qquad \widehat t=\frac{tc^3}{GM},\qquad \widehat\omega=\frac{\omega GM}{c^3},$$

then the physical quantities are

$$\begin{aligned} L&=\widehat L\,\mathcal M r_{g,\odot}^{\mathrm N}, &t&=\widehat t\,\mathcal M t_{g,\odot}^{\mathrm N},\\ \omega&=\frac{\widehat\omega} {\mathcal M t_{g,\odot}^{\mathrm N}}, &f&=\frac{\widehat\omega} {2\pi\mathcal M t_{g,\odot}^{\mathrm N}}. \end{aligned}$$

Numerically, with the exact nominal inputs rounded only for display,

$$\begin{aligned} L&=(1.476625038\ldots\ \mathrm{km})\,\mathcal M\widehat L,\\ t&=(4.925490948\ldots\ \mathrm{\mu s})\,\mathcal M\widehat t,\\ \omega&=(2.030254467\ldots\times10^5\ \mathrm{s^{-1}}) \frac{\widehat\omega}{\mathcal M},\\ f&=(3.231250342\ldots\times10^4\ \mathrm{Hz}) \frac{\widehat\omega}{\mathcal M}. \end{aligned}$$

The last line assumes the code reports dimensionless angular frequency $\widehat\omega$. If it instead reports $\widehat f=fGM/c^3$, then $f=\widehat f/(\mathcal M t_{g,\odot}^{\mathrm N})$ with no additional $2\pi$. The same time map converts waveform timestamps, damping times, and orbital periods once their normalization is declared.

Two factors that must not be guessed

The first trap is the factor of two. The geometric mass length is $r_g=GM/c^2$, whereas the Schwarzschild horizon radius is $r_s=2GM/c^2$. A code statement such as “$r=2M$” normally uses $M=r_g$; replacing that $M$ by a quoted Schwarzschild radius doubles the answer again. For rotating black holes, neither $2r_g$ nor the Boyer–Lindquist coordinate $r_+$ should be treated as a universal invariant size; use the convention- controlled Black-Hole Horizon Scales record.

The second trap is $2\pi$. Angular frequency obeys $\omega=2\pi f$. Before using a tabulated prefactor, inspect both the symbol and the defining equation: a number labeled merely “frequency” does not reveal whether it is in $\mathrm{s^{-1}}$ or cycles per second. The broader restoration rules are in Units and Dimensional Conventions, and the standards inputs are collected in Fundamental Constants Used in Gravity.

Kilograms, mass definitions, and frames

Converting the exact nominal parameter to kilograms requires a measured value of $G$. Using the 2022 CODATA recommendation $G=6.67430(15)\times10^{-11}\ \mathrm{m^3\,kg^{-1}\,s^{-2}}$ Mohr et al. 2025, Table XXXIII, p. 025002-45, NIST-hosted PDF gives

$$\frac{(GM)_\odot^{\mathrm N}}{G} =1.988410(45)\times10^{30}\ \mathrm{kg}.$$

The relative standard uncertainty is that of $G$, about $2.25\times10^{-5}$ or $22.5$ parts per million. This kilogram value is therefore measured-model dependent; it is not another exact nominal constant, and changing the adopted $G$ changes it.

Finally, the conversion does not decide which mass belongs in $GM$. A binary calculation may use total mass, a component mass, chirp mass, or final remnant mass, while a perturbation calculation may normalize by the background mass. Source-frame and detector-frame masses differ when cosmological redshift is relevant, so the mass frame must match the frame of the reported time or frequency. If a detector-frame mass scales a detector-frame waveform, do not apply the redshift factor a second time. Object inference and the meanings of astrophysical mass parameters belong to Identifying Astrophysical Black Holes.

References

• Bureau International des Poids et Mesures. The International System of Units (SI). 9th ed., version 4.01. Sèvres: BIPM, 2026. DOI. Official PDF.
• International Astronomical Union. “Resolution B3 on Recommended Nominal Conversion Constants for Selected Solar and Planetary Properties.” XXIX General Assembly, 2015. Official PDF.
• Mohr, Peter J., David B. Newell, Barry N. Taylor, and Eite Tiesinga. “CODATA Recommended Values of the Fundamental Physical Constants: 2022.” Reviews of Modern Physics 97 (2025): 025002. DOI. NIST-hosted PDF.