Calibrations of SFR indicators have been presented in the literature for almost 30 years, derived across the full electromagnetic spectrum, from the X-ray, through the ultraviolet (UV), via the optical and infrared (IR), all the way to the radio, and using both continuum and line emission (see review by Kennicutt 1998, and, e.g., Donas & Deharveng 1984; Yun et al. 2001; Kewley et al. 2002, 2004; Ranalli et al. 2003; Bell 2003; Calzetti et al. 2005, 2007, 2010; Schmitt et al. 2006; Alonso-Herrero et al. 2006; Moustakas et al. 2006; Salim et al. 2007; Persic & Rephaeli 2007; Rosa-González et al. 2007; Kennicutt et al. 2007, 2009; Rieke et al. 2009; Lawton et al. 2010; Boquien et al. 2010; Verley et al. 2010; Li et al. 2010; Treyer et al. 2010; Murphy et al. 2011; Hao et al. 2011). The most recent review on the subject is by Kennicutt & Evans (2012).
Recent findings that most of the star formation at redshift z ~ 1-3 was enshrouded in dust (Le Floch et al. 2005; Magnelli et al. 2009; Elbaz et al. 2011; Murphy et al. 2011b; Reddy et al. 2012) have renewed interest in IR SFR indicators, particularly in the monochromatic (single-bands) ones, which can be in principle as straightforward to use as those already available at UV and optical wavelengths. This interest has been aided by the advent of high-angular resolution, high-sensitivity IR space telescopes (Spitzer, Herschel), that have enabled the calibration of monochromatic SFR indicators in nearby galaxies. The IR investigations complement efforts at UV and optical wavelengths to chart the SFR evolution of galaxies from redshift ~ 7-10 to the present (e.g., Giavalisco et al. 2004; Bouwens et al. 2009, 2010). The UV and optical may be the preferred SFR indicators at very high redshift, when galaxies contained little dust (e.g., Wilkins et al. 2011; Walter et al. 2012). The calibration of SFR indicators remains, however, a central issue for studies of distant galaxies (e.g., Reddy et al. 2010; Lee et al. 2010; Wuyts et al. 2012), since it can be affected by differences in star formation histories, metal abundances, content and distribution of stellar populations and dust between low and high redshift galaxies (Elbaz et al. 2011), and, possibly, by cosmic variations in the cluster mass function and stellar initial mass function (IMF, Wilkins et al. 2008; Pflamm-Altenburg et al. 2009).
Throughout this chapter, I refer to two categories of SFR calibrations: (1) `global', i.e., defined for whole galaxies, thus they are luminosity-weighted averages across local variations in star formation history and physical conditions within each galaxy; and (2) `local', i.e., defined for measuring SFRs in regions within galaxies, on sub-galactic/sub-kpc scale (e.g., Wu et al. 2005; Alonso-Herrero et al. 2006; Calzetti et al. 2005, 2007, 2010; Zhu et al. 2008; Rieke et al. 2009; Kennicutt et al. 2009; Lawton et al. 2010; Boquien et al. 2010, 2011; Verley et al. 2010; Li et al. 2010; Treyer et al. 2010; Liu et al. 2011; Hao et al. 2011; Murphy et al. 2011). While global SFR calibrations have received most of the attention in the past, both for objective limitations in the spatial resolution of the data and for their broader applicability to distant galaxy populations, local SFR calibrations have become increasingly prominent in the literature as important tools to investigate the physical processes of star formation.
The definition of a local SFR can, however, be problematic if referring to too small a region: for instance a single star cluster that formed almost instantaneously 15 Myr ago has a current SFR=0 (it is no longer forming stars), although stars were clearly formed in the recent past. To avoid such extreme situations, local SFRs are meant to refer to measurements performed on areas that include multiple star-forming regions, so that star formation can be considered constant over the relevant timescale for the SFR indicator used. For all practical purposes, such regions tend to be a few hundred pc across or larger.
In general, global calibrations are not necessarily applicable to local conditions and vice versa. The fundamental reason is that while the stellar and dust emission from entire galaxies can be treated, in first approximation, as if the galaxy were an isolated system, the same is not necessarily true for a sub-galactic region. Stellar populations mix within galaxies on timescales that are comparable to those of their UV light lifetime. The stellar IMF (i.e., the distribution of stellar masses at birth) may or may not be fully sampled locally. The star formation history may vary from region to region. Both young and old stars can heat the dust in a galaxy, and the dust spectral energy distribution (SED) and features provide little discrimination as to the source of the heating. Because of all these reasons, local SFR indicators are by far less settled than the global ones.
In what follows, I will discriminate between global and local SFR indicators, when appropriate. All calibrations are given for a Solar metallicity stellar population, when models are used.
1.2.1. General characteristics
Techniques for measuring the rate at which stars are being formed vary enormously, also depending on whether the target system is resolved into individual units (e.g., young stars) or not. In all cases, however, the basic goal is to identify emission that probes newly or recently formed stars, while avoiding as much as possible contributions from evolved stellar populations.
The timescale over which `recent' is a valid word also varies between
different applications and among different systems, but free-fall times
ff probably
provide a reasonable ballpark
scale. Most researchers would agree that `recent' refers to timescales
10-100 Myr when
considering whole galaxies, and
1-10 Myr when
considering regions or structures within
galaxies (e.g., giant molecular clouds, etc.).
The most common approach for measuring SFRs in resolved regions, such as regions within the Milky Way, is to count individual objects or events (e.g., supernovae) that trace the recent star formation (Chomiuk & Povich 2011). In the molecular clouds within 0.5-1 kpc of the solar system, this is accomplished by counting young stellar objects (YSOs), i.e., protostars at different stages of evolution, which, because they are still embedded in their natal clouds, are optimally identified in the IR. The total number of YSOs is converted to a SFR via:
![]() |
(1.1) |
where the mean YSO mass,
< M > depends weakly on the adopted
stellar IMF (see Section 1.2.2), and the lifetime
of a YSO is, with some uncertainty,
~ 2 Myr
(Evans et al.
2009;
Heidermann et al.
2010;
Gutermuth et al.
2011).
The SFR(YSO) is in units of
M
yr-1.
In unresolved systems, SFR indicators are merely measures of luminosity, either monochromatic or integrated over some wavelength range, with the goal of targeting continuum or line emission that is sensitive to the short-lived massive stars. The conversion from the luminosity of massive stars to a SFR is performed under the assumption that: (1) the star formation has been roughly constant over the timescale probed by the specific emission being used; (2) the stellar IMF is known (or is a controllable parameter) so that the number of massive stars can be extrapolated to the total number of high+low mass stars formed; and (3) the stellar IMF is fully sampled, meaning that at least one star is formed in the highest-mass bin, and all other mass bins are populated accordingly with one or more stars (see discussion in Section 1.2.2).
SFR indicators in the UV/optical/near-IR range (~ 0.1-5 µm) probe the direct stellar light emerging from galaxies, while SFR indicators in the mid/far-IR ( ~ 5-1000 µm) probe the stellar light reprocessed by dust. In addition to direct or indirect stellar emission, the ionising photon rate, as traced by the gas ionised by massive stars, can be used to define SFR indicators; photo-ionised gas usually dominates over shock-ionised gas in galaxies or large structures within galaxies (e.g., Calzetti et al. 2004; Hong et al. 2011). Tracers include hydrogen recombination lines, from the optical, through the near-IR, all the way to radio wavelengths, forbidden metal lines, and, in the millimetre range, the free-free (Bremsstrahlung) emission. The X-ray emission produced by high-mass X-ray binaries, massive stars, and supernovae can also, in principle, be used to trace SFRs. Finally, the synchrotron emission from galaxies can be calibrated as a SFR indicator (Condon 1992), since cosmic rays are produced and accelerated in supernova remnants, and core-collapse supernovae represent 70% or more of the total supernovae in star-forming galaxies (Bossier & Prantzos 2009).
The following five subsections describe in more detail a few of these SFR indicators for unresolved systems. The emission contribution to the galaxy luminosity from a potential active galactic nuclei (AGN) can be large, depending on the galaxy type and the wavelength of the SFR indicator. I assume that this potential contribution has been recognised and removed from the emission that is being used as a SFR indicator.
1.2.1.1. Indicators based on direct stellar light
The youngest stellar populations emit the bulk of their energy in the
restframe UV (< 0.3 µm); in the absence of dust attenuation,
this is the wavelength range `par excellence' to investigate star
formation in galaxies over timescales of
100-300 Myr, since
both O and B stars are brighter in the UV than at longer wavelengths. As
a reference, the lifetime of an O6 star is ~ 6 Myr, and that of a
B8 star is ~ 350 Myr. The luminosity ratio at 0.16 µm of an
O6 to a B8 star is ~ 90, but, if the stellar population follows a
Kroupa (2001)
IMF (see Section 1.2.2), for every O6
star formed, about 150 B8 stars are formed. Thus, at age zero, the UV
emission from the collective contribution of B8 stars is comparable to
that of O6 stars.
For a Kroupa stellar IMF, with constant star formation over 100 Myr, the
non-ionising UV (0.0912 µm <
<
0.3µm) stellar continuum can be converted to a SFR:
![]() |
(1.2) |
with SFR(UV) in
M
yr-1,
in
Å, and
L(
) in
erg/s. The stellar SED used for this calibration is
from Starburst99, with solar metallicity
(Leitherer et al.
1999).
The accuracy of the calibration constant is ± 15%, which takes into
account small variations as a function of
.
For constant star formation over timescales longer than 100 Myr, the
calibration constant only decreases by a few percent. However, for
shorter timescales, changes are more significant. For
= 10 Myr and 2
Myr, the constant is about 42% and a factor 3.45, respectively, higher
than in Equation 1.2
(Table 1.1). This shows that if star formation has
been active in a region on a timescale shorter than about 100 Myr, the
cumulative UV emission of massive stars is still increasing in
luminosity, and the calibration of any SFR(UV) indicator has to take
this fact into account.
Luminositya | Cb | Assumptionsc |
L(UV) | 3.0 × 10-47
![]() |
0.1-100
M![]() ![]() ![]() |
L(UV) | 4.2 × 10-47
![]() |
0.1-30
M![]() ![]() ![]() |
L(UV) | 4.3 × 10-47
![]() |
0.1-100
M![]() ![]() |
L(UV) | 1.0 × 10-46
![]() |
0.1-100
M![]() ![]() |
L(TIR) | 1.6 × 10-44 | 0.1-100
M![]() ![]() |
L(TIR) | 2.8 × 10-44 | 0.1-100
M![]() ![]() |
L(TIR) | 4.1 × 10-44 | 0.1-30
M![]() ![]() |
L(TIR) | 3.7 × 10-44 | 0.1-100&
M![]() ![]() |
L(TIR) | 8.3 × 10-44 | 0.1-100
M![]() ![]() |
L(H![]() |
5.5 × 10-42 | 0.1-100
M![]() ![]() ![]() |
L(H![]() |
3.1 × 10-41 | 0.1-30
M![]() ![]() ![]() |
L(Br![]() |
5.7 × 10-40 | 0.1-100
M![]() ![]() ![]() |
a Luminosity in erg s-1. Stellar and
dust continuum luminosities are given as
![]() ![]() |
||
b The constant C appears in the
calibration as: SFR(![]() ![]() ![]() ![]() |
||
c Assumptions for mass range of the stellar
IMF, which we adopt to have the expression derived by
Kroupa (2001),
see Section 1.2.2, and for
the timescale ![]() |
A more subtle, but not less important, effect is caused by the length of
time over which a stellar SED remains relatively bright in the UV. This
is due to the significant UV emission of mid-to-late B stars. For
example, a constant star formation event of 10 Myr duration, which, at
constant SFR = 1
M
yr-1, accumulates 107
M
in stars, has the same UV luminosity and a similar UV
SED over the range 0.13-0.25 µm of a 50 Myr old,
2.5 × 108
M
instantaneous burst of star formation. In the absence
of dust attenuation and if only observed in the UV, the two populations
would be attributed the same SFR(UV) = 1
M
yr-1. While this number is correct for
the first population, it would be incorrect, and possibly misleading,
for the second population (which has not been forming stars since 50
Myr). If dust attenuation is also present, the potential of
misclassifying an ageing population for an active star-forming one
increases.
As the vast majority of galaxies contain at least some dust, the use of SFR(UV) becomes complicated, since dust attenuation corrections are usually required, and are uncertain. For the most part dust corrections only work on ensembles of systems, rather than individual objects. In a show of Cosmic Conspiracy, the most active and luminous systems are also richer in dust, implying that they require more substantial corrections for the effects of dust attenuation (Wang & Heckman 1996; Calzetti 2001; Hopkins et al. 2001; Sullivan et al. 2001; Calzetti et al. 2007). As a reference number, a modest optical attenuation AV = 0.9 produces a factor ten reduction in the UV continuum at 0.13 µm, if the attenuation curve follows the recipe of Calzetti et al. (2000).
1.2.1.2.Indicators based on dust-processed stellar light
The IR luminosity of a system will depend not only on its dust content, but also on the heating rate provided by the stars. To first order, the shape of the thermal IR SED will depend on the starlight SED, in the sense that UV-luminous, young stars will heat the dust to higher mean temperatures than old stellar populations (e.g., Helou 1986).
Because of the properties of the Planck function, hotter dust in thermal
equilibrium has higher emissivity in the IR than cooler
dust. Furthermore, the cross-section of the dust grains for stellar
light is higher in the UV than in the optical, as inferred from the
typical trend of interstellar extinction curves. Thus, qualitatively,
the dust heated by UV-luminous, young stellar populations will produce
an IR SED that is more luminous and peaked at shorter wavelengths
(observationally
60 µm) than the dust heated by UV-faint, old or low-mass stars
(observationally with an IR SED peak at
100-150
µm). This is the foundation for using the IR emission ( ~ 5-1000
µm) as a SFR indicator.
The thermal IR emission is, however, a `blunt tool' for measuring SFRs,
in the sense that there is not a one-to-one mapping between UV photons
and IR photons, and a monochromatic heating source will produce a
modified Planck function for the thermal equilibrium dust
emission. Hence the use of `bolometric' IR measures for SFRs, where the
IR emission is integrated over the full wavelength range; in practice,
most of the emission is located in the wavelength range ~
5-1000 µm. The bolometric IR luminosity is often indicated with
L(TIR) (where TIR is the total infrared emission), and a star
formation rate calibration for a stellar population undergoing constant
star formation over = 100
Myr is:
![]() |
(1.3) |
with SFR(TIR) in
M
yr-1, and L(TIR) in erg/s. For this
calculation I have assumed that the Starburst99, Solar-metallicity stellar
bolometric emission is completely absorbed and re-emitted by dust, i.e.,
Lstar(bol) = L(TIR).
Not all the stellar emission in a galaxy is generally absorbed by dust. A ballpark number is given by the cosmic background radiation (e.g., Dole et al. 2006), which shows about half of the light emerging at UV-optical-near-IR wavelengths and half at IR wavelengths. Thus, a simplified approach would be to assume that in a typical galaxy only about half of its stellar light is absorbed by dust. This fraction is, however, strongly dependent on the dust content and distribution within the galaxy itself. The application of the SFR(TIR) calibration derived in this section to actual galaxies, which is based on models and the assumption that all of the stellar emission is absorbed by dust and re-emitted in the IR, will therefore result in a lower limit to the true SFR.
The main reason for giving a theoretical expression for SFR(TIR) is to
show how dependent the calibration is on assumptions on the stellar
population's characteristics. If
= 10 Myr and 2 Myr, the
calibration constant has
-dependent
variations that are not too dissimilar from
those of SFR(UV) (Table 1.1). However, unlike
SFR(UV), the calibration constant of SFR(TIR) keeps changing for star
formation timescales longer than 100 Myr, and for
= 10 Gyr it is about
57% of the 100 Myr calibration constant. The difference relative to the
SFR(UV) case is due to the accumulation over time of long-lived,
low-mass stars in the stellar SED. These contribute to the TIR emission,
but not to the UV one. The heating of dust by multiple-age stellar
populations has the additional effect of producing a thermal equilibrium
IR SED that is significantly broader than that produced by a
single-temperature modified blackbody function. This has been modelled
in the past with at least two approaches: (1) two or more dust
components with different temperatures, or (2) one single-temperature
dust component with a small absolute value of the dust emissivity
index. Physically-motivated models are now available
(Draine & Li 2007),
which describe the dust emission from galaxies with large accuracy
(Draine et al.
2007;
Aniano et al.
2012).
Progress over the past ~ 10-20 years in dissecting the various dust components that contribute to the IR SED has helped refining the original simple picture (of which a summary can be found in, e.g., Draine 2003, 2009). Since this chapter is about SFR indicators and not dust properties, I will only summarise the salient traits that connect dust characteristics to wavelength regions in the TIR emission.
The short-wavelength mid-IR range ( ~ 3-20 µm) dust emission arises from a combination of broad emission features, generated by the bending and stretching modes of polycyclic aromatic hydrocarbons (PAHs), and continuum. The latter is due to emission from both single-photon, stochastically-heated small dust grains and thermally emitting hot (T > 150 K) dust: which of these two components dominates depends on the nature of the heating sources, although single-photon heating is predominant in the general interstellar medium of the Milky Way.
The long-wavelength mid-IR range ( ~ 20-60 µm) is emission
continuum dominated by hot/warm (T
50 K) dust in thermal
equilibrium and single-photon
heated small-grain dust. This is the region where, in most galaxies, the
dust emission transitions from being dominated by emission from
stochastically heated grains to being dominated by large grains in
thermal equilibrium.
Finally, the far-IR range
(
60 µm) is mainly due to thermal emission
from large grains. The mean temperature of the dust decreases for longer
wavelengths (termed `cool' or `cold' dust, depending on the author),
although typical temperatures are about 15-20 K or above.
Both massive, short-lived stars and low-mass, long-lived stars can heat the dust contributing to each of the spectral regions identified above. However, UV-bright stars will likely heat the surrounding dust to relatively high effective temperatures. The ~ 20-60 µm IR wavelength region, where the emission transitions from stochastic heating to thermal heating, has thus been targeted as a promising region for defining monochromatic (single-band) SFR indicators. The advantage of such indicators is the ease of use: instead of obtaining multi-point measurements along the IR SED and/or perform uncertain extrapolations, monochromatic IR SFR indicators only require a single wavelength measurement.
Owing to the uncertainty of assigning a given waveband to a specific dust emission component, monochromatic IR SFRs have been calibrated across a wide range of IR wavelengths, including the IR Space Observatory (ISO) 7 and 15 µm bands, the Spitzer Space Telescope 8, 24, 70 µm bands, and, currently, the Herschel Space Telescope 70 µm and longer wavelength bands. The range of angular resolutions offered by each facility has been and is enabling the calibration of both global and local SFR indicators.
Monochromatic SFR indicators shortward of 15-20 µm require care of use: stochastically-heated dust can trace both young and evolved stellar populations (e.g., Boselli et al. 2004; Calzetti et al. 2007; Bendo et al. 2008; Crocker et al. 2012). The PAHs may be better tracers of B stars than current SFR (Peeters et al. 2004), and the emission features show strong dependence on the metal abundance of the system (e.g., Madden et al. 2000, 2006; Engelbracht et al. 2005, 2008; Draine et al. 2007; Smith et al. 2006; Galliano et al. 2008; Gordon et al. 2008; Muñoz-Mateos et al. 2009; Marble et al. 2010). Only about 50% of the emission at 8 µm from a galaxy is dust-heated by stellar populations 10 Myr or younger, and about 2/3 by stellar populations 100 Myr or younger (Crocker et al. 2012). Thus, a significant fraction of the 8 µm emission is unrelated to current star formation. While this is likely to affect mainly studies of sub-galactic regions or structures, some effect may be expected on the global SFR indicators. Various calibration efforts have usually recovered a roughly linear or slightly sub-linear relation between the global 8 µm luminosity, providing reference SFR indicators for metal-rich, star-forming galaxies. However, the peak-to-peak scatter tends to be large, a factor of about three (Treyer et al. 2010), with larger deviations observed for compact starbursts (Elbaz et al. 2011).
Longward of around 70 µm, the contribution to the IR SED of thermal dust at increasingly lower temperature, and therefore heated by stars that are low-mass and long-lived, becomes more and more prominent, thus compromising the ability of the IR emission to trace exclusively or almost exclusively recent star formation.
For the above two reasons, I offer here only two empirical monochromatic IR SFR calibrations: in the 24 µm and 70 µm restframe bands. I distinguish between local and global SFR indicators, since, as discussed earlier in this section, the bolometric luminosity of a stellar population undergoing constant star formation increases with time. Therefore, to the extent that the IR emission traces the bolometric emission of the stellar population, the calibration constant will be different for a global, galaxy-wide SFR indicator and a local SFR indicator, since the former includes the Hubble-time-integrated stellar population of a galaxy, while the latter is generally derived from regions that are dominated by stellar populations with short star formation timescales (Hii regions, large star-forming complexes, etc.).
At 24 µm, the local (spatial scale ~ 500 pc) calibration offered by Calzetti et al. (2007) is:
![]() |
(1.4) |
with SFR(24) in
M
yr-1, and L(24) =
L(
) in
erg s-1. The uncertainty is 0.02 in the exponent, and 15% in the
calibration constant. The non-linear correlation between L(24)
and SFR is a common characteristic of this tracer at the local scale
(Alonso-Herrero et
al. 2006;
Pérez-González
et al. 2006;
Calzetti et al.
2007;
Relaño et
al. 2007;
Murphy et al.
2011),
and may be a
manifestation of the increasing transparency of regions for decreasing
L(24) luminosity, of the increasing mean dust temperature for
increasing L(24) luminosity, or a combination of the two.
At the galaxy-wide scale, both linear and non-linear correlations between SFR and L(24) have been derived (Wu et al. 2005; Zhu et al. 2008; Kennicutt et al. 2009; Rieke et al. 2009), perhaps owing to differences in sample selections and in the reference SFR indicators used to calibrate SFR(24), and sufficient scatter in the data that both linear and non-linear fits can be accommodated (Wu et al. 2005; Zhu et al. 2008). The linear calibration of Rieke et al. (2009), reported using the same IMF as all the other indicators in this presentation, is:
![]() |
(1.5) |
where a small correction for self-absorption is included at the high luminosities. The linear calibrations in the literature tend to be within 30% of each other, suggesting a general agreement.
At 70 µm, the local calibration over ~ 1 kpc scales derived by Li et al. (2010) using over 500 star-forming regions is:
![]() |
(1.6) |
with SFR(70) in
M
yr-1, and L(70) =
L(
) in erg s-1.
The formal uncertainty on the calibration constant is about 2%, although the
scatter in the datapoints is about 35%. The global calibration provided by
Calzetti et al.
(2010)
is:
![]() |
(1.7) |
with a scatter of the datapoints of about 60%. The SFR(70) calibration constant thus increases when going from whole galaxies to 1 kpc regions, i.e., for decreasing region sizes. Li et al. (2012) obtain a tantalising result: the calibration constant for SFR(70) becomes even larger for regions smaller than ~ 1 kpc. These constants can be interpreted in terms of star formation timescale within each region size (Fig. 1.1), for simple assumptions on the star formation history, the fraction of stellar light re-emitted by dust in the IR, and the fraction of IR emission contained in the 70 µm band (Draine & Li 2007).
![]() |
Figure 1.1. The calibration constant,
C70, between SFR and the 70 µm
luminosity, expressed as SFR(70) = C70 L(70),
as a function of the physical size of the regions used to derive the
calibration. The filled red triangles are observed values from
Li et al. (2010,
2012)
and
Calzetti et al.
(2010),
using both Spitzer and Herschel data. The
black stars are from stellar population synthesis models, for constant star
formation and a Kroupa IMF, in the stellar mass range 0.1-100
M |
Gas-rich, but metal-poor, galaxies offer little opacity to stellar emission (the same is true for metal-rich, but gas-poor galaxies, e.g., ellipticals, but for these galaxies few or no stars form). Metal content correlates with galaxy luminosity (Tremonti et al. 2004, and references therein), and faint galaxies are faint IR emitters. In this case, SFR(IR) becomes a highly uncertain tool. SFR tracers that mix tracers of both dust-obscured and dust-unobscured star formation have recently been calibrated, and will be presented in Section 1.2.1.4.
1.2.1.3. Indicators based on ionised gas emission
Young, massive stars produce copious amounts of ionising photons that
ionise the
surrounding gas. Hydrogen recombination cascades produce line emission,
including the well-known Balmer series lines of
H (0.6563
µm) and
H
(0.4861
µm), which, by virtue of being strong and located in the
optical wavelength range, represent the most traditional SFR indicators
(Kennicutt 1998).
Only stars more massive than ~ 20
M
produce a measurable ionising photon flux. In a stellar population
formed through an instantaneous burst with a Kroupa IMF the ionising
photon flux decreases by two orders of magnitude between 5 Myr and 10
Myr after the burst.
The relation between the intensity of a hydrogen recombination line and
the ionising photon rate is dictated by quantum mechanics, for a nebula
that is optically thick to ionising photons (case B,
Osterbrock & Ferland
2006).
Case B is usually assumed for most astrophysical
situations where SFRs are of interest. Although typical interstellar
hydrogen densities are sufficient to ensure that most
Hii regions should be
radiation-bound, inhomogeneities in the interstellar medium cause some
fraction of the ionising photons to leak out of the regions (but usually
not out of galaxies). More discussion on this issue is given below. The
relation between the luminosity of the
H emission line and the
ionising photon rate is given by
Osterbrock & Ferland
(2006):
![]() |
(1.8) |
where L(H) is in
erg s-1,
H
eff is the
effective recombination coefficient at
H
,
B
is the case B recombination coefficient, Q(Ho)
is the ionising photon rate in units of s-1, and the constant
at the right-hand side of the
equation is the resulting coefficient for electron temperature
Te = 10000 K and density
ne = 100 cm-3. For a Kroupa IMF
(Section 1.2.2), the relation between ionising
photon rate and SFR is:
![]() |
(1.9) |
with SFR(Q(Ho)) in
M
yr-1. Combining
Equations 1.8 and 1.9, we obtain the well-known calibration:
![]() |
(1.10) |
again, with SFR(H) in
M
yr-1 and
L(H
) in
erg s-1. The variation of the calibration constant is ~ 15% for
variations in electron temperature in the range Te =
5000-20000 K ,
and is < 1% for electron density variations in the range
ne = 100-106 cm-3
(Osterbrock & Ferland
2006).
Star formation needs to have remained constant over timescales > 6
Myr for the calibration constant to
be applicable (Table 1.1), but there is no
dependency on long timescales, unlike SFR(UV) or SFR(TIR).
All SFR indicators that use the ionisation of hydrogen to trace the
formation of massive stars are sensitive to the effects of dust. The
most commonly treated effect is that of dust attenuation of the line or
continuum. As we will see in Section 1.4,
various techniques have been developed to try to remove this effect;
furthermore, dust attenuation decreases for increasing wavelength. A
far more difficult effect to treat is the direct absorption of Lyman
continuum photons by dust. In this case, the ionising photons are
removed altogether from the light beam and are no longer available to
ionise hydrogen. Thus, no emission from either recombination lines or
free-free continuum emission will result. The actual impact of Lyman
continuum photon (Lyc) absorption by dust has been notoriously difficult
to establish from an empirical point of view, owing to the absence of a
`ground truth' (or reference) with which to compare measurements.
Models have to be involved, and these show that the level of Lyc
absorption depends on the assumption for the geometry of the nebulae
(Dopita et al.
2003,
and references therein). The parametrisation
of Dopita et al., where the ratio of
H line
luminosity with
and without Lyc absorption is given as a function of the product of
metal abundance and ionisation parameter, shows that most normal disk
galaxies fall into the regime of low Lyc absorption, typically less than
15%-20%; however, Lyc absorption by dust can become significant at large
ionisation parameters and metallicities, such as those typical of local
luminous and ultra-luminous IR galaxies (LIRGs and ULIRGs, galaxies with
bolometric luminosity > a few 1011
L
), and
of some high-density central regions of galaxies.
A somewhat opposite effect is represented by leakage of ionising photons, i.e., case B recombination does not fully apply. Leakage of ionising photons from galaxies is likely negligible, at the level of a few percent or less (e.g., Heckman et al. 2011), although the jury is still out in the case of low-mass, low-density galaxies (Hunter et al. 2010; Pellegrini et al. 2012). Star-forming regions within galaxies tend, on the other hand, to be leaky, and lose about 25%-40% of their ionising photons (see recent work by Pellegrini et al. 2012; Relaño et al. 2012; Crocker et al. 2012). Thus, the use of ionising photon tracers for local SFRs may be biased downwards by about 1/3 of their true value because of this effect. This correction is not included in Table 1.1.
Adopting for the moment that Lyc absorption by dust and leakage are not
issues, emission lines still need to be corrected for the effects of
dust attenuation. As an example, a modest attenuation of
AV = 1 mag by foreground dust depresses the
H luminosity by a factor ~
2. At longer wavelengths,
Br
(2.16
µm) is depressed by only
11%, for the same AV. Recent advances in the
linearity, stability, and field-of-view size of infrared detectors' are
making it possible to collect significant samples of galaxies observed
in the IR Hydrogen recombination lines.
Table 1.1 shows a calibration for
Br
,
derived under the same assumptions as
SFR(H
). Calibrations for
other lines can be inferred from those of
H
and
Br
and
the emissivity ratios published in
Osterbrock & Ferland
(2006).
Recombination lines at wavelengths longer than the optical regime, while
offering the advantage of lower sensitivity to dust attenuation, have
the dual disadvantage of being progressively fainter and more sensitive
to the physical conditions of the gas, especially the temperature; these
are natural consequences of transition probabilities and conditions of
thermal equilibrium, respectively. The luminosity of
Br is about
1/100th of that at H
,
and it changes by about 35% for Te in the range
5000-20000 K, and by ~ 4% for density in the range
ne = 102-106 cm-3. For
Br
(4.05
µm), the
variations are 58% and 13% for changes in Te and
ne, respectively. The sensitivity of
Br
and
Br
to
Te is a factor 2.4 and 3.9 larger, respectively, than
that of L(H
).
New or greatly improved radio and millimetre facilities such as
ALMA (the Atacama Large Millimetre/submillimetre Array) or
EVLA (the Expanded Very Large Array) are opening the window for
exploring millimetre and/or radio recombination lines as ways to measure
SFRs unimpeded by effects of dust attenuation, albeit using lines that
are intrinsically extremely weak. I will cumulatively refer as RRLs all
(sub)millimetre and radio recombination lines from hydrogen quantum
levels n > 20. At high quantum numbers (n > 80-200,
depending on electron density), i.e., at wavelengths of a few cm or
longer, stimulated emission is no longer negligible and adds extra
parameters in the expression of the line luminosity
(Brown et al.
1978).
Even within the regime where stimulated emission is not a
concern, the line luminosity is dependent on the electron temperature,
producing SFR(RRL)
Te0.7
(Gordon & Sorochenko
2009).
This translates into a variation in the line
intensity of a factor 2.6 for Te in the range
5000-20000 K.
The traditional approach of measuring the electron temperature from the radio line-to-continuum ratio relies on the assumption that the underlying continuum is free-free emission. The true level of free-free emission from a galaxy, or from a large star-forming region embedded in a galaxy, needs to be carefully disentangled from both dust emission (dominant < 2-3 mm) and synchrotron emission (dominant > 1-3 cm, depending on the source). Often, multi-wavelength observations are used to accomplish this (e.g., Murphy et al. 2011), and add an additional layer of complication to the use of SFR(RRL). Exploratory work is still ongoing to test how efficiently RRLs can be detected in external galaxies (e.g., Kepley et al. 2011), and what advantage they can bring relative to more classical and efficient methods. If the high-density, and heavily dust-obscured, regime turns out to be the main niche for these tracers (Yun 2008), they will need to be carefully weighted against SFR(IR), in light of the potentially heavy impact of the Lyc absorption by dust.
The free-free emission from galaxies or regions itself is a SFR tracer,
being the product of the Coulomb interaction between free electrons and
ions in thermal equilibrium. The electron temperature dependence of this
SFR tracer, SFR(ff)
Te-0.45
(Condon 1992),
is shallower than that of the
SFR(RRL), implying less than a factor of two change for a factor of four
variation in Te. A calibration of SFR(ff) consistent
with our IMF choice is given in
Murphy et al.
(2011).
SFR tracers that use forbidden metal line emission will not be discussed in this review, as they suffer from the same limitations as the hydrogen recombination lines, and have additional dependencies on the metal content and ionisation conditions of a galaxy or region. A review is found in Kennicutt (1998) and a recent calibration in Kennicutt et al. (2009).
1.2.1.4. Indicators based on mixed processes
The necessity to capture both dust-obscured and dust-unobscured star formation has led to the formulation of SFR indicators that attempt to use the best qualities of each indicator above. This advantage compensates for the slight disadvantage of having to obtain two measures at, generally, two widely separated wavelengths: one that measures the direct stellar light and one that measures the dust-processed light. These mixed indicators have been calibrated empirically over the past ~ 5-7 years, using combinations of data from space and the ground (e.g., Calzetti et al. 2005, 2007; Kennicutt et al. 2007, 2009; Liu et al. 2011; Hao et al. 2011), and are usually expressed as:
![]() |
(1.11) |
with SFR(1,
2) in
M
yr-1.
1 is usually
a wavelength probing either direct stellar light (e.g., the GALEX FUV
at 0.153 µm) or ionised gas tracers (e.g.,
H
), and
2 is a
wavelength or range of wavelengths where dust emission dominates (e.g.,
24 µm, 25 µm, TIR, etc.). The constant
C(
1)
is the calibration for the direct stellar light probe, often derived
from models (see Table 1.1). The luminosities
L(
1)obs and
L(
2)obs are in units of erg
s-1 and are the observed
luminosities (i.e., not corrected for effects of dust attenuation or other
effects). The proportionality constant
a
2,
Type depends on both the dust emission tracer used and whether the
calibration is for local or
global use (Type = local, global). This latter characteristic is due to the
sensitivity of dust emission to heating from a wide range of stellar
populations (see Section 1.2.1.2). An example of
a mixed indicator calibration is given in Fig. 1.2.
![]() |
Figure 2. An example of the calibration for
a mixed SFR indicator, from
Calzetti et al.
(2007).
This specific example is for a local SFR indicator:
the data points include star-forming regions in nearby galaxies (red
triangles, green squares, blue crosses) and local LIRGs (black stars,
from
Alonso-Herrero et
al. 2006).
The horizontal axis is the luminosity/area of the
regions/galaxies in the hydrogen recombination line
P |
Table 1.2 summarises a few of the published
calibrations from the references above; a more complete set of global
calibrations can be found in the recent review by
Kennicutt & Evans
(2012).
Local calibrations show systematically higher values of
a2, Type than global ones. At
24-25 µm, a24, local /
a25,global ~ 1.55. This difference
cannot be attributed to the difference between L(24) and
L(25), which is around 2% typically
(Kennicutt et al.
2009;
Calzetti et al.
2010).
The larger fraction of 24 µm emission that
needs to be added to either FUV or
H
in local SFR
measurements may simply reflect the fact that regions within galaxies
are probing stellar populations over shorter timescales
(
~ 100 Myr or smaller) than
global SFR measurements
(
> many Gyr). The dust
emission traces this difference accordingly
(Kennicutt et al.
2009).
From Table 1.1, the
ratio of the calibration constants for SFR(TIR) at 100 Myr and 10 Gyr is
1.75, close to the observed value of 1.55. Differences in the mean dust
temperature, that is likely to boost the L(24) in sub-galactic
regions, may account for the remaining discrepancy.
![]() |
![]() |
C![]() |
a![]() |
Typed |
FUV (0.153 µm) | 24 µm | 4.6 × 10-44 | 6.0 | local |
FUV (0.153 µm) | 25 µm | 4.6 × 10-44 | 3.89 | global |
FUV (0.153 µm) | TIR | 4.6 × 10-44 | 0.46 | global |
H![]() |
24 µm | 5.5 × 10-42 | 0.031 | local |
H![]() |
25 µm | 5.5 × 10-42 | 0.020 | global |
H![]() |
TIR | 5.5 × 10-42 | 0.0024 | global |
a The calibrations are expressed as:
SFR(![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
||||
b
![]() ![]() ![]() |
||||
c The constant
C(![]() |
||||
d The constant
a![]() ![]() ![]() ![]() |
1.2.1.5. Indicators based on other processes
SFR indicators based on non-thermal (synchrotron) radio and X-ray emission represent more indirect ways of probing star formation in galaxies.
In the case of synchrotron emission, the basic mechanism is the production and acceleration of cosmic rays in supernova explosions; since the supernova rate is directly related to the SFR, we should be able to use the synchrotron luminosity as a proxy for the SFR. There is, however, an added complication in that the non-thermal luminosity depends not only on the mean cosmic ray production per supernova, but also on the galaxy's magnetic field (e.g., Rybicki & Lightman 2004). The case for SFR(sync) is helped by the well-known IR-radio correlation (e.g., Yun et al. 2001): if the IR is correlated with both SFR and radio emission, then SFR and radio emission are correlated among themselves. SFR(sync) calibrations can only be derived empirically (Condon 1992; Schmitt et al. 2006; Murphy et al. 2011), because of the complexity of the relation between the SFR and the underlying physical mechanism; a recent derivation consistent with our IMF can be found in Murphy (2011).
A similarly indirect relation exists between SFR and X-ray luminosity. In star-forming galaxies, the X-ray luminosity is produced by high-mass X-ray binaries, massive stars, and supernovae, but non-negligible contributions from low-mass X-ray binaries are also present. The latter are not directly related to recent star formation, and represent a source of uncertainty in the calibration of SFR(X-ray). Because of the difficulty of establishing the frequency and intrinsic luminosity of each X-ray source (related or unrelated to current star formation) from first principles, the SFR(X-ray) calibrations have been derived empirically, and examples are given in Ranalli et al. (2003), Persic & Raphaeli (2007), and Mineo et al. (2012). Care should be taken when comparing these published calibrations, obtained for a Salpeter IMF, with those reported in this summary, which are based on a Kroupa IMF.
1.2.2. A word about the stellar initial mass function and stochastic sampling
All calibrations listed in this section make the implicit assumption that the stellar IMF is constant across all environments and given by the double-power law expression (Kroupa 2001):
![]() |
(1.12) |
where (M) is the
number of stars between M and
M + dM. The stellar mass
distribution and total stellar mass produced by this expression are not
significantly different from that produced by the log-normal expression
proposed by
Chabrier (2003).
The Kroupa IMF expression produces a smaller number
of low-mass stars than the
Salpeter (1955)
IMF, which has been customarily
represented with a single power law with slope -2.35 between 0.1 and
100 M
.
Since the majority of SFR indicators trace massive stars, a
calibration based on the Kroupa IMF can be converted to one using the
Salpeter IMF simply by multiplying the calibration constant by 1.6.
The assumption that the IMF is constant and universal is justified by
many observational results, although these are generally rather
uncertain, especially at the high-mass end (review by
Bastian et al.
2010).
There is still the possibility of variation in some
extreme (in terms of density, SFR, or other) environments, and arguments
both in favour of and against variations have been brought forward by
many different authors. To gauge the impact of a different IMF
assumption on our SFR calibrations, we can adopt a modified Kroupa IMF,
with the maximum stellar mass set to 30
M,
instead of 100
M
.
The new calibration constants, for selected
timescales, are listed in
Table 1.1. The constants change by factors 1.4,
1.5, and 5.6 for SFR(UV), SFR(TIR), and
SFR(H
),
respectively. The change
for the H
calibration is
the largest of all; it is larger than the UV one by a factor of four,
simply because significant UV emission is produced by stars down to ~
5 M
,
but significant ionising photon flux is produced only
by stars more massive than ~ 20
M
. In
addition, it takes slightly longer (10 Myr for the upper mass limit of
30 M
versus 6 Myr for 100
M
) for
the ionising photons to reach their asymptotic value. The changes for
the UV and TIR calibration constants are similar to each other.
In contrast with the results just discussed, the mean stellar
mass for the Kroupa IMF is < M > ~ 0.6
M, with
less than 10% difference between using 100
M
or
30 M
as
stellar upper mass limit. This makes tracers based on the mean stellar
mass of a system (Equation 1.1) more robust than those based on tracing
the most massive stars.
Even if the IMF is universal, individual systems may show departures from this condition, based on simple arguments of sampling.
If we consider a single-age, very young stellar cluster, we can ask what the
minimum mass is that this cluster needs to have so that at least one
star with mass 100
M is
formed. The mass is 2.8 × 105
M
, which
is a large value, only achieved by some of the most massive star
clusters known. As a comparison, if the maximum stellar mass is 30
M
, full
sampling of the IMF, meaning that at least one 30
M
star
is formed, is achieved with a cluster mass of 1.7 × 104
M
.
Under these circumstances, it is not uncommon that studies that involve low
SFRs, either because the region considered is small and/or inefficient at
forming stars, or because the galaxy has a low overall SFR, are subject
to the effects of stochastic sampling, i.e., the stellar IMF is
randomly, not fully, sampled. The impact of stochastic sampling is
higher for the most massive stars, since there are proportionally less
massive stars than low-mass ones. From the Kroupa IMF expression above,
only 11% of all stars, by number, have masses above 1
M,
although these stars represent 56% of the total mass.
Stochastic sampling has a larger impact on tracers of ionising photons
than on tracers of UV continuum light, for the same reason that a low
upper limit in stellar mass has. Clear evidence for this is shown by the
so-called extended UV (XUV) disks of galaxies, as revealed by
GALEX. The original hypothesis that these XUV disks, bright in
the UV but faint in H,
could be due to peculiar IMFs (e.g., deficient in high-mass stars) has
been replaced by the finding that the IMF is stochastically sampled in
these low-SFR areas
(Goddard et al
2010;
Koda et al.
2012).
The models of
Cerviño et
al. (2002),
renormalised to the Kroupa IMF (Equation 1.12), show
that a star cluster with mass ~ 1 × 104
M
will be subject to sufficient stochastic sampling that
a scatter as large as 20% can be expected in the measured ionising
photon flux. The scatter increases dramatically for decreasing cluster
mass, and becomes as large as 70% for a cluster mass ~ 1 ×
103
M
.
This poses a practical limitation of SFR
0.001
M
yr-1 for the use of SFR indicators
based on the ionising photon flux if a 20% or less uncertainty is
desired; a similar uncertainty value for the UV is obtained at SFR
0.0003 M
yr-1, or about 3.5 times lower than
when using tracers of ionising photons
(Lee et al. 2009,
2011).