AN EMPIRICAL SEQUENCE OF DISK GAP OPENING REVEALED BY ROVIBRATIONAL CO

We build high signal-to-noise CO line profiles for each science target by stacking several transitions observed in the 4.7 μm spectra. Nominally, the maximum spectral coverage obtained in this survey includes rovibrational CO transitions from P(1)–P(32) for ¹²CO v = 1–0, P(1)–P(26) and R(0)–R(6) for ¹²CO v = 2–1, and P(1)–P(23) and R(0)–R(15) for ¹³CO v = 1–0. In practice, the number of transitions available for each disk depends on the spectral coverage obtained for individual targets and on the gaps caused by detector chips and telluric absorption, which in turn depends on the time of year of a given observation. For stacking and averaging line profiles, we use the lines that are most commonly available, which have similar integrated flux, and which are not significantly blended with other transitions. These criteria preserve both the line flux and shape in the stacked profiles, and enforce homogeneity in the analysis method for the entire disk sample. Note that, within the relatively narrow spectral range 4.7–4.9 μm, lines out of a given vibrational level have very similar fluxes and excitation energies (within 10%–30%) and line profiles depend only very weakly on the rotational quantum number (e.g., Pontoppidan et al. 2011a; Salyk et al. 2011; van der Plas et al. 2015). For high rotational quantum numbers, this may no longer be true ($J\gtrsim 25$).

Given these constraints, the ¹²CO v = 1–0 lines that are most suitable for stacking are those between P(7) and P(12), while the ¹²CO v = 2–1 lines are those between P(3) and P(6). As for the ¹³CO lines, several can typically be used over the entire spectrum, where not blended with ¹²CO. Each line is normalized to its local continuum, re-sampled on a common velocity grid, and the stacked line profiles are built by weighted average of the pixel values from the individual transitions. While the stacked v = 2–1 line profiles typically have continuous spectral sampling, v = 1–0 lines often have spectral gaps on one side due to the counterpart telluric absorption lines at the Doppler velocity of the target at the time of observation (see for instance the gap on the red side of the v = 1–0 line in Figure 2). In some disks (CW Tau, DoAr 24E S, HT Lup, VSSG 1), an absorption component is present on top of ¹²CO emission in the v = 1–0 lines. This is typically due to absorption from unrelated foreground gas (Herczeg et al. 2011; Pontoppidan et al. 2011a; Brown et al. 2013). In these targets, we exclude pixels affected by the absorption component from the stacked profiles and further analysis.

The presence of separate broad and narrow components is apparent by visual inspection of many disk spectra (e.g., Figure 1). In practice, the difference is formalized by comparison of the v = 1–0 and v = 2–1 stacked line profiles. We scale the v = 1–0 to match the broad wings of the v = 2–1 profile, and then subtract one from the other. The best scaling factor is found by ${\chi }^{2}$ minimization between the scaled v = 1–0 line profile and the v = 2–1 line profile, excluding the central region of narrow emission, if present. The NC is typically much narrower than BC, such that most of the line profile can be used for the ${\chi }^{2}$ minimization. The choice of the wavelength exclusion region is determined iteratively, starting by using the full line profile. If the residuals after subtraction of the two line profiles are consistent with zero over the entire spectral range of the v = 2–1 line, the NC is not detected and the procedure stops there. In the opposite case, the fit is iterated after exclusion of increasingly large portions at the center of the line profile, until the residuals are composed of a flat continuum and of the NC emission line. Figure 2 shows a demonstrative sketch of the outcome, while spectral decomposition plots for all disks are shown in Appendix B.

Two basic parameters are measured from the decomposition procedure: the line width and the vibrational flux ratio $v2/v1={F}_{v=2-1}/{F}_{v=1-0}$. Line widths are measured at 50% of the line peak (FWHM) from fits to the line profiles, allowing up to two Gaussian functions to account for double-peaked profiles. The vibrational flux ratio is measured from the scaling factor that matches the v = 1–0 to the v = 2–1 lines as explained above, and is exactly equivalent to the integrated line intensity ratio between the v = 2–1 and the v = 1–0 lines in each spectrum (further details are provided in the appendix). It is possible to measure vibrational ratios in all single-component disks and for the BC in all double-component disks. For the NC, ${F}_{v=2-1}/{F}_{v=1-0}$ can be measured in only three disks: AS 205 N, DR Tau, and S CrA N. In these disks, the NC can also be identified in the v = 2–1 lines through fits of multiple Gaussians to the line profile, one for each of the broad and narrow components (plots are shown in the appendix). The non-detection of the NC in the v = 2–1 lines of all other double-component disks is generally consistent with an upper limit on the vibrational flux ratio of $\lesssim 0.2$. Tables 2 and 3 report line widths and vibrational ratios for the entire disk sample.

Table 2.  Disks with Two CO Components

	BC				NC
Name	FWHM	$v2/v1$	${R}_{\mathrm{CO}}$	${T}_{\mathrm{vib}.\mathrm{ex}.}$	FWHM	${R}_{\mathrm{CO}}$
	(km s−1)		(AU)	(K)	(km s−1)	(AU)
AS205 N	52	0.38	0.17	970	13	2.3
AS209	160	0.91	0.07	1054	33	1.7
CrA-IRS2	53	0.37	0.11	960	17	1.1
CVCha	102	0.25	0.18	790	47	0.9
CWTau	54	0.32	0.32	890	45	0.5
DFTau	89	0.43	0.20	1060	53	0.6
DoAr24E S	81	0.30	0.04	860	29	0.4
DRTau	37	0.39	0.06	990	13	0.5
EXLup 08	155	0.61	0.06	1420	32	1.3
EXLup 14	150	0.51	0.07	1220	40	1.0
GQLup	107	0.28	0.20	830	62	0.6
HH100	80	1.02	0.04	1200	12	1.9
HTLup	190	0.65	0.05	1510	35	1.6
IMLup	200	1.16	0.05	1400	40	1.3
RNO90	92	0.21	0.23	730	52	0.7
RULup	111	0.50	0.07	1200	18	2.5
SCrA S	101	0.40	0.04	1000	15	1.8
SCrA N	57	0.30	0.05	860	12	1.1
TTau N	110	0.57	0.07	1340	28	1.1
VSSG1	66	0.36	0.27	940	35	1.0
VWCha	80	0.32	0.16	890	33	0.9
VZCha	53	0.44	0.18	1080	24	0.9
WaOph6	133	0.33	0.07	900	32	1.2
WXCha	140	0.45	0.11	1100	67	0.5

Note. ${R}_{\mathrm{CO}}$ is estimated from Kepler's law using the line velocity at HWHM, the stellar mass, and the disk inclination (Table 1). ${T}_{\mathrm{vib}.\mathrm{ex}.}$ is estimated from $v2/v1={F}_{{\rm{v}}=2-1}/{F}_{{\rm{v}}=1-0}$ assuming a CO column density of 10¹⁸ cm⁻², apart from for AS209, IMLup, and HH100 where 10¹⁹ cm⁻² is used (see the appendix for details). $v2/v1$ and ${T}_{\mathrm{vib}.\mathrm{ex}.}$ can be measured for NC only in three disks, where it is detected also in the ¹²CO v = 2–1 lines: AS205 N (0.06, 520 K), DRTau (0.13, 620 K), and SCrA N (0.06, 520 K).

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Table 3.  Disks with a Single CO Component

Name	FWHM	$v2/v1$	${R}_{\mathrm{CO}}$	${T}_{\mathrm{vib}.\mathrm{ex}.}$
	(km s−1)		(AU)	(K)
AATau	115	0.14	0.2	630
DoAr44	72	0.11	0.4	590
EC82	49	0.03	0.9	420
FNTau	9	0.01	1.8	330
FZTau	31	0.20	1.0	720
HD 135344B	14	0.05	1.7	420
HD 142527	25	0.04	2.3	460
HD 144432S	40	0.09	0.8	560
LkHa330	9	0.09	4.7	420
TWCha	65	0.16	0.7	660
TWHya	6	0.09	0.3	500
VVSer	55	0.16	2.9	660
IRS48	14	0.38	16.	970
SR21	9	0.64	6.5	1480
HD 100546*	16	0.27	15.	810
HD 97048*	17	0.33	14.	900
HD 179218*	18	0.37	21.	950
HD 141569*	16	0.56	17.	1300
HD 190073*	16	0.27	6.0	810
HD 98922*	21	0.29	9.3	840

Notes. ${R}_{\mathrm{CO}}$ is estimated from Kepler's law using the line velocity at HWHM, the stellar mass, and the disk inclination (Table 1). ${T}_{\mathrm{vib}.\mathrm{ex}.}$ is estimated from $v2/v1$ assuming a CO column density of 10¹⁸ cm⁻². Disks marked with ^* are included from van der Plas et al. (2015).

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By implementing this decomposition procedure, we find that all disks fall into two phenomenological classes: the "single-component" disks (14 of 37), where the v = 1–0 line profile, once scaled down, is consistent with that of the v = 2–1 line, and the "double-component" disks (23 of 37), where the v = 2–1 profile matches only the wings of the v = 1–0 profile, revealing an additional narrower component of the v = 1–0 profile. In the double-component disks, the ¹³CO lines are typically significantly narrower than the BC, but match the profile of the NC (Figure 2). Only AS 209 and IM Lup have ¹³CO lines detected in the BC only, rather than in the NC, while a few disks tentatively show both the BC and NC in ¹³CO (CrA-IRS 1, AS 205 N, and RU Lup). In double-component disks, the BC profile shows properties distinct from those of the single-component disks: broader lines (≈50–200 km s⁻¹), a clear separation in terms of line flux ratios with a boundary at ${F}_{v=2-1}/{F}_{v=1-0}\approx 0.2$, and weak or undetected ¹³CO lines. Conversely, the NC shares the same line widths (≈10–70 km s⁻¹), same vibrational ratios ${F}_{v=2-1}/{F}_{v=1-0}\lt 0.2$, and same ¹³CO detection rate as for CO in 12/14 single-component disks (SC) of our sample.

When using line widths to infer radial properties of Keplerian disks, it is necessary to correct for inclination effects. Consequently, we collect values of disk inclinations from the literature, where available, for the disks in our sample. We carefully searched for the most reliable estimates obtained using spatially resolved disk images, either from millimeter interferometry, optical/near-infrared (NIR) scattered-light imaging, or spectro-astrometry from CRIRES. Note that for spectro-astrometry the derived disk inclination is degenerate with an assumed stellar mass. For consistency, we therefore adopt both the stellar masses and the disk inclinations reported by Pontoppidan et al. (2011a). Table 1 shows the adopted disk inclinations, with notes on the technique used to measure the inclination for each disk. Using 10 double-component disks with known inclinations (7 from spectro-astrometry, and 3 from interferometry), we find strong correlation between the CO line width and disk inclination, with Pearson coefficients of 0.7 for the BC and of 0.9 for the NC. Strong correlation between NIR CO line widths and disk inclination has been reported previously by Blake & Boogert (2004) from a sample of five Herbig disks. Correlation between these parameters is expected when the gas is in Keplerian rotation around the star, a condition that is suggested by the high detection rate of double-peaked line profiles in our disk sample ($\gt 70$%) and that has been proposed since the first surveys of NIR CO emission in disks (e.g., Brittain et al. 2003; Najita et al. 2003; Blake & Boogert 2004). Note that the two-component decomposition significantly decreases the scatter in this relation relative to what is seen when the line profiles are not fully velocity-resolved as with CRIRES, or if the line profiles were all modeled with a single Gaussian profile to derive average line widths (i.e., without separating BC from NC).

Gas in Keplerian rotation at a disk radius R produces emission at a maximal velocity ${\rm{\Delta }}v$ from the rest frequency of a given CO line following Kepler's law as ${\rm{\Delta }}v=\mathrm{sin}(i)\sqrt{{{GM}}_{\star }/R}$, where i is the disk inclination, G is the gravitational constant, and ${M}_{\star }$ is the stellar mass. Using this relation, it is possible to derive an emitting radius for the gas in a rotating disk from the observed line widths. In general, emission from higher velocities originate from smaller orbital radii. These velocities, usually taken at the 50% or 10% of the line peak value, therefore provide estimates of the disk radii where CO is emitting. We refer to the innermost emitting radius of CO, ${R}_{\mathrm{in}}$, when derived from $1.7\times \mathrm{HWHM}$ as proposed by Salyk et al. (2011). In comparison, the velocity at $1.0\times \mathrm{HWHM}$ provides an estimate of the disk radius ${R}_{\mathrm{CO}}$ that emits the peak line flux (in agreement with the emitting radii derived from spectro-astrometry by Pontoppidan et al. 2008, 2011a). Figure 3 shows the measured CO line widths divided by $\sqrt{{M}_{\star }}$. Removing the stellar mass dependence on the inclination relation leaves only the dependence on the Keplerian radius R, which can be compared to a simple model of gas emission in Keplerian rotation around a $1\;{M}_{\odot }$ star (shown in Figure 3 at three different disk radii).

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In these terms, the BC has ${R}_{\mathrm{in}}$ between 0.01 and 0.1 AU. This is consistent with the disk region of the corotation and magnetospheric accretion radii for solar-mass stars (e.g., Bouvier et al. 2007), as found by previous studies of rovibrational CO emission from disks (e.g., Najita et al. 2003; Salyk et al. 2011). The NC, instead, emits from larger disk radii, with ${R}_{\mathrm{in}}$ between 0.1 and 1 AU. The single-component disks show CO emission at radii similar to, or larger than, those of the NC.

In Figure 3 we show linear fits to the BC and NC in disks with known inclinations, obtained using the Bayesian method by Kelly (2007). The regression takes into account the uncertainties on both variables, as well as the intrinsic scatter in the regression relation due to physical quantities not explicitly accounted for. Typical errors on the line widths are $\lesssim 7\;\mathrm{km}\;{{\rm{s}}}^{-1}$ for the BC and $\lesssim 4\;\mathrm{km}\;{{\rm{s}}}^{-1}$ for the NC, the stellar masses are assumed to be known to 20% and the disk inclinations are typically known to accuracies better than 10%. The best-fit parameters and their errors are taken as the median and standard deviation of the posterior distributions of the regression parameters. The median intercept and slope are used to plot the lines in Figure 3, while the posterior distributions are plotted in the appendix.

The single linear fits shown in Figure 3 assume that differences in ${R}_{\mathrm{in}}$ due to the dependence on stellar and disk luminosity for either the BC and the NC is weak relative to the inclination dependence. This is supported by the Bayesian regression of line widths versus disk inclinations, which confirms strong correlation ($\gt 0.8$) for both the BC and NC, although the uncertainties in the regression parameters for BC are 3–4 times larger than for NC. This is also consistent with previous results for disks around solar-mass stars where ${R}_{\mathrm{in}}$ was found to vary only within a factor of a few (Salyk et al. 2011), and with the theoretical expectation that the inner radius increases only with $\sqrt{{L}_{\star }}$. The residual intrinsic scatter on the linear relation for the BC is four times larger than for the NC, which may point to a remaining dependence on the stellar and (variable) accretion luminosity.

Given the empirical line width–inclination relation, it is possible to estimate inclinations for disks with high resolution CO spectroscopy even when resolved imaging is not yet available. In doing so, it is assumed that the two relations found for the BC and NC represent a general property of rovibrational CO emission in the double-component disks. The advantage of having two relations, one for each velocity component, is that for a given double-component disk there are two constraints to derive one parameter. We take the measured values of the BC and NC line widths in each disk and use them to derive the disk inclination by using the regression coefficients $a,b$ from the empirical linear relations:

$$\begin{eqnarray}&&i=(\mathrm{FWHM}/\sqrt{{M}_{\star }}-a)/b,\end{eqnarray} \tag{ 1 }$$

where $a=31\pm 30$ and $b=1.8\pm 0.9$ for the BC, and $a=0.6\pm 8.8$ and $b=1.0\pm 0.3$ for the NC. We derive two inclination values, one for the BC and one for the NC, and adopt the mean of the 1σ range in common between the two. Among the single-component disks, only three have unknown inclination: EC 82, FZ Tau, and TW Cha. For these disks, we adopt the NC relation, since single-component disks with known inclinations span disk radii similar to those of the NC of double-component disks (Figure 3). Following this procedure, we estimate an inclination for a total of 15 new disks. These values are included in Table 1.

Using the inclinations described in the previous section, we derive the CO emitting radii for all disks in the sample. In Figure 4 we show a diagram made with the vibrational line flux ratios plotted against the CO emitting radii. The diagram at the left of the Figure shows a strong logarithmic anti-correlation between ${R}_{\mathrm{CO}}$ and ${F}_{v=2-1}/{F}_{v=1-0}$, with a Pearson coefficient of −0.9. In addition to spanning spatially separated ${R}_{\mathrm{CO}}$ ranges, as already shown in Figure 3, the BC and NC clearly span distinct values in ${F}_{v=2-1}/{F}_{v=1-0}$. Here too, the single-component disks have similar values to those of the NC. Two obvious outliers are SR 21 and IRS 48, both transitional disks with large inner gaps, which have vibrational excitation temperatures as high as the BC at much smaller disk radii (they will be discussed in Section 4). Excluding them, Figure 4 demonstrates that the vibrational flux ratio decreases with disk radius as

$$\begin{eqnarray}&&\displaystyle \frac{{F}_{v=2-1}}{{F}_{v=1-0}}\propto {R}^{-0.7\pm 0.1}.\end{eqnarray} \tag{ 2 }$$

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The vibrational line flux ratios measure how much the vibrational level v = 2 is populated as compared to v = 1, which in turn is a measure of the vibrational excitation temperature ${T}_{\mathrm{vib}.\mathrm{ex}.}$ (e.g., Thi et al. 2013). The relation between ${F}_{v=2-1}/{F}_{v=1-0}$ and ${R}_{\mathrm{CO}}$ can thus be converted into a radial profile for the CO vibrational excitation temperature in inner disks. The conversion of the vibrational flux ratio to a temperature depends on the CO column density, so the model-independent relation is that using the vibrational flux ratio (Equation (2)). To estimate the corresponding temperature relation, we use a model of a slab of gas in thermal equilibrium, where the level populations are defined by a single temperature following the Boltzmann distribution. Other analyses of CO rovibrational emission from disks consistently find a relatively narrow range of column densities. Consequently, we adopt a single ¹²CO column density of 10¹⁸ cm⁻², as an order-of-magnitude value representative of the range found in previous studies (10¹⁷–10¹⁹ cm⁻², Blake & Boogert 2004; Herczeg et al. 2011; Salyk et al. 2011; Banzatti et al. 2015; van der Plas et al. 2015). The details of the conversion are described in Appendix A. The resulting temperature radial profile is

$$\begin{eqnarray}&&{T}_{\mathrm{vib}.\mathrm{ex}.}\propto {R}^{-0.3\pm 0.1}.\end{eqnarray} \tag{ 3 }$$

If disks without previously known inclinations are excluded, the exponent becomes −0.2 ± 0.1, within the error, indicating that the result is not biased by our empirical derivation of disk inclinations. The excitation temperatures estimated for each disk are reported in Tables 2 and 3. The nominal uncertainty on the temperatures, once the column density is fixed, is less than 50 K as due to the uncertainty on the vibrational ratios (which is typically $\lt 10\%$).

Herczeg et al. (2011) proposed that the non-detection of ¹³CO lines in the BC, combined with the curvature seen in rotation diagrams, constrained the BC column density to within ${10}^{17.5}$–10¹⁸ cm⁻². On the other hand, rotation diagrams of ¹³CO lines were found to require column densities of ${10}^{18.5}$–10¹⁹ cm⁻² for the NC. If we relax our assumption and allow the BC to have columns as low as 10¹⁷ cm⁻², and NC as large as 10¹⁹ cm⁻², then the temperatures will decrease/increase for the NC/BC, respectively, and the overall temperature profile may exhibit a steeper index up to $q\approx 0.5$ (see Appendix A).

The vibrational excitation temperatures of the BC match those previously found from rotational analyses of ¹²CO lines (700–1700 K; e.g., Salyk et al. 2011), while the CO in the NC and SC disks typically have lower temperatures, corresponding to those determined for ¹³CO line fluxes (200–600 K; e.g., Brown et al. 2013). This is consistent with our finding that the ¹³CO line profiles match those of the NC, as indicative of probing a similar disk region. In turn, this suggests that the NC is formed in an optically thick layer, leading to high $F{(}^{13}\mathrm{CO})/F{(}^{12}\mathrm{CO})$ ratios, while the BC must be more optically thin. A similar conclusion was reached for disks in embedded young stars (Herczeg et al. 2011).

The radii derived from the BC span the range found previously for disks around solar-mass stars (0.01–0.1 AU) while the NC and SC span the range found for Herbig and transitional disks (0.1–10 AU). This is consistent with the finding that double-component disks (where BC is detected) are primarily disks around low-mass stars, while the single-component disks are primarily Herbig and transitional disks. The NC and part of SC share similar properties in terms of line widths, orbital radii, and vibrational ratios (Section 3 and Table 4), and they may trace the same gas component, regardless of the presence of a BC.

Table 4.  Summary of Rovibrational CO Emission Properties

CO comp.	FWHM	$v2/v1$	R${}_{\mathrm{CO}}$	T${}_{\mathrm{vib}.\mathrm{ex}.\ }$	13CO
	(km s−1)		(AU)	(K)
Broad (BC)	50–200	0.2–0.6	0.04–0.3	800–1500	3/23
Narrow (NC)	10–70	$\lt 0.2$	0.2–3	$\lt 700$	18/23
Single I (SC)	6–70	0.01–0.2	0.2–5	300–700	9/12
Single II (SC)	9–20	0.3–0.6	6–20	800–1500	8/8

Notes. Single-component disks are separated into two groups based on their vibrational ratios, with ${F}_{v=2-1}/{F}_{v=1-0}\lt 0.2$ found for the IR regime and $0.3\lt {F}_{v=2-1}/{F}_{v=1-0}\lt 0.6$ found for the UV regime shown in Figure 5, including the six Herbig disks from van der Plas et al. (2015). The last column reports ¹³CO detection fractions.

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While a disk origin for the BC is supported by the common double-peak profiles, as well as by a Keplerian signal seen in spectro-astrometry (Pontoppidan et al. 2011a), the origin of the NC is less clear. A disk wind has been proposed for single-peak ¹²CO v = 1–0 line profiles (those dominated by the NC, e.g., AS205 N) to reproduce the sub-Keplerian motions of the line peak, observed with spectro-astrometry (Bast et al. 2011; Pontoppidan et al. 2011a). In one case, a corresponding large-scale disk wind has possibly been seen with ALMA (Salyk et al. 2014). An observational characteristic of a disk wind launched at small radii, and preserving angular momentum as it expands, is that the gas contributing to the line becomes increasingly sub-Keplerian closer to the line center. Specifically, the line wings are exactly Keplerian, while the inner few km s⁻¹ are strongly sub-Keplerian, tracing radii significantly smaller than what their velocity would suggest. This could explain why observables constructed from the line wings of both the broad and narrow components, as we do here, yield results consistent with a nearly Keplerian disk.

Broad and narrow components that may be similar to those of the rovibrational CO lines that have been found in the 6300 Å[O i] line by Rigliaco et al. (2013). These authors suggest a disk origin for the broad component and an unbound cool disk wind as the origin of the narrow component. It is not clear whether the narrow components of [O i] and CO are generally tracing the same disk wind, but they share close resemblance in at least a few disks. For the disks in common between this work and Rigliaco et al. (2013), the FWHM of NC in CO and [O i] agrees to within $\lt 3$ km s⁻¹ in DR Tau, VZ Cha, RU Lup, TW Hya, and CW Tau; in DR Tau, the same is true also for the BC of CO and [O i].

In Section 3.5 we found a strong logarithmic anti-correlation between the CO vibrational excitation temperature and emitting radius. This relation corresponds to an empirical temperature profile of a typical inner disk surface. However, it was noted that two disks from the sample with large CO emitting radii, the transitional disks SR 21 and IRS 48, clearly depart from the relation, showing significantly higher vibrational ratios at their respective CO emitting radii than the relation predicts. To explore this further, we add six disks from the CRIRES survey of 12 disks around intermediate-mass (${M}_{\star }\approx 2-3\;{M}_{\odot }$) stars from van der Plas et al. (2015). This survey presents 4.7 μm spectroscopy with a similar experimental design and wavelength coverage as our survey. Specifically, we include the full subset of disks for which lines are detected from both ¹²CO v = 1–0 and v = 2–1 transitions in a wavelength range similar to that of our survey. Following the same procedure as for other disks, we derive ${R}_{\mathrm{CO}}$ and ${T}_{\mathrm{vib}.\mathrm{ex}.}$ for the disks around HD 100546, HD 179218, HD 190073, HD 97048, HD 141569, and HD 98922 (included in Table 3). Their locations in the temperature–radius diagram are included in Figure 5, which shows that disks around intermediate-mass stars have vibrational temperatures that are typically higher at a given radius than the relation defined by the lower-mass stars at smaller disk radii. It is clear that SR 21 and IRS 48 belong to this vibrationally hot class of disks. The combination of CRIRES surveys of disks spanning a wide range of stellar masses therefore demonstrates the existence of two trends: a vibrational temperature decrease out to radii of $\approx 3$ AU followed by a temperature increase at radii $\gt 3$ AU. We note that the remaining six disks from van der Plas et al. (2015) lack v = 2–1 measurements due either to poor telluric corrections and low signal-to-noise, or the line widths and disk inclinations put them in the IR regime of our diagram ($\lt 3$ AU), where low v = 2–1 line fluxes are expected.

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Analyses by Brittain et al. (2003), Blake & Boogert (2004), and van der Plas et al. (2015) included two possible excitation regimes for CO rovibrational emission: in the inner disk, IR pumping due to the local dust continuum radiation, and at larger disk radii, where the dust temperature is too low to generate sufficient amounts of 4.7 μm photons, pumping by UV photons from the star and/or accretion shock. The specificity of UV pumping is to populate very high vibrational levels ($v\geqslant 6$) by decay of an excited electronic state (Krotkov et al. 1980), while IR pumping drives the CO level population to equilibrate to the local color temperature of the dust. At dust temperatures of 500–1000 K, this will only populate the v = 1 and v = 2 levels. Indeed, Figure 5 can be interpreted as evidence for a transition between the IR and UV pumping regimes near 3 AU. The "IR regime," at small disk radii, and the "UV regime," at larger disk radii, are correspondingly marked in the figure. Further evidence is provided by comparison to rotational temperatures ${T}_{\mathrm{rot}.\mathrm{ex}.}$ derived in the same disks from previous studies (Salyk et al. 2009, 2011; Herczeg et al. 2011). Figure 5 shows that ${T}_{\mathrm{vib}.\mathrm{ex}.}\simeq {T}_{\mathrm{rot}.\mathrm{ex}.}$ in the IR regime, indicating that the CO rovibrational population is thermalized to the local dust temperature in the inner disk (Blake & Boogert 2004). Conversely, in the UV regime, ${T}_{\mathrm{rot}.\mathrm{ex}.}\lt {T}_{\mathrm{vib}.\mathrm{ex}.}$ (Thi et al. 2013). This is consistent with the finding of others; for instance, Brittain et al. (2003) found ${T}_{\mathrm{rot}.\mathrm{ex}.}=200$ K and ${T}_{\mathrm{vib}.\mathrm{ex}.}\gt 2000$ K in the disk of HD 141569. Note that, while the vibrational temperatures of the v = 2–1 transitions are higher than the rotational temperature in the UV regime, they are typically much lower than the vibrational temperature of even higher-lying levels (v = 6–5 can be as high as 6000 K). Therefore, while we use the v = 2–1 temperature since it is easy to measure in most disks, our results are consistent with previous work citing very high temperatures of high-lying vibrational levels.

In Figure 6 we illustrate how the temperature–radius diagram of rovibrational CO emission resembles key aspects of the current theoretical picture of protoplanetary disk dispersal. In the phenomenological class of the double-component disks, the BC is observed from a hot disk region extending to the magnetospheric accretion radius at $\approx 0.05$ AU (Bouvier et al. 2007), demonstrating the presence of abundant gas at the smallest radii dynamically permissible for Keplerian orbits. These disks are likely to be in a primordial phase, before giant planets have fully formed (sketch 1). DR Tau is an example of a primordial disk as based on its CO emission.

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The next stage in the CO sequence leads to the suppression and eventual disappearence of BC (sketch 2), strongly suggesting the opening of partly devoid inner gaps or an inner region depleted in gas compared to the primordial disks. In fact, it has been observed by monitoring CO spectra of a strong episodic accretor, EX Lup, that the BC gets significantly weaker relative to NC after an accretion outburst that depleted the gas mass in the inner disk by an order of magnitude (Banzatti et al. 2015). In this stage, double- and single-component disks overlap in the diagram. NC is detected from fractions of an AU up to $\approx 3$ AU. At similar radii, and even larger, CO is observed in single-component disks, where the BC is not detected. TW Hya is an example of this second class of disks, which probably has significant overlap with the "pre-transitional" disks as defined from dust emission (Espaillat et al. 2007). Indeed, for TW Hya, an inner gap in the optically thick dust disk has been detected (Calvet et al. 2002). Note that a non-detection of the BC does not directly imply the complete absence of dust or gas at $\lesssim 0.1$ AU. Residual gas and dust in the innermost region, which absorbs most of the direct UV radiation from the star, may likely explain why the temperature profile in some single-component disks continues along an IR-pumped power law. Photo-evaporation probably plays a role in opening disk gaps, and it is interesting to note that UV photo-evaporation is expected to take over at a few AU (Alexander et al. 2006, 2014), close to where the transition between the IR and UV regimes is found in the CO diagram. Other clearing mechanisms may dominate the disk dissipation at smaller disk radii, including disk accretion and dust evolution to growing planetesimals and protoplanets (see, e.g., Gorti et al. 2015).

The transition to a UV-dominated regime of CO rovibrational excitation defines a third class of disks, as discussed in Section 4.2. In order for UV radiation to excite the gas to high vibrational temperatures beyond a few AU, the inner disk region must be largely devoid of shielding dust. We identify these disks as having large inner gaps, with very little amounts of residual material in the inner disk (sketch 3). This interpretation is consistent with that proposed by van der Plas et al. (2015), who also suggested that UV-pumped CO emission may be a distinct signature of devoid inner gaps, and Maaskant et al. (2013), who imaged large inner dust gaps in two Herbig disks in the UV branch of our diagram, one of which was previously unknown (in HD 97048). In total, large inner gaps in the dusty disks have been detected in up to six out of eight disks in the UV branch so far, and for the remaining two (HD 98922 and HD 190073) the current lack of resolved imaging may simply be due to their large distance ($\gtrsim 300$ pc). Potentially, our analysis suggests that all disks that show a single rovibrational CO component are developing or have developed gaps in their inner 10 AU region, making them prime targets for future imaging detections.

A stellar-mass dependence of ${R}_{\mathrm{CO}}$ may be implied by Figure 6, where higher-mass stars tend to have larger ${R}_{\mathrm{CO}}$ relative to lower-mass stars. CO gas from $\lesssim 2$ AU is rarely detected in disks around stars of $\gt 1.5{M}_{\odot }$. This cannot be explained by stellar magnetospheric truncation of the disk, which is expected to happen at radii $\lt 0.1$ AU across the stellar mass spectrum (Bouvier et al. 2007), and an inner disk radius set by dust sublimation may be marginally consistent only in a few disks, and only at $\lt 1$ AU given their bolometric luminosities (Dullemond & Monnier 2010; van der Plas et al. 2015). Alternatively, a stellar-mass-dependent inner disk lifetime would be consistent with the CO temperature–radius diagram, with disks around intermediate-mass stars dispersing faster than those around low- and solar-mass stars (e.g., Muzerolle et al. 2010; Yasui et al. 2014; Ribas et al. 2015). That is, we detect fewer broad CO components in disks with short dispersal timescales. In this scenario, a mass dependence in Figure 6 would be directly linked to the timescale for gas dispersal in inner disks, and therefore to planet-formation processes and timescales.

In summary, we suggest that the CO temperature–radius diagram shows a sequence of gap opening in protoplanetary disks, where formation of smaller gaps (interpreted as an early transition, in evolutionary terms) is marked by the disappearance of the CO broad component, while opening of larger gaps (or a more evolved phase) is marked by an inversion in the trend of the temperature profile. The UV pumping regime may represent the very last step before the primordial gas in the disk is dispersed, setting the beginning of the debris disks phase (see the Beta Pic point in Figure 6, from rovibrational CO v = 1–0 lines detected by Troutman et al. 2011). A high-resolution spectrum of CO emission at 4.7 μm may therefore be a powerful discriminator between primordial, pre-transitional, and transitional disk stages at radii between 0.1 and 10 AU. Observations of the relative radial profiles of CO gas and of dust grains of different sizes in inner disks may provide a powerful tool to distinguish between competing gap formation mechanisms.

The lifetime of the planet-forming region of protoplanetary disks is linked to the observed orbital distribution of exoplanets. In particular, the semimajor axis distribution of giant exoplanets is known to contain structure indicating the presence of multiple populations, including the "hot Jupiters," orbiting very close to the star at a small fraction of an AU (Marcy et al. 2005), and a deficit of giant planets out to 0.5 AU (Cumming et al. 2008). It is generally accepted that hot Jupiters migrated inwards from a natal orbit at a few AU, either due to angular momentum exchange with protoplanetary gas (Lin et al. 1996; Kley & Nelson 2012) during the lifetime of the disk, or due to a later dynamical process, such as the Kozai mechanism (Kozai 1962; Eggleton & Kiseleva-Eggleton 2001).

To explore this in the context of our observed inner disk radii, we show the distribution of ${R}_{\mathrm{in}}$ compared to the semimajor axes of known exoplanets¹ with $M\mathrm{sin}(i)\gt 0.5\;{M}_{\mathrm{Jup}}$ in Figure 7. The BC radius distribution matches the orbital distribution of hot Jupiters (0.01–0.1 AU), as expected in the Type II migration scenario. That is, we observe that gas is indeed often available at the right disk radii within 1–2 Myr to allow hot Jupiters to migrate inwards to the inner gas disk truncation radius. However, hot Jupiters probably comprise a small fraction of giant exoplanets (Marcy et al. 2005). If the inner disk dissipates rapidly, most planets may not have time to migrate, and the orbital distribution of "cold Jupiters" will be determined by the inner gas radius of transitional disks, as traced by the narrow CO component. Indeed, Figure 7 shows that the NC distribution peaks slightly inward of the bulk of the exo-Jupiter distribution (1–10 AU).

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A stellar-mass dependence of the inner disk gas radius may also be reflected in the exoplanet population. It is known that intermediate-mass stars have very few hot Jupiters compared to lower-mass stars (Johnson et al. 2007; Sato et al. 2008), although a possible overestimation of the intermediate-mass stellar masses is currently under debate (Lloyd 2011; Johnson et al. 2013). Currie (2009) proposed that the paucity of hot Jupiters around intermediate-mass stars may be due to a stellar-mass dependent disk lifetime, where inner disks around intermediate-mass stars dissipate before any giant planets have time to migrate to the disk corotation radius. In the lower panel of Figure 7, we compare the distributions of innermost gas disk radii separated into two stellar mass bins at ${M}_{\star }=1.5\;{M}_{\odot }$. This distribution suggests that the BC component is rarely found in disks around intermediate-mass stars, relative to disks around low- and solar-mass disks. That is, the paucity of the BC within 0.1 AU in disks around intermediate-mass stars is reflected by a similar paucity of hot Jupiters in the same stellar mass bin. A likely interpretation of our CO observations is that the gaseous disk at 0.01–0.5 AU around stars with ${M}_{\star }\gt 1.5\;{M}_{\odot }$ indeed dissipates faster than the planet migration timescale. Larger samples of CO rovibrational spectra of disks and exoplanets around intermediate-mass stars are needed to confirm this link between the natal disk properties and planetary architectures.