Time Variations in the Flux Density of Sgr A* at 230 GHz Detected with ALMA

To characterize the variability of Sgr A*'s flux in the 230 GHz band, we employed the first-order structure function, which is widely used to measure the characteristic timescale in light curves of active galactic nuclei and Sgr A* (Simonetti et al. 1985; Falcke 1999; Meyer et al. 2009; Yusef-Zadeh et al. 2011; Dexter et al. 2014; Witzel et al. 2018). The first-order structure function V(τ) is defined as the variance:

$$\begin{eqnarray}&&V(\tau )=\langle {\left[F(t+\tau )-F(t)\right]}^{2}\rangle ,\end{eqnarray} \tag{ 1 }$$

where F(t) is an arbitrary function of time t. The time lag τ at a maximum value of V(τ) corresponds to a characteristic timescale. We divided the variances into 15 bins equally spaced in logarithmic time lags up to 60 minutes. The structure functions derived from the concatenated light curve of all epochs at 234.0 GHz are shown in the top panel of Figure 2 in blue. Each bin contains at least 48 variances. The structure functions of 217.5 and 219.5 GHz behave almost identically, although they are not shown in the figure.

Zoom In Zoom Out Reset image size

The variance for the light curve of all epochs increases for time lags of up to 60 minutes. This indicates that variability of longer than 60 minutes exists in the light curves. Characteristic timescales of several hours were found in the submillimeter and NIR fluxes of Sgr A* (Meyer et al. 2009; Dexter et al. 2014; Witzel et al. 2018). These hourly timescales are interpreted as being related to viscous timescales near the ISCO rather than orbital periods (Dexter et al. 2014). The observed long-timescale variation may be of the same origin as the previously observed hourly scale variations.

Long-timescale variations are clearly seen in the light curves of epochs 1 and 3, and also possibly epoch 8. However, short-timescale variations have been superposed on long-timescale variations, being dominant in the other epochs. To identify the characteristic time of the short-timescale variation, we calculated the structure function from the light curves excluding epochs 1, 3, and 8. The result is also shown in the top panel of Figure 2. The variance increases up to the time lag of ∼20 minutes, with almost the same slope of that derived from all epochs. The slope becomes flat for time lags larger than ∼30 minutes. This changing point corresponds to the characteristic timescale. The variance asymptotically approaches the value of ∼10⁻² Jy², indicating the amplitude of the short-timescale variations of ∼0.1 Jy.

We also tried to derive the characteristic timescale by searching for periodicity in the light curves because structure functions may not be reliable for quantitative determination of timescales (Emmanoulopoulos et al. 2010). To identify periodicity in the unevenly sampled light curves, we computed the Lomb–Scargle periodogram (Lomb 1976; Scargle 1982; VanderPlas 2018). The bottom panel of Figure 2 shows the resultant periodograms for the light curves of epochs 1, 2, 4, and 7. The two periodograms of epochs 4 and 7, in which the light curves show sinusoidal shapes, exhibit prominent peaks at ∼30 minutes. The heights of the peaks well exceed the 3σ level of the false-alarm probability (Baluev 2008). The epoch 1 periodogram shows a faint peak at ∼40 minutes. The epoch 2 periodogram also has a faint peak at ∼20 minutes and a significant peak at ∼60 minutes. The periodograms from the other epochs have no significant peaks. Except for the 60 minute peak in epoch 2, each of these peaks is consistent with that expected from the number of intensity peaks in the corresponding light curve. We therefore conclude that the quasi-periodicity with timescales of a few tens of minutes up to an hour appears in the light curves. The periodogram analysis supports the results from the structure functions.

Here we consider the meaning of the few tens of minutes quasi-periodicity, assuming that it reflects a certain physical process. The orbital period at the ISCO of Sgr A* is ∼30 minutes if the black hole does not rotate. Such orbital motions around Sgr A* have recently been detected in NIR flares with a period of ∼40 minutes (Gravity Collaboration et al. 2018b). Therefore, our observed short-timescale variations possibly originate from the orbital motions near the ISCO of Sgr A*. In this case, its timescale varies depending on the orbital radius. The difference in the quasi-period between epochs may reflect different orbital radii of millimeter-wave emitting hotspots.

Previous studies have suggested that a flare observed at lower frequencies may be delayed from those at higher frequencies (Yusef-Zadeh et al. 2006, 2008). To examine any time lag between the observed frequencies, we used the z-transformed discrete correlation function (ZDCF; Alexander 1997). This method enabled us to estimate a cross-correlation function from unevenly sampled light curves. We obtained 10 ZDCFs for the light curves at 217.5 and 234.0 GHz of each epoch. We estimated the time lags by fitting a quadratic function to the peaks of the ZDCFs.

All ZDCFs except for epochs 6 and 10 show absolute time lags of less than 1 minute. The variation during epoch 6 at 234.0 GHz is delayed by 2.6 ± 0.3 minutes from that at 217.5 GHz. On the other hand, the variation within epoch 10 at 234.0 GHz precedes that at 217.5 GHz by 1.4 ± 0.4 minutes. These two lags seemingly do not depend on the flux densities or spectral indices. Considering the near-zero lags of the eight ZDCFs, we conclude that there is no coherent time lag between fluxes across the different observed frequencies.

The origin of time lags at lower frequencies is proposed to be due to the change in the optical depth of adiabatically expanding plasma blobs. We estimated an expected time lag between 234.0 and 217.5 GHz based on the model of van der Laan (1966) and the following analyses of Yusef-Zadeh et al. (2006) and Miyazaki et al. (2013). The expected time lag ${\rm{\Delta }}{T}_{4-3}$ between ${\nu }_{4}=234.0\,\mathrm{GHz}$ and ν₃ = 217.5 GHz is calculated as

$$\begin{eqnarray}&&{\rm{\Delta }}{T}_{4-3}=\displaystyle \frac{{\nu }_{4}^{A}-{\nu }_{3}^{A}}{{\nu }_{2}^{A}-{\nu }_{1}^{A}}\times {\rm{\Delta }}{T}_{2-1},\end{eqnarray} \tag{ 2 }$$

where A = −(p + 4)/(4p + 6) and p is the index of the relativistic electron energy spectrum [n(E) ∝ E^−p] (Yusef-Zadeh et al. 2006). Using the time lag ΔT₂₋₁ between ${\nu }_{2}=43\,\mathrm{GHz}$ and ${\nu }_{1}=22\,\mathrm{GHz}$ of 30 minutes (Yusef-Zadeh et al. 2006, 2008) and p = 2, ΔT₄₋₃ becomes 1.4 minutes, meaning that the variation at 234.0 GHz precedes by 1.4 minutes from that at 217.5 GHz. This expected time lag is comparable to the time resolution of the ALMA observations and could be detectable.

One likely reason why we could not detect the time lag was that our observations were performed during the quiescent state of Sgr A*, and time delay is often detected in the flaring states. Miyazaki et al. (2013) found no time lag between light curves at 90 and 102 GHz. They suggested the reason is that the blob is initially optically thin, or that the emission does not originate in expanding plasma but instead comes from orbiting hot plasma spots on the accretion disk. The observed orbital timescale variations are in favor of the orbiting hotspots model.

The time lag between different frequencies could also be explained in the context of a jet model (Falcke et al. 2009; Brinkerink et al. 2015). Following the analysis of Brinkerink et al. (2015), we can constrain the gas flow velocity v_f by combining the time lag with the source size difference between two observed frequencies. Given the upper limit of the time lag of 1.5 minutes (an average of the time resolution of our observations), and the source size difference of 3.6 μas, we found v_f ≥ 0.16c. This is consistent with v_f = 0.77c that was derived from the data of the Very Large Array and ALMA (Brinkerink et al. 2015). Therefore, the jet model cannot be ruled out by our time lag analysis. Further simultaneous multiwavelength observations are necessary to examine the jet model.

We derived spectral indices between the flux densities at 217.5 and 234.0 GHz at all data points. The mean spectral index of all the data is 0.12 ± 0.10. The maximum and minimum values are 0.31 ± 0.01 and −0.21 ± 0.01, respectively. In the NIR, several authors reported a variation in the spectral index dependent on the flux density (Ghez et al. 2005; Bremer et al. 2011; Witzel et al. 2018). Figure 3 shows the spectral index dependence on the flux density at 217.5 GHz. A clear correlation that a higher flux density has a higher spectral index can be seen.

Zoom In Zoom Out Reset image size

In the radio band, some authors have already reported a trend that the spectral index becomes larger near the peak of the flux density in the centimeter (Falcke 1999) and millimeter (Miyazaki et al. 2013) bands. In comparison with the spectral indices between 218.0 and 233.8 GHz derived from the ALMA Cycle 1 observations (Bower et al. 2015), seven out of eight spectral indices agree with the trend of our result.

The similar trend in the NIR suggests that the emissions in the 230 GHz and NIR bands originate from the same source. The dependence of the NIR spectral index on the flux density can be explained by the synchrotron emission in the optically thin frequency regime, with a variable exponential cutoff in the energy of the electron population due to synchrotron losses (Eckart et al. 2006; Bremer et al. 2011; Witzel et al. 2018). This interpretation is, however, not applicable to the 230 GHz trend because a change in the cutoff frequency ν₀ appears as a term of $\exp [-{(\nu /{\nu }_{0})}^{1/2}]$, and ν₀ of ∼10–100 THz does not alter spectral indices at the millimeter band coherently.

Thermal synchrotron emission with variable electron temperature may also explain the change of the spectral index in the submillimeter band by shifting the peak of the spectrum (Bower et al. 2015, 2019; von Fellenberg et al. 2018). However, the thermal model solely cannot account for the NIR flux of Sgr A*, and an additional nonthermal component may contribute to it. Measuring variable SED must be necessary to unveil the emission mechanism of Sgr A*.