# Ejection Patterns in the DG Tau Jet Over the Last 40 Years:
Insights into Mass Accretion Variability111Based on observations obtained at
the international Gemini Observatory, a program of NSF’s NOIRLab, which is
managed by the Association of Universities for Research in Astronomy (AURA)
under a cooperative agreement with the National Science Foundation on behalf
of the Gemini Observatory partnership.

Tae-Soo Pyo Subaru Telescope, National Astronomical Observatory of Japan,
National Institutes of Natural Sciences, 650 North A‘ohōkū Place, Hilo, HI
96720, USA  School of Mathematical and Physical Science, SOKENDAI (The
Graduate University for Advanced Studies), Hayama, Kanagawa 240-0193, Japan
<EMAIL_ADDRESS>Masahiko Hayashi National Astronomical Observatory of Japan,
2-21-1 Osawa, Mitaka, Tokyo 181-8588, Japan  JSPS Bonn Office, Ahrstr. 58,
53175 Bonn, Germany  Michihiro Takami Institute of Astronomy and Astrophysics,
Academia Sinica, 11F of Astronomy-Mathematics Building, No.1, Sec. 4,
Roosevelt Rd, Taipei 10617, Taiwan, R.O.C.  Tracy L. Beck The Space Telescope
Science Institute, 3700 San Martin Dr., Baltimore, MD 21218, USA

###### Abstract

We aim to clarify the link between mass accretion and ejection by analyzing DG
Tau’s jet observations from optical and near-infrared data spanning 1984 to
2019, alongside photometric variations between 1983 and 2015. We classified 12
moving knot groups among 17 total knot groups based on their constant proper
motions and comparable radial velocities. A strong correlation emerges between
deprojected flow velocities of the knots and the photometric magnitudes of DG
Tau. From 1983 to 1995, as the deprojected ejection velocities surged from
$\sim$ 273 $\pm$ 15 km s-1 to $\sim$ 427 $\pm$ 16 km s-1 the photometric
magnitudes ($V$) concurrently brightened from 12.3 to 11.4. Notably, when DG
Tau became brighter than 12.2 in the $V$ band, its (B V) color shifted bluer
than its intrinsic color range of K5 to M0. During this period, the launching
point of the jet in the protoplanetary disk moved closer to 0.06 AU from the
star in 1995. Following a $V$ magnitude drop from 11.7 to 13.4 in 1998, the
star may have experienced significant extinction due to a dust wall created by
the disk wind during the ejection of the high-velocity knot in 1999. Since
then, the magnitude became fainter than 12.2, the (B V) and (V R) colors
became redder, and the deprojected velocities consistently remained below 200
km s-1. The launching point of the jet then moved away to $\sim$ 0.45 AU by
2008. The prevailing factor influencing photometric magnitude appears to be
the active mass accretion causing the variable mass ejection velocities.

ISM: Herbig-Haro objects — ISM: individual (DG Tauri) —
ISM: jets and outflows — stars:formation — stars: pre-main-sequence — AAVSO

##  1 INTRODUCTION

Brightness variability is intrinsic to young stellar objects (YSOs). Various
phenomena, such as outbursts, mass ejection (loss), and mass accretion, drive
this variability. For example, FUors display a 4 to 6 mag increase in visual
magnitude, often attributed to outbursts resulting from significant upticks in
the mass accretion rate (Hartmann & Kenyon, 1996). Numerous studies have
investigated the correlation between mass ejection events associated with
knots in YSO jets and changes in photometric magnitude. Garufi et al. (2019)
noted that two knots in the jet of RY Tau were ejected during local
photospheric peaks in the $V$ magnitude. On the other hand, Babina, Artemenko,
& Petrov (2016) discovered a correlation between stellar magnitude and the
radial velocity of the wind from RY Tau. Takami et al. (2020) identified a
potential correlation between the ejections of four knots and the dimming
events observed in the optical photometric data of RW Aur.

DG Tau is one of the first young stellar objects for which optical jets were
discovered in the early 1980s (Mundt & Fried, 1983). Mundt & Fried (1983)
reported that the DG Tau jet (also known as HH 158) extends up to $\sim$
10$\arcsec$ in the southwest direction from the star (PA $\sim$
225${}^{{}^{\circ}}$). Eislöffel & Mundt (1998) estimated the proper motions
and radial velocities for the four knots. Since their work, only two papers
have observed the entire length of the jet (Whelan, Ray, & Davis, 2004; Oh et
al., 2015). Oh et al. (2015) demonstrated that the jet extended up to
20$\arcsec$ by 2014. Other studies have focused their observations primarily
around the star within a span of several arcseconds (e.g. Bacciotti et al.,
2000; Lavalley-Fouquet, Cabrit, & Dougados, 2000; Agra-Amboage et al., 2011;
White et al., 2014; Maurri et al., 2014; Takami et al., 2023).

Pyo et al. (2003) reported from observations of the [Fe II] $\lambda$1.644
$\mu$m line that the DG Tau jet possesses two distinct velocity components
characterized by their radial velocities ($V_{r}$) and spatial distributions.
The high-velocity component (HVC) has velocities of $V_{\rm LSR}~{}=~{}-$220
$\pm$ 50 km s-1 and an intensity peak at $\sim$0$\hbox
to0.0pt{.\hss}^{{}^{\prime\prime}}$75 away from the star222The stellar radial
velocity is $V_{\rm LSR}~{}=~{}+6.4$ km s-1., while the low-velocity component
(LVC) is measured at $-$100 $\pm$ 100 km s-1 and shows an intensity peak at
$\sim$0$\hbox to0.0pt{.\hss}^{{}^{\prime\prime}}$45 from the star. They
concluded that the HVC is associated with a fast, well-collimated jet
originating from either the innermost disk, the star-disk interface, or the
star surface. In contrast, the LVC represents a broader wind, likely arising
from the outer radii near the inner edge of the disk. White et al. (2014) and
Agra-Amboage et al. (2011) also provided the position velocity diagrams (PVDs)
of the [Fe II] $\lambda$ 1.644 $\mu$m line, acquired on 11th November and 15th
October 2005, respectively. White et al. (2014) identified two velocity
components: one located at $\sim$0$\hbox to0.0pt{.\hss}^{{}^{\prime\prime}}$4
with a velocity of $-$180 km s-1 and another at $\sim$1$\hbox
to0.0pt{.\hss}^{{}^{\prime\prime}}$24 with a velocity of $-$250 km s-1. These
components correspond to their knots B and C, which we will refer to as knots
WB and WC, respectively, in this paper.

Notable peaks of this high-velocity flow, surpassing $-$200 km s-1, were
observed from 1988 to 2002, as evidenced in Figure 1o of Hamann (1994)
(observed in February 1990), Figures 5 and 7 of Hartigan, Edwards, & Ghandour
(1995) (observed on January 9, 1988), and Figures 2a and 2b of Whelan, Ray, &
Davis (2004) (observed on September 17, 2002). The PVDs by Maurri et al.
(2014) also revealed dominant emissions in the $|V_{r}|>$ 200 km s-1 range
from $\sim$ 0$\hbox to0.0pt{.\hss}^{{}^{\prime\prime}}$75 (knot A1) to
5$\arcsec$ (knot B1) in January 1990. This high-velocity component was
similarly noted in the [Fe II] emissions of Takami et al. (2002), Pyo et al.
(2003), and White et al. (2014) for the period between 2000 and 2005. However,
the flux levels at $|V_{r}|>$ 200 km s-1 were minimal in Figure 1 of Chou et
al. (2013), which detailed line profiles of [S II] $\lambda$6731 Å, [O I]
$\lambda$6300 Å, and [Fe II] $\lambda$7155 Å obtained from October to November
2010. Furthermore, Takami et al. (2023) reported the presence of a single
collimated jet with $|V_{r}|=$ 120$-$160 km s-1 over 2013-2019. This jet
exhibited a notably reduced velocity when compared to previous observations.
This indicates that the high-velocity flow exceeding $|V_{r}|>$ 200 km s-1 was
present from 1988 to 2005 but has been weakened since 2006. This variation
implies changes in the inner disk structure, given that the high-velocity flow
is likely associated with the inner boundary layer between the stellar surface
and the inner edge of the circumstellar disk (Pyo et al., 2003).

Knots may form due to internal shocks between ejecta of varying velocities,
exhibiting periodic spacing. The ejection of knots in HH 158 has been
extensively studied. Pyo et al. (2003) proposed a knot ejection cycle of 5
years, determined from the proper motions of the knots. This was later
corroborated by Rodríguez et al. (2012) using radio, optical, and X-ray data.
Agra-Amboage et al. (2011) proposed a shorter period of 2.5 years, deduced
from the presence of an intermediary knot and assuming a typical proper motion
ranging from 0$\hbox to0.0pt{.\hss}^{{}^{\prime\prime}}$27 to 0$\hbox
to0.0pt{.\hss}^{{}^{\prime\prime}}$34 yr-1. White et al. (2014) suggested that
the ejection period might be variable, citing the lack of a new knot for 4
years between November 2005 and November 2009, as well as the atypical proper
motion of 0$\hbox to0.0pt{.\hss}^{{}^{\prime\prime}}$17 yr-1 observed for knot
WB.

Beyond the variability associated with knots, T Tauri stars show significant
photometric variability. Grankin et al. (2007) summarized the optical
photometric variations of DG Tau over a period of 20 years, spanning from 1984
to 2004. They reported that DG Tau experienced large changes in its mean
brightness ($\sigma_{V_{\rm m}}$) over this timescale. Specifically, DG Tau’s
brightness gradually increased, fluctuating from 12.5 to 11.5 mag ($\Delta
V_{\rm m}=-$1.0) between 1984 and 1996. There was a pronounced drop in the
mean brightness from 11.71 to 13.31 mag ($\Delta V_{\rm m}=+$1.6) between 1997
and 1998.

In this paper, we provide a comprehensive summary of knot ejection events in
DG Tau over the past 40 years, detailing their proper motions and radial
velocities, and discussing their correlation with the star’s photometric
variability.

##  2 OBSERVATIONS AND DATA REDUCTION

We incorporated new data of the [Fe II] $\lambda$1.257 $\mu$m emission line
observed on 13th February 2007 (UT), using the Near-Infrared Integral Field
Spectrometer (NIFS) mounted on the Gemini-North Telescope atop Mauna Kea,
Hawai‘i. NIFS, an image-slicing integral field unit, is complemented by the
adaptive optics (AO) system ALTAIR333ALTtitude conjugate Adaptive optics for
the InfraRed. It offers a high spatial resolution utilizing either a natural
guide star (NGS) or a laser guide star (LGS) (McGregor et al., 2002).

Given its sufficient brightness with $R$ $=$ 12.28 mag, we employed DG Tau as
the NGS for AO correction. After the AO correction, the full width at half
maximum (FWHM) of the stellar image was $\sim 0\farcs 14$. The Integral Field
Unit (IFU) offered a spectral resolution of $R$ $\sim$ 5000, covering a
3${}^{\prime\prime}\times$3′′ field of view at a pixel scale of
0$\farcs$1$\times$0$\farcs$04\. The observations were conducted under
photometric conditions, with the average natural seeing ranging between
0$\farcs$6 and 0$\farcs$85\. Further details on the observations and data
reduction can be found in Pyo et al. (2014) and Beck et al. (2008). The
reduced data was subsequently re-sampled to a square grid with a pixel scale
of 0$\farcs$04 $\times$ 0$\farcs$04 pixels.

##  3 RESULTS

###  3.1 Knots and velocity structure in the new data

The left panel of Figure 1 shows the continuum-subtracted image of the [Fe II]
emission, integrated over the velocity range of $-$600 km s-1 to +600 km s-1.
The positive direction of the vertical axis aligns with the blue-shifted jet
direction, at PA $=$ 222∘. It reveals a well-collimated jet. The red-shifted
jet becomes visible at Y $<-$0$\hbox to0.0pt{.\hss}^{{}^{\prime\prime}}$6,
traversing a gap likely induced by the protoplanetary disk (Pyo et al., 2003;
Agra-Amboage et al., 2011; White et al., 2014). Notably, the intensity of the
red-shifted [Fe II] emission is 4–5 times less than the peak brightness of the
blue-shifted jet located at Y = 0$\farcs$72.

The blue-shifted jet reveals three intensity peaks at positions 0$\arcsec$,
0$\hbox to0.0pt{.\hss}^{{}^{\prime\prime}}$30, and 0$\hbox
to0.0pt{.\hss}^{{}^{\prime\prime}}$72, which we have designated as knots PA,
PB, and PC, respectively. We compared these to the knots A, B, and C of White
et al. (2014) (hereinafter WA, WB, and WC). White et al. (2014) observed a
comparable area as we did three times between 2005 and 2009, while we
conducted our observations in 2007. The position of the knot PA is situated
closely to the star’s position within a spatial resolution of 0$\farcs$14\.
Consequently, it may be influenced by residual effects following the
subtraction of the star’s continuum. Previous work by White et al. (2014)
identified a stationary knot WA, situated approximately at 0$\farcs$2,
spanning the years 2006 to 2010. Considering the aforementioned reasons, we
have excluded knots PA and WA from the list of moving knots. Knot PB, found at
0$\hbox to0.0pt{.\hss}^{{}^{\prime\prime}}$30, when hypothesized to align with
knot WA, reveals a proper motion of 0.06 $\arcsec$ yr-1, significantly smaller
than other identified knots. Figure 10 in White et al. (2014) does not exhibit
any knot structure roughly 0$\hbox to0.0pt{.\hss}^{{}^{\prime\prime}}$47 away
from the star either. This obscures the relationship between knots WA and PB.
Finally, the location of knot PC overlaps with the trajectory of knot WB from
White et al. (2014). It indicates that the knot PC belongs to the knot WB.

The right panel of Figure 1 presents the PVD of the jet, detailing the
velocity characteristics of the [Fe II] $\lambda$1.257 $\mu$m emission. The
blue-shifted [Fe II] emission spans velocities from $-$215 to 0 km s-1,
delineating a clear boundary at $\sim-$220 km s-1, aligning with the LVC of
the [Fe II] $\lambda$1.644 $\mu$m emission described by Pyo et al. (2003). The
peak velocity fluctuates between $-$150 to $-$100 km s-1, with no discernible
higher velocity component exceeding $-$200 km s-1. The absence of HVC
contrasts sharply with Figure 1 from Pyo et al. (2003), which displays a
pronounced HVC extending to $\sim-$300 km s-1 peaking at $V_{\rm LSR}=-$220 km
s-1. Agra-Amboage et al. (2011) and White et al. (2014) presented PVDs of the
[Fe II] $\lambda$ 1.644 $\mu$m line, obtained in 2005, roughly two years prior
to our [Fe II] observations. When considering the proper motion of the knots
and the interval between observations, the features they observed within
1$\arcsec$ of the star largely align with ours. The PVD in Figure 4 of Agra-
Amboage et al. (2011) displays a broader velocity range, extending to $-$400
km s-1 attributed to a lower wavelength resolution of $R\sim$ 3,000.
Meanwhile, Figure 15 of White et al. (2014) reveals the emission spanning up
to $-$280 km s-1 in their NIFS dataset. Their data shows the emission
stretching to velocities beyond $-$200 km s-1. In contrast, the PVDs taken
after 2013 by Takami et al. (2023) echo the velocity features of ours,
indicating a single lower-velocity flow at $V_{\rm LSR}=$ $-170$ to $-140$ km
s-1 within 2$\arcsec$ from the star.

###  3.2 Identification of knots

Table Ejection Patterns in the DG Tau Jet Over the Last 40 Years:  Insights
into Mass Accretion Variability111Based on observations obtained at the
international Gemini Observatory, a program of NSF’s NOIRLab, which is managed
by the Association of Universities for Research in Astronomy (AURA) under a
cooperative agreement with the National Science Foundation on behalf of the
Gemini Observatory partnership. and Figure 2 compile all the moving knots
observed in the DG Tau jet from 1984 to 2019, as reported in the literature. A
total of 51 knots have been documented. These knots have been observed in
optical and near-infrared forbidden emission lines. Notably, over half of the
knots are within 2$\arcsec$ from the star (approximately 51% within 1$\arcsec$
and about 57% within 2$\arcsec$). This reminds us that most observations have
been concentrated on detecting knots close to the star with a narrow field of
view using high spatial resolutions with AO.

Assuming constant velocity trajectories following ejection from the star, we
have categorized these observations into 17 unique knot groups, each defined
by a consistent proper motion (see Table Ejection Patterns in the DG Tau Jet
Over the Last 40 Years:  Insights into Mass Accretion Variability111Based on
observations obtained at the international Gemini Observatory, a program of
NSF’s NOIRLab, which is managed by the Association of Universities for
Research in Astronomy (AURA) under a cooperative agreement with the National
Science Foundation on behalf of the Gemini Observatory partnership.). The
knots associated with the HVC and LVC in the study by Pyo et al. (2003) are
denoted as kHVC and kLVC, respectively, to prevent any ambiguity with general
high- and low-velocity components. For several knots, namely B, B0, A2, WC,
and PB, subsequent observations did not identify their presence further down
the jet, suggesting they might have been dispersed, merged with other knots,
or no longer be shocked as they traveled downstream. Our analysis will
concentrate on knot groups that exhibit long-term stability. Of the 17 knot
groups, we estimated the proper motions for 12 of them as discussed below. The
12 moving knot groups are listed in Table 2.

Mundt et al. (1987) reported for the first time an optical jet from DG Tau,
which corresponds to knot C of Eislöffel & Mundt (1998) who identified four
knots, labeled A, B, C, and D, in their [S II] images. While knots B and C
were distinctly identified by peaks enclosed by closed contours, knots A and D
lacked such defining characteristics. Knot D is positioned ahead of the
luminous knot C, making it seem like an extension of knot C without an
individual peak. Extrapolating from its observed proper motion of $\sim$
0$\farcs$32 yr-1, as reported by Eislöffel & Mundt (1998), the projected
location of knot D in 2015 would be $\sim$ 20$\arcsec$ distant from the star.
Yet, Oh et al. (2015) observed a continuous emission trail from the central
star to knot C, not discerning a distinct knot corresponding to D. It is
plausible to deduce that knot D characterizes the jet’s leading edge and
invariably precedes knot C. Of the four knots, knot C was the most distinct at
the jet’s forefront from 1984 to 2014, demonstrating a proper motion of
0$\farcs$18 $\pm$ 0$\farcs$01 yr-1. Knot B, a faint peak observed by Eislöffel
& Mundt (1998) from 1983 to 1990, was not distinctly identified as an
independent knot in subsequent long-slit observations after 2002. Using the
proper motion measurement of 0$\farcs$23 $\pm$ 0$\farcs$02 yr-1 provided by
Eislöffel & Mundt (1998), it appears that knot B is converging with Knot C,
which has a proper motion of approximately 0$\farcs$18 yr-1. In the 2002
observation by Whelan, Ray, & Davis (2004), knot C spanned over 5$\arcsec$
(extending between approximately 10$\arcsec$ and 15$\arcsec$). This extension
is likely due to the merging of knots B and C, as the predicted position for
knot B was around $\sim$ 10$\farcs$2\. Since then, the separation between
knots B and C has remained under 1$\farcs$2\. As a result, knot B has become
indistinct, especially in proximity to the brighter knot C, particularly in
the low spatial resolution observations (with a seeing of roughly 2$\arcsec$)
by Oh et al. (2015). Knot A was located 2$\farcs$5 from the star in the images
of Eislöffel & Mundt (1998). Meanwhile, Solf & Bohm (1993) and Lavalley et al.
(1997) identified two knots, which were labeled as B1 and B2, in their [S II]
PVDs. The position of knot A well matches the trail of knot B2, as shown in
Figure 2. We thus posit that knots A and B2 are indistinguishable and
henceforth refer to them as knot A-B2. The proper motion of knot A-B2 is
0$\farcs$25 yr-1.

Knot B1 has been observed multiple times by many observers from 1992.62 to
2014.89, exhibiting a proper motion of 0$\farcs$23 $\pm$ 0$\farcs$01 yr-1.
Knot B0 was located at 3$\farcs$3, which was 0$\farcs$5 behind knot B1, in
1999 (Maurri et al., 2014). It exhibited a greater radial velocity of $-$255
km s-1 compared to the $-$210 km s-1 of knot B1. In 2002, Whelan, Ray, & Davis
(2004) showed a long knot at 4$\farcs$6, called knot B, whose radial velocity
varies from $-$240 km s-1 at 4$\arcsec$ to $-$200 km s-1 at 6$\arcsec$. This
velocity gradient along the long knot is probably due to the merge effect of
the faster knot B0 at the behind of the knot B1. The two close knots could not
be separated with low spatial resolution $\geq$ 1$\arcsec$. Any fainter knots,
should they exist, interspersed between knots B1, B2, and C remain unresolved
due to a spatial resolution of $\sim 2^{\prime\prime}$, as depicted in Figure
3 by Oh et al. (2015).

A knot at 0$\farcs$25 found by Kepner et al. (1993), knot C by Solf & Bohm
(1993), and the extension of the high-velocity branch by Lavalley et al.
(1997) align well in a constant proper motion, as illustrated in Figure 2.
These are collectively categorized as knot KHY1 in Table Ejection Patterns in
the DG Tau Jet Over the Last 40 Years:  Insights into Mass Accretion
Variability111Based on observations obtained at the international Gemini
Observatory, a program of NSF’s NOIRLab, which is managed by the Association
of Universities for Research in Astronomy (AURA) under a cooperative agreement
with the National Science Foundation on behalf of the Gemini Observatory
partnership. with the proper motion of 0$\farcs$21 $\pm$ 0$\farcs$01 yr-1.

Bacciotti et al. (2000) identified the two distinct features A1 at 1$\farcs$45
and A2 at 0$\farcs$75 in 1999. They interpreted the positions of A1 and A2 as
the edges of the extended emission structure, rather than peak positions.
Maurri et al. (2014) presented position-velocity diagrams from seven
contiguous slits, covering an extended area up to 5$\farcs$5 from the star.
The peak position for knot A1 was identified slightly closer to the star, at
1$\farcs$35\. Knot A2 is associated with diffuse continuous jet emission
extended to 0$\farcs$75 from the star. We selected the midpoint at $\sim$
0$\farcs$45 for knot A1, with an uncertainty of $\pm$0$\farcs$2\. Our analysis
indicated that the peak position of knot A1 aligns well with a constant proper
motion when considering three subsequent observations: a knot structure peak
at 0$\farcs$6 in January 1997, 1$\farcs$0 in December 1997 as reported by
Dougados et al. (2000), and the high-velocity peak at 0$\farcs$93 in January
1998 by Lavalley-Fouquet, Cabrit, & Dougados (2000). The radial velocities
measured in these studies were consistently similar, averaging around $-$347
$\pm$ 5 km s-1. As a result, we categorized these observations under knot A1.

For the knots kHVC and kLVC initially identified by Pyo et al. (2003) in the
[Fe II] $\lambda$ 1.644 $\mu$m line, we observed a consistent match in
locations and radial velocities with the knots identified by Takami et al.
(2002) in their Figure 1 and by Whelan, Ray, & Davis (2004) in their Figure
3b. The calculated proper motions for knots kHVC and kLVC are 0$\farcs$31 yr-1
and 0$\farcs$21 yr-1, respectively, based on the data from these three
observations.

Knot WC is situated along the path of kLVC in Figure 2. However, its radial
velocity of $-$250 km s-1 aligns more closely with that of kHVC. If knot WC
were a successor to kHVC, its peak would be expected at 1$\farcs$98 in 2005,
placing it outside the field of view of NIFS. Instead, knot WC was observed at
1$\farcs$24, near the edge, and appeared to extend beyond that point. One
possibility is that WC represents the tail end of kHVC, truncated by the field
of view. Alternatively, knot WC might be a distinct high-velocity knot with a
velocity of $-$250 km s-1. Given these uncertainties, we have chosen not to
incorporate WC into either kHVC or kLVC.

Knot WB, referred to as knot B by White et al. (2014), was documented in three
separate instances by White et al. (2014). In this paper, we have included
knot PC at 0$\farcs$72 within the knot WB group due to its alignment with knot
WB’s trajectory, as detailed in Section $\S$2. Knot PB was identified in this
study (see Section $\S$2). No subsequent observations have captured a
corresponding knot since our initial identification, and as a result, we have
chosen not to consider it further.

Takami et al. (2023) recently identified knots labeled A, B, C, D, and E. This
paper refers to them as TA, TB, TC, TD, and TE, respectively. Among these,
knots TA and TB vanished by 2017 and 2014, respectively. Earlier observations
did not detect these two knots either. We hence have chosen not to consider
them in the current study. The remaining knots, TC, TD, and TE exhibit
distinct peaks and discernible proper motions. Consequently, we have included
them in the current study.

Figure 3 shows the radial velocities of the knots plotted against their
distances from the star. Out of the 51 knots observed, radial velocities were
determined for 41 of them. All recorded radial velocities are presented as
absolute values, $|V_{r}|$, relative to DG Tau’s radial velocity of $V_{\rm
LSR}=+$6.4 km s-1 (Kimura, Kawabe, & Saito, 1996). The accuracy of the radial
velocity for each knot is within a few ten km s-1.

Within 1′′ of the star, knot velocities span a wide range, from 47 km s-1 up
to 350 km s-1. Conversely, at distances exceeding 2′′, we observed velocities
predominantly clustered around the mid-range value of 206 $\pm$ 37 km s-1.
Figure 4 displays the distribution of $|V_{r}|$ for all 41 observations.
Notably, 51% (21 out of 41) of these observations indicate velocities ranging
between 150 and 250 km s-1, with a mean velocity of 199 $\pm$ 73 km s-1. This
velocity distribution implies that knots with either very high or very low
velocities are confined to within 2′′ of the star. Knots related KHY1 and A1
have very high radial velocities over 300 km s-1 . Their locations are
expected around 2$\hbox to0.0pt{.\hss}^{{}^{\prime\prime}}$6$-$2$\hbox
to0.0pt{.\hss}^{{}^{\prime\prime}}$7 from the star on 2002.71. However,
Whelan, Ray, & Davis (2004) did not show any knot-like structures around that
position and only showed weak diffuse emissions. Knot kLVC showed the lowest
radial velocity, which was no longer discernible in the data of White et al.
(2014) on 2005.87. The vanishing of knot TA within $\sim$ 2′′ was reported by
Takami et al. (2023). Further investigation will be required to provide
explanations for these instances of knot disappearance.

The radial velocities of knots beyond 2′′ decline at a rate of $-$4.8 km s-1
arcsec-1 while the velocity gradient is much smaller as $-$0.3 km s-1 arcsec-1
including the inner knots within 2$\arcsec$. Each knot displays, however,
unique characteristics. Knot B1 exhibits a pronounced deceleration trend of
$-$5.4 km s-1 arcsec-1 spanning from 2$\farcs$3 to 6$\farcs$7\. Knot C
manifests a subtle increase in speed, transitioning from 136 km s-1 at
8$\farcs$5 to 172 km s-1 at 13$\farcs$7\. Likewise, knot A-B2 demonstrates a
mild velocity ascent, climbing from 200 km s-1 at 3$\farcs$3 to 220 km s-1 at
8$\farcs$8.

##  4 EJECTION DATES, PROPER MOTIONS, VELOCITIES, AND INCLINATION ANGLES of
TWELVE KNOT GROUPS

In Table 2, we provide a summary of the observations for the 12 knot groups
observed multiple times. This table lists the ejection dates, proper motions,
average radial velocities ($|\bar{V}_{r}|$), tangential velocities ($V_{t}$),
deprojected knot velocities ($V_{f}$), and inclination angles ($I$) relative
to the line of sight. The proper motion for each knot group was determined
using least squares fitting. Based on these values, we deduced the ejection
dates of the knots from the star position. Additionally, we computed the
average radial velocity for each knot group. To transform the proper motion
into tangential velocity ($V_{t}$), we employed the distance to the TMC, which
is 141 $\pm$ 7 pc, as given by the GAIA DR2 data (Zucker et al., 2019). The
deprojected knot velocities were calculated using the formula
$V_{f}=\sqrt{|\bar{V}_{r}|^{2}+V_{t}^{2}}$. The inclination angles of the knot
ejections were estimated from the formula $I=\arctan(V_{t}/|\bar{V}_{r}|)$.

Figure 5a and b show the proper motions and tangential velocities as functions
of the ejection dates. The average proper motion for all the knot groups is
0$\farcs$21 $\pm$ 0$\farcs$08 yr-1, corresponding to a tangential velocity
$V_{t}$ of 140 $\pm$ 53 km s-1. The proper motions remained approximately
consistent with this average for the knots ejected from 1937 to 1990. However,
there is a notable peak at 1995.44 associated with the knot A1, which has an
elevated proper motion of 0$\farcs$37 $\pm$ 0$\farcs$03 yr-1, or a tangential
velocity $V_{t}$ of 249 $\pm$ 26 km s-1. After 2003, the recorded proper
motions have indicated slower movements less than 0$\farcs$15 yr-1
($V_{t}\sim$ 100 km s-1).

In Figure 5b, the radial velocity ($|V_{r}|$) is between 160 and 255 km s-1
for the knots ejected between 1937 and 1983. There is then a sharp increase in
1990, reaching 335 $\pm$ 15 km s-1 and peaking at 347 $\pm$ 5 km s-1 in 1995.
By 2000, the radial velocity had significantly reduced, hovering around $\sim$
140 km s-1 from 2003 onwards, with knot kLVC marking a low point at 82 $\pm$
25 km s-1. The temporal variation in the radial velocity mirrors that of the
tangential velocity, with both showing a pronounced rise around 1990 followed
by a notable drop around 2000. Figure 5c displays a temporal variation in the
deprojected knot velocity $V_{f}$. It peaked at 427 $\pm$ 16 km s-1 in 1995,
reflecting the trends in radial and tangential velocities. The radial
velocities are typically $\sim$ 1.5 times greater than the tangential
velocities. The knot flow velocity $V_{f}$ was 200 to 300 km s-1 between 1930
and 1983. In the ejection dates of the 1990s, the velocity exceeded 300 km
s-1. However, it dropped to below 200 km s-1 after 2000. We will discuss the
potential link between the rise in flow velocity during the 1990s and changes
in the star-disk system activities. Such changes might have spurred rapid mass
accretion onto the stellar surface while concurrently driving mass ejection
from the star-disk system. Figure 5d shows the inclination angle ($I$). It
exhibits a roughly constant value of
$\bar{I}=\overline{\arctan(V_{t}/|\bar{V}_{r}|)}$ =35$\hbox
to0.0pt{.\hss}^{\circ}$6 $\pm$ 10$\hbox to0.0pt{.\hss}^{\circ}$9.

In Figure 6, the radial and tangential velocities of the knots are plotted and
annotated with their respective ejection years for detailed examination of
ejection angles. The dash-dotted line indicates the angle of $\bar{I}$. It
appears that the direction of jet ejection varies over time. Misaligned or
wiggling shocked regions often occur due to variations in ejection angle over
time (Frank et al., 2014). In 1990, knot KHY1 was identified as the closest to
the line of sight with an inclination angle of 22$\hbox
to0.0pt{.\hss}^{\circ}$6 $\pm$ 1$\hbox to0.0pt{.\hss}^{\circ}$8\. In 1999,
knot kLVC registered an angle of 59$\hbox to0.0pt{.\hss}^{\circ}$6 $\pm$
29$\hbox to0.0pt{.\hss}^{\circ}$2\. Since 2000, the inclination angle has
shifted by $\sim$ 6∘ closer to the line of sight, averaging at 29$\hbox
to0.0pt{.\hss}^{\circ}$6 $\pm$ 7$\hbox to0.0pt{.\hss}^{\circ}$0, which is
still within 1$\sigma$ of the mean inclination angle of 35$\hbox
to0.0pt{.\hss}^{\circ}$6\. It is reasonable to assume that the direction of
jet ejection varies across a certain range of angles. This tendency can also
be seen in the sky plane. The early four knots (A, B, C, D) deviated
$\pm$12$\arcdeg$ from the mean jet direction (PA = 225$\arcdeg$) in proper
motion vectors (Eislöffel & Mundt, 1998). The jet wiggles about its mean
direction, showing a bow shock or arc-shape feature at its tip (Lavalley et
al., 1997; Dougados et al., 2000; Takami et al., 2023). The wiggle amounts to
0$\hbox to0.0pt{.\hss}^{{}^{\prime\prime}}$2 deviated from the mean jet
direction at $\sim$ 2$\hbox to0.0pt{.\hss}^{{}^{\prime\prime}}$6 in 1997
(Dougados et al., 2000) and $\sim$2$\arcsec$ in 2014 (Takami et al., 2023).
The wiggling is corresponding to $\sim$4$-$6 degrees from the star on the sky
plane. The ejection directions both on the sky plane and with respect to the
line of sight vary within 1$\sigma$ level. The irregular knot structure may be
attributed to the variation in jet direction when the shock front in the knot
is not fully resolved in space.

##  5 Launching radii and mass accretion

Numerous models propose that the velocity of a directly driven outflow,
whether a jet, wind or another form, is roughly proportional to the Keplerian
velocity at its launching radius (Shu et al., 2000; Pudritz et al., 2007).
This suggests that high-velocity jets originate near the inner boundary of a
disk, while low-velocity winds emerge from a slightly more outer region of the
inner disk (Pyo et al., 2009; Frank et al., 2014).

To estimate the jet launching radius from $V_{f}$, we utilize the equation
given by White et al. (2014), operating under the assumption of a magneto-
centrifugally driven wind.

$(r_{\rm o}/0.1~{}AU)\simeq~{}[(109~{}{\rm
km\,s}^{-1}/{V}_{f})~{}(r_{A}/r_{\rm o})]^{2},$ (1)

where $r_{A}$ denotes the Alfvén radius, and $r_{\rm o}$ represents the
launching radius. We assumed parameters for the magnetic lever arm such that
$(r_{A}/r_{\rm o})^{2}=\lambda=$ 9\. This is based on the scenario where the
mass ejection rate from the outflow, $\dot{M}_{w}$, is approximately 10% of
the mass accretion rate, $\dot{M}_{a}$, i.e., $\dot{M}_{w}\simeq$
0.1$\dot{M}_{a}$ (Pudritz et al., 2007; Königl & Salmeron, 2011; Ellerbroek et
al., 2013). DG Tau has $\dot{M}_{w}\sim 0.07\dot{M}_{a}$ (Coffey, Bacciotti, &
Podio, 2008). We also used $M_{*}~{}=~{}$0.67 M☉ for the stellar mass of DG
Tau (Hartigan, Edwards, & Ghandour, 1995). The launching radii for the 12
identified knot groups, as detailed in Figure 7 and Table 2, fall within a
range of 0.06 to 0.45 AU. Coffey et al. (2007) reported similar radii (0.2-0.5
AU) for the higher velocity components estimated from the possible rotation
signatures in the jet, assuming steady magneto-centrifugal acceleration.

Following the ejection of knot C in 1937 at a radius of 0.27 $\pm$ 0.04 AU,
$r_{\rm o}$ progressed inward by over 0.2 AU between 1958 and 1999. Knot B,
ejected in 1958, had a launching point at $\sim$ 0.12 AU with a velocity of
297 $\pm$ 10 km s-1. From the late 1970s onward, ro neared the stellar
surface, shifting from 0.16 AU to a close 0.06 AU by mid-1995. This radius
equates to $\sim$ 5.5 R∗ when considering the stellar radius of DG Tau as 2.3
R☉ (Hartigan, Edwards, & Ghandour, 1995). This inward progression culminated
with knot A1 achieving a maximum jet velocity of 427 $\pm$ 16 km s-1. From
late 1999 to 2008, spanning approximately 10 years, $r_{\rm o}$ swiftly moved
outward again, reaching 0.45 AU. Since 2008, however, it has decreased back to
0.35 AU. The variation in the launching radius suggests unsteady mass
accretion for the last 80 years in the protoplanetary disk of DG Tau. The
launching radii of the jets approached the stellar surface in 1995 and then
expanded away, which may mean that the mass located in the radial range
between 0.06 AU (5.5 $R_{*}$) and 0.45 AU accreted onto the star during the
period.

Ferreira, Dougados, & Cabrit (2006) delineate distinct regions from which
extended disk winds, X-winds, and stellar winds originate. As illustrated in
Figure 7, these phenomena occupy specific radial zones: extended disk winds
manifest beyond $r_{\rm o}\gtrsim$ 0.07 AU, X-winds are situated between 0.05
AU $\lesssim r_{\rm o}\lesssim$ 0.07 AU, and stellar winds dominate the
innermost region, existing at $r_{\rm o}\lesssim$ 0.05 AU. The range of
$r_{\rm o}$ associated with X-winds is informed by Shu et al. (1994, 1995) and
Shang, Shu, & Glassgold (1998). Notably, the A1 knot was launched at 0.06 AU,
marking within the X-winds’ launching domain. Subsequently, the launching
radii retracted further from the star. This behavior might be linked to the
efficient disruption of accretion and contraction in young stars by the
phenomenon termed “Reconnection X-wind” (Ferreira, Dougados, & Cabrit, 2006).

From the fundamental energy and angular momentum conservation requirement, the
following equation (See Coffey et al., 2015, Eq. 4) is valuable in verifying
consistency:

$\frac{\dot{M}_{w}}{\dot{M}_{a}}\leq\frac{1}{4(\lambda-1)}\ln{\frac{r_{out}}{r_{in}}}$
(2)

where $\lambda$ is the magnetic lever arm parameter ($=(r_{A}/r_{\rm
o})^{2}$), $r_{out}$ is the maximum launching radius, $r_{in}$ is the minimum
launching radius. During 40 years, we can set $r_{out}=$ 0.45 AU and $r_{in}=$
0.06 AU in this paper. Agra-Amboage et al. (2011) summarized that the range of
the mass accretion rate ($\dot{M}_{a}$) is 1$-$5 $\times 10^{-7}M_{\sun}$ yr-1
and the range of the mass ejection rate ($\dot{M}_{w}$) falls within 2.2$-$4.4
$\times 10^{-8}M_{\sun}$ yr-1. It gives the ratio of mass ejection and
accretion rates of 0.04$-$0.44 for DG Tau. Applying these parameters implies
that $\lambda\leq$ 2.1$-$12.4 is consistent with our assumed $\lambda\sim$ 9.

##  6 Photometric variation and the flow velocities

In Figure 8, we plotted the photometric variation of DG Tau in the $V$, $\ub$,
$\bv$, and $\vr$ magnitude (left vertical axis) from 1983 to 2015 and the
deprojected flow velocities ($V_{f}$, right vertical axis) at their respective
ejection dates. The photometric data from Grankin et al. (2007) from 1983 to
2003 is augmented by incorporating AAVSO $V$ magnitude observations from 2009
to 2015, predominantly undertaken by James McMath and Kenneth Menzies until
March 2015 (Kafka, 2015). We computed average magnitudes over ten rotational
periods to exclusively discern long-term photometric variations and negate
rotation-induced signals. DG Tau’s rotational period is 6.3 days, so each data
point reflects an average magnitude spanning 63 days.

The local maximum of the observed brightness rose from a magnitude of 12.1 to
11.1 in the $V$-band between 1985 and 1997, mirroring the rise in flow
velocity. Between 1997 and 1998, it experienced a sharp decline, remaining
dimmer than 12.4 in magnitude from 1998 to 2015. Overall, the photometric
variation in $V$-band aligns well with the changes in flow velocity. A notable
exception may be the kHVC in 1999, which followed the most pronounced dip in
brightness in 1988. The $\bv$, and $\vr$ color variations align also with the
changes in the flow velocity while $\ub$ shows sharp decline (redder) in 1992
between the ejections of KHY1 and A1 knots.

Grankin et al. (2007) emphasized that DG Tau displays significant fluctuations
in its average brightness over the years, beyond the abrupt seasonal
variations. These pronounced fluctuations hint at considerable changes either
in the accretion rate from the disk to the star or in the circumstellar
extinction across multiple years. As previously discussed, a higher outflow
velocity indicates that the outflow stems from regions nearer to the stellar
surface. The strong correlation between DG Tau’s brightness and outflow
velocity, as depicted in Figure 8, indicates that the brightness surge in the
1990s is more likely due to an increase in the accretion rate rather than a
reduction in circumstellar extinction. Assuming constant mass density and gas
volume, an increase in flow velocity suggests a corresponding increase in mass
ejection rate. The higher mass ejection rate must be related to the higher
mass accretion rate because they are proportional to each other as shown in
the previous section, i.e. $\dot{M}_{w}/\dot{M}_{a}=0.04-0.44$.

The rise in Br$\gamma$ emission-line luminosity during the 1990s further
suggests an increase in the mass accretion rate. Muzerolle, Hartmann, & Carvet
(1998) demonstrated that the luminosities of Br$\gamma$ emission lines are
closely correlated with the accretion luminosity ($L_{acc}$). They noted that
in January 1998, when DG Tau’s brightness was around $V$ $\sim$ 11.7 mag,
which was slightly dimmer than the brightest peak at $V$ $\sim$ 11.4 mag in
October 1995, the equivalent width of the Br$\gamma$ emission, W(Br$\gamma$),
was 14.1 Å. This corresponds to a mass accretion rate of $\dot{M}_{a}=$ 7.9
$\times$ 10-7 M☉ yr-1. Coffey, Bacciotti, & Podio (2008) estimated the mass
ejection rate of 1.3 $\times$ 10-7 M☉ yr-1 with the data obtained by Bacciotti
et al. (2000) in 1999 when knot A1 ejected in 1995, was located within
1$\arcsec$ from the star. For 1995$-$1999, the mass ejection and accretion
rates marked the highest points. The ratio of $\dot{M}_{w}/\dot{M}_{a}$ was
$\sim 0.2$. However, in October 2005, following a dimming of DG Tau about one
magnitude to $V$ $\sim$ 12.5$-$13.0 mag, Agra-Amboage et al. (2011) reported a
reduced equivalent width of W(Br$\gamma$) = 5.6 Å, indicating a lower mass
accretion rate of $\dot{M}_{a}=$ 1 $\times$ 10-7 M☉ yr-1. Correspondingly,
they estimated a decreased mass ejection rate of 3.3 $\pm$ 1.1 $\times$ 10-8
M☉ yr-1 using [Fe II]emission. These rates were notably smaller compared to
those observed in 1998. The ratio $\dot{M}_{w}/\dot{M}_{a}$ was 0.2$-$0.4.
Beck, Bary, & McGregor (2010) observed a similar mass accretion rate, 9.6
$\times$ 10-8 M☉ yr-1, and estimated a lower limit for the mass ejection rate
at 1.2 $\times$ 10-8 M☉ yr-1. Consequently, the ratio
$\dot{M}_{w}/\dot{M}_{a}$ was inferred to be $>$ 0.13.

The variation in DG Tau’s photometric color further indicates that its
increased brightness can be attributed to heightened accretion. The color-
magnitude diagrams depicted in Figure 9 illustrate the variations of the
colors $\ub$, $\bv$, and $\vr$ in relation to the photometric variation, based
on the data from Grankin et al. (2007). The two dotted vertical lines
represent the intrinsic color of DG Tau, falling between K5 and M0, as
inferred from previously published spectral types: K5 (Hartigan, Edwards, &
Ghandour, 1995), K8.5 (Hamann & Persson, 1992), and M0 (Herbst et al., 1994).
The observed $\ub$ color indices are noticeably bluer than the expected
intrinsic color range. Furthermore, the $\bv$ color indices are bluer than the
intrinsic color when the star is at a brighter magnitude (11.2 $<V<$ 12.5),
whereas they align with the intrinsic color when the star’s brightness
approaches its minimum magnitude (12.5 $<V<$ 13.7). The large U B excess in
the left panel of Figure 9 should be related to accretion from the inner edge
of the disk onto the stellar surface because the $U$ band luminosity is
proportional to the accretion luminosity (Calvet et al., 2000). Vrba et al.
(1993) showed that CTTSs typically have intrinsic colors of the normal
photospheres at the minimum brightness and bluer colors at maximum brightness,
regardless of whether the variability is periodic. They pointed out that
accretion-driven hot spots are the predominant cause of increasing brightness.

The color indices of V R for DG Tau are notably redder at minimum brightness.
This pronounced redness in V R can be ascribed to circumstellar extinction,
analogous to the $V-I$ observed in the case of RW Aur (Petrov & Kozack, 2007).
Both the $\bv$ and V R color-magnitude diagrams align best with a ratio of
absolute to selective extinction $R_{\rm v}$ of approximately 6.9, as
indicated by the arrow in the middle panel of Figure 9. This value surpasses
the standard interstellar extinction value of $R_{\rm v}=$ 3.1. A larger
$R_{\rm v}$ may suggest the presence of larger grain sizes (Cardelli, Clayton,
& Mathis, 1989). Isella et al. (2010) proposed that dust particles around DG
Tau underwent reprocessing and grew in size, potentially reaching radii of at
least 20 $\mu$m.

To quantitatively assess the hot and cold spots impact on the brightness, we
calculated the magnitude difference, $\Delta m(\lambda)$, attributable to cool
and hot spots, using formulas derived from Carpenter et al. (2001) and Vrba et
al. (1993):

$\Delta
m(\lambda)=-2.5\log\\{1-f[1.0-B_{\lambda}(T_{spot})/B_{\lambda}(T_{\ast})]\\}$
(3)

Here $f$ represents the fraction of the stellar photosphere covered by spots,
$T_{spot}$ and $T_{\ast}$ denote the effective temperatures of the spot and
star, respectively, and $B_{\lambda}(T)$ is the Planck function. For DG Tau,
which has a spectral type ranging from K5 to M0, we selected $T_{\ast}=4000$
K, corresponding to temperatures between 3800 K and 4340 K (Hartigan, Edwards,
& Ghandour, 1995; Hamann & Persson, 1992; Herbst et al., 1994). Figure 10
illustrates the magnitude and color variations across the $UBVR$ bands. For
cool spots, we consider $T_{spot}$ values of 2000 K and 2500 K. Meanwhile, for
hot spots, $T_{spot}$ is taken as 8000 K and 12000 K. These variations are
plotted for spot coverage fractions ($f$) of 0.01, 0.05, 0.1, 0.15, 0.2, and
0.3.

For the typical values of $T_{\ast}-T_{spot}\leq$ 2000 K and $f\leq$ 30 % for
cool spots, the magnitude and color variations are $\lesssim$ 0.4 mag and
$\lesssim$ 0.009 mag, respectively. As the coverage fraction of the cool spot
increases, the star’s magnitude becomes fainter and its color shifts to a
redder value. With greater coverage of the stellar surface by cool spots, the
shift in color becomes more pronounced at longer wavelengths:
$\Delta(\ub)<\Delta(\bv)<\Delta(\vr)$.

For typical parameters, with $T_{spot}\leq 8000$ K and $f\leq 10\%$ for hot
spots, the variations in magnitude and color are $\Delta V\lesssim$ 1.5 mag
and $\Delta(\bv)\lesssim$ 0.7, respectively. More significant variations, such
as $\Delta V\geq$ 3 mag and $\Delta(\bv)\geq$ 1 mag, can be expected if the
hot spot has a high temperature ($T_{spot}=12000$ K) and covers a more
substantial portion of the stellar surface, with $f\geq 20\%$. When hot spots
maintain a consistent temperature, $T_{spot}$, and occupy a more extensive
region on the stellar surface, the star appears brighter and exhibits a bluer
color. The variation in magnitude becomes more pronounced with decreasing
wavelength: $\Delta R<\Delta V<\Delta B<\Delta U$. Similarly, color variations
follow the trend: $\Delta$($\vr$) $<$ $\Delta$($\bv$) $<$ $\Delta$($\ub$).

Figure 11 displays the variations $\Delta V$ and $\Delta$($\bv$) caused by
both cool and hot spots. While both types of spots occupy the positive $\Delta
V/\Delta(\bv)$ space, they differ in their variation vector directions. From
this figure, it is evident that the cool spot primarily accounts for the minor
magnitude and color variation found in the upper right corner of the $\Delta
V$ and $\Delta$($\bv$) space. As the value of $f$ increases, the rate of color
change diminishes, even though the $V$ magnitude undergoes rapid alterations
at a constant $T_{spot}$. The dash-dotted lines link the data points across
various $T_{spot}$ values but maintain a consistent $f$ value. Should
$T_{spot}$ undergo changes without any alterations in spot coverage, the
$\Delta V$ and $\Delta(\bv)$ would shift along these dash-dotted lines. We,
therefore, deduce that the observed rise in the DG Tau’s $V$ magnitude in the
1990s can be attributed to an increase in the coverage fraction of hot spots,
in line with the elevated accretion rate, rather than a decrease in the
coverage fraction of cold spots.

Figure 12a displays a plot comparing the relative magnitude $dV_{rel}$ with
the color $d(\bv)_{rel}$, based on the averaged data points from Figure 8.
These relative values are determined by the difference in magnitude and color
relative to their preceding data points. Data positioned in the lower left and
upper right quadrants yield a positive value for $dV_{rel}/d(\bv)_{rel}$,
which suggests that they are either becoming brighter and bluer or fainter and
redder compared to their preceding data points.

Increased disk mass accretion typically results in a trend toward brighter and
bluer values, whereas a reduction in accretion leads to the converse trend, as
pointed out by (Carpenter et al., 2001). Variations in brightness, often
spanning several magnitudes and exhibiting prolonged, irregular fluctuations,
can arise from extinction. This effect occurs when a clump of circumstellar
gas and dust obstructs the line of sight, making an object appear both fainter
and redder. Both scenarios—brightening due to mass accretion characterized by
hot spots with a varying $f$ but consistent $T_{spot}$, and dimming from
extinction effects—can explain the observed behavior in the two aforementioned
quadrants.

Data in the lower right and upper left quadrants display a negative value for
$dV_{rel}/d(\bv)_{rel}$, suggesting they either became brighter yet redder or
dimmed while turning bluer relative to the previous measurements. This pattern
does not align with typical extinction or mass accretion models. For hot
spots, such a trend can emerge if the temperature $T_{spot}$ decreases while
its coverage fraction $f$ increases (e.g., transitioning from $T_{spot}=$
12000 K to 8000 K, alongside an $f$ shift from 0.01 to 0.06, yields a brighter
and redder outcome). Conversely, an increase in $T_{spot}$ coupled with a
decrease in its coverage (e.g., moving from $T_{spot}=$ 8000 K to 12000 K, and
$f$ from 0.06 to 0.01) produces a fainter, bluer look. Hence, the observed
negative shift in $dV_{rel}/d(\bv)_{rel}$ underscores active mass accretion
episodes marked by changes in both hot spot temperature and extent. Figure 12b
plots $dV_{rel}$ and $d(\bv)_{rel}$ along the observation dates. It shows all
nine fainter and bluer data are located from 1984 to 1995 when the brightness
of DG Tau was generally increasing with some fluctuation.

As discussed, the robust correlation between the variations in photometric
magnitudes and flow velocities points to the predominant influence of mass
accretion accompanied by hot spots. On the other hand, the observed reddening
in the $\vr$ color for $V$ $\geq$ 12.5 suggests that DG Tau underwent
significant extinction after its sharp drop in brightness from 11.7 mag to
13.1 mag in the $V$ magnitude in 1998. This rapid dimming of V magnitude may
be the result of the star being obscured by a dust barrier caused by the
development of extensive disk winds. As shown in Figure 8, when knot kHVC was
ejected at 1999.48, its brightness measured approximately $V\sim 12.6$ mag.
This value is notably dimmer than the anticipated $V\sim 12$ mag, based on the
apparent correlation observed in other knots between brightness and flow
velocity. At that time, both kHVC and kLVC were ejected nearly concurrently at
0.11 $\pm$ 0.01 AU and 0.4 $\pm$ 0.1 AU, respectively. Pyo et al. (2003)
postulated that the broad velocity range of kLVC was indicative of a disk wind
originating over an extensive radius. Between 1990 and 2002, a low-velocity
component with a broad velocity range was consistently detected in the PVDs of
these knots (Solf & Bohm, 1993; Lavalley et al., 1997; Takami et al., 2002;
Pyo et al., 2003; Maurri et al., 2014). Although these components are not
counted among the 12 knots because of their diffuse emissions and undetermined
proper motion, they may represent a pair, consisting of a high-velocity jet
from the disk’s inner edge and a disk wind covering a significant part of the
outer disk. This expansive disk wind might have led to the formation of a dust
wall, inducing more pronounced extinction than anticipated, akin to the
phenomenon detailed by Ellerbroek et al. (2014).

##  7 Summary and Conclusion

We have compiled an extensive dataset of the blueshifted jet knots from DG Tau
spanning the last 40 years. By doing so, we were able to trace the mass
ejection event back to 1937. This dataset has been compared with the recorded
photometric magnitude and color variations from 1983 to 2015.

The key findings can be summarized as follows:

  1. 1.

Our analysis of [Fe II] data from 2007 does not reveal any distinct higher
velocity component exceeding -200 km s-1. This stands in stark contrast to the
findings presented in Pyo et al. (2003). We found that a high-velocity flow,
surpassing $|V_{r}|$ $>$ 200 km s-1, was consistently detected from 1988 to
2005. However, its intensity notably diminished after 2006.

  2. 2.

We categorized 51 observed knots spanning 40 years along DG Tau’s blueshifted
jet, grouping them into 17 distinct knot groups. This classification was
established under the assumption of ballistic motion, relying on consistent
proper motions and comparable radial velocities. Our analysis focused on 12
moving knot groups, excluding five single-observation knots. Leveraging their
proper motions and radial velocities, we derived estimations for ejection
dates, inclination angles, deprojected flow velocities, and the launching
radii of these knot groups.

  3. 3.

We discovered a clear correlation between the deprojected flow velocities and
the photometric magnitudes throughout the ejection dates. Initially, the
deprojected flow velocity ranged between 200 and 300 km s-1from 1937 to 1983.
However, by the 1990s, the velocity surged beyond 300 km s-1, peaking at over
420 km s-1. This increase was followed by a steep decline to below 200 km
s-1between 1996 and 1999, maintaining lower velocities throughout the 2010s.
The variation in photometric magnitudes corresponded closely to these velocity
changes from 1983 to 2015. The local maximum of the observed $V$ magnitude
ascended from 12.1 to 11.1 between 1985 and 1997, mirroring the rise in the
velocity during that period. However, between 1997 and 1998, a sharp decline
in magnitude occurred, persisting dimmer than 12.4 from 1998 to 2015. Notably,
the mass accretion rate around 1998 was approximately eight times higher than
that in 2005, while the ratio of $\dot{M}_{w}/\dot{M}_{a}$ remained nearly
constant at about 0.2. These variations in magnitude predominantly stem from
active mass accretion, coupled with hot spot activities on the stellar
surface.

  4. 4.

The estimation of launching radii was derived from the deprojected velocity
utilizing the magneto-centrifugal wind model. The launching radius of the jets
varied between 0.06 and 0.45 AU. Across 58 years, spanning from 1937 to 1995,
the launching radius gradually reduced from 0.27 AU to 0.06 AU, reaching the
”X-winds” launching region by 1995. However, within a decade, by 2008, it
swiftly retracted to a larger distance.

  5. 5.

Between 1997 and 1999, when the kHVC ejection occurred, DG Tau displayed a
significantly fainter magnitude ($V$ $\sim$ 12.6 mag) compared to anticipated
levels ($V$ $\sim$ 12.0 mag). This notable dimming likely resulted from
substantial extinction caused by a dense dust wall, aligning with the sharp
decrease in brightness observed during that specific time frame.

  6. 6.

Several knots appear transiently, lasting only within a 2$\arcsec$ span. Thus,
ejection events occur more frequently than the intervals between these knots
indicate.

We are grateful for the valuable input provided by the anonymous referee,
which significantly contributed to enhancing the quality of this paper. We
express our sincere appreciation to the global community of observers for
their contributions to the AAVSO International Database. The variable star
observations they provided were instrumental in this study. Additionally, our
research was enriched by the resources of NASA’s Astrophysics Data System and
the SIMBAD database, maintained by CDS in Strasbourg, France.

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Table 1: Distances and radial velocities of knots

Obs year | D | C | B | A-B2 | B1 | B0 | KHY1 | Reference
---|---|---|---|---|---|---|---|---
(yr) | ($\arcsec$) | (km s-1) | ($\arcsec$) | (km s-1) | ($\arcsec$) | (km s-1) | ($\arcsec$) | (km s-1) | ($\arcsec$) | (km s-1) | ($\arcsec$) | (km s-1) | ($\arcsec$) | (km s-1) |
1984.77 |  $\cdots$ | $\cdots$ | 8.50 | -136 |  $\cdots$ | $\cdots$ |  $\cdots$ | $\cdots$ |  $\cdots$ | $\cdots$ |  $\cdots$ | $\cdots$ |  $\cdots$ | $\cdots$ | 1
1986.99 | 10.70 | $\cdots$ | 8.70 | -156 | 6.60 | -255 | 2.50 | -161 |  $\cdots$ | $\cdots$ |  $\cdots$ | $\cdots$ |  $\cdots$ | $\cdots$ | 2
1991.23 |  $\cdots$ | $\cdots$ |  $\cdots$ | $\cdots$ |  $\cdots$ | $\cdots$ |  $\cdots$ | $\cdots$ |  $\cdots$ | $\cdots$ |  $\cdots$ | $\cdots$ | 0.25 | $\cdots$ | 3
1992.62 |  $\cdots$ | $\cdots$ |  $\cdots$ | $\cdots$ |  $\cdots$ | $\cdots$ | 3.25 | -200 | 2.25 | -255 |  $\cdots$ | $\cdots$ | 0.52 | -350 (a) | 4
1994.84 |  $\cdots$ | $\cdots$ |  $\cdots$ | $\cdots$ |  $\cdots$ | $\cdots$ | 3.90 | $\cdots$ | 2.70 | -210 |  $\cdots$ | $\cdots$ | 1.00 | -320 (b) | 5
1997.04 |  $\cdots$ | $\cdots$ |  $\cdots$ | $\cdots$ |  $\cdots$ | $\cdots$ |  $\cdots$ | $\cdots$ | 3.30 | -235 |  $\cdots$ | $\cdots$ |  $\cdots$ | $\cdots$ | 6
1997.99 |  $\cdots$ | $\cdots$ |  $\cdots$ | $\cdots$ |  $\cdots$ | $\cdots$ |  $\cdots$ | $\cdots$ | 3.40 | $\cdots$ |  $\cdots$ | $\cdots$ |  $\cdots$ | $\cdots$ | 6
1998.07 |  $\cdots$ | $\cdots$ |  $\cdots$ | $\cdots$ |  $\cdots$ | $\cdots$ |  $\cdots$ | $\cdots$ | 3.60 | -235 |  $\cdots$ | $\cdots$ |  $\cdots$ | $\cdots$ | 7
1998.10 |  $\cdots$ | $\cdots$ | 10.65 | $\cdots$ |  $\cdots$ | $\cdots$ |  $\cdots$ | $\cdots$ |  $\cdots$ | $\cdots$ |  $\cdots$ | $\cdots$ |  $\cdots$ | $\cdots$ | 8
1999.04 |  $\cdots$ | $\cdots$ |  $\cdots$ | $\cdots$ |  $\cdots$ | $\cdots$ |  $\cdots$ | $\cdots$ | 3.80 | -210 | 3.30 | -255 |  $\cdots$ | $\cdots$ | 9, 10
2002.71 |  $\cdots$ | $\cdots$ | 12.00 | -175 |  $\cdots$ | $\cdots$ |  $\cdots$ | $\cdots$ | 4.60 | -220 (c) |  $\cdots$ | $\cdots$ |  $\cdots$ | $\cdots$ | 13
2012.87 |  $\cdots$ | $\cdots$ | 13.70 | -172 |  $\cdots$ | $\cdots$ | 8.75 | -220 | 6.74 | -200 |  $\cdots$ | $\cdots$ |  $\cdots$ | $\cdots$ | 8
2014.89 |  $\cdots$ | $\cdots$ | 13.50 | $\cdots$ |  $\cdots$ | $\cdots$ |  $\cdots$ | $\cdots$ | 7.65 | $\cdots$ |  $\cdots$ | $\cdots$ |  $\cdots$ | $\cdots$ | 8

Note. — (a) knot C in the Solf & Bohm (1993), (b) an extension of the HVB of
Lavalley et al. (1997), (c) knot B in the Whelan, Ray, & Davis (2004).

References. — (1) Mundt et al. (1987), (2) Eislöffel & Mundt (1998), (3)
Kepner et al. (1993), (4) Solf & Bohm (1993), (5) Lavalley et al. (1997), (6)
Dougados et al. (2000), (7) Lavalley-Fouquet, Cabrit, & Dougados (2000), (8)
Oh et al. (2015), (9)Bacciotti et al. (2000), (10) Maurri et al. (2014), (11)
Takami et al. (2002), (12) Pyo et al. (2003), (13) Whelan, Ray, & Davis
(2004), (14) White et al. (2014), (15)Agra-Amboage et al. (2011), (16) This
paper, (17) Takami et al. (2023)

Table 1: Continued

Obs year | A1 | A2 | kHVC | kLVC | WC | WB | PB | Reference
---|---|---|---|---|---|---|---|---
(yr) | ($\arcsec$) | (km s-1) | ($\arcsec$) | (km s-1) | ($\arcsec$) | (km s-1) | ($\arcsec$) | (km s-1) | ($\arcsec$) | (km s-1) | ($\arcsec$) | (km s-1) | ($\arcsec$) | (km s-1) |
1997.04 | 0.60 | -350 |  $\cdots$ | $\cdots$ |  $\cdots$ | $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ | $\cdots$ |  $\cdots$ | $\cdots$ |  $\cdots$ | $\cdots$ | 6
1997.99 | 1.00 | $\cdots$ |  $\cdots$ | $\cdots$ |  $\cdots$ | $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ | $\cdots$ |  $\cdots$ | $\cdots$ |  $\cdots$ | $\cdots$ | 6
1998.07 | 0.93 | -350 |  $\cdots$ | $\cdots$ |  $\cdots$ | $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ | $\cdots$ |  $\cdots$ | $\cdots$ |  $\cdots$ | $\cdots$ | 7
1999.04 | 1.35 | -340 | 0.45 | -230 |  $\cdots$ | $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ | $\cdots$ |  $\cdots$ | $\cdots$ |  $\cdots$ | $\cdots$ | 9,10
2000.95 |  $\cdots$ | $\cdots$ |  $\cdots$ | $\cdots$ | 0.45 | -220 | 0.23 | -100 |  $\cdots$ | $\cdots$ |  $\cdots$ | $\cdots$ |  $\cdots$ | $\cdots$ | 11
2001.83 |  $\cdots$ | $\cdots$ |  $\cdots$ | $\cdots$ | 0.75 | -226 | 0.45 | -100 |  $\cdots$ | $\cdots$ |  $\cdots$ | $\cdots$ |  $\cdots$ | $\cdots$ | 12
2002.71 |  $\cdots$ | $\cdots$ |  $\cdots$ | $\cdots$ | 1.0 | -266 | 0.60 | -47 |  $\cdots$ | $\cdots$ |  $\cdots$ | $\cdots$ |  $\cdots$ | $\cdots$ | 13
2005.87 |  $\cdots$ | $\cdots$ |  $\cdots$ | $\cdots$ |  $\cdots$ | $\cdots$ |  $\cdots$ |  $\cdots$ | 1.24 | -250 (d) | 0.40 | -180 (e) |  $\cdots$ | $\cdots$ | 14, 15
2006.98 |  $\cdots$ | $\cdots$ |  $\cdots$ | $\cdots$ |  $\cdots$ | $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ | $\cdots$ | 0.61 | $\cdots$ |  $\cdots$ | $\cdots$ | 14
2007.12 |  $\cdots$ | $\cdots$ |  $\cdots$ | $\cdots$ |  $\cdots$ | $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ | $\cdots$ | 0.72 | -154 (f) | 0.30 | -154 | 16
2009.88 |  $\cdots$ | $\cdots$ |  $\cdots$ | $\cdots$ |  $\cdots$ | $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ | $\cdots$ | 1.07 | $\cdots$ |  $\cdots$ | $\cdots$ | 14

Note. — (d) and (e) knots A4 and A5 of Agra-Amboage et al. (2011)
respectively, (f) knot PC of this paper.

Table 1: Continued  Obs year | TC | TD | TE | Reference
---|---|---|---|---
(yr) | ($\arcsec$) | (km s-1) | ($\arcsec$) | (km s-1) | ($\arcsec$) | (km s-1) |
2014.16 | 0.53 | -142 |  $\cdots$ | $\cdots$ |  $\cdots$ | $\cdots$ | 17
2014.99 | 0.61 | -134 | 0.31 | -120 |  $\cdots$ | $\cdots$ | 17
2017.13 |  $\cdots$ | $\cdots$ | 0.51 | -152 |  $\cdots$ | $\cdots$ | 17
2017.98 |  $\cdots$ | $\cdots$ | 0.69 | -134 |  $\cdots$ | $\cdots$ | 17
2018.91 |  $\cdots$ | $\cdots$ | 0.82 | -136 | 0.28 | -141 | 17
2019.79 |  $\cdots$ | $\cdots$ | 0.94 | -140 | 0.38 | -155 | 17

Table 2: Proper motions, velocities, inclination angles, and launching radii

Knot Group | Proper motion |  $|\bar{V}_{r}|$ (a) |  $V_{t}$ (b) |  $V_{f}$ (c) |  $I$ (d) | Launching radius ($r_{o}$) | Ejection Date
---|---|---|---|---|---|---|---
| ($\arcsec$ yr-1) | (km s-1) | (km s-1) | (km s-1) | (∘) | (AU) | (yr)
C | 0.18 $\pm$ 0.01 | 160 $\pm$ 16 | 120 $\pm$ 9 | 200 $\pm$ 14 | 36.9 $\pm$ 5.3 | 0.27 $\pm$ 0.04 | 1937.70 $\pm$ 1.45
B | 0.23 $\pm$ 0.02(e) | 255 $\pm$ 8 | 153 $\pm$ 14 | 297 $\pm$ 10 | 31.0 $\pm$ 3.3 | 0.12 $\pm$ 0.01 | 1958.17 $\pm$ 2.18
A-B2 | 0.25 $\pm$ 0.02 | 194 $\pm$ 24 | 168 $\pm$ 16 | 256 $\pm$ 21 | 40.9 $\pm$ 7.7 | 0.16 $\pm$ 0.03 | 1978.50 $\pm$ 1.21
B1 | 0.23 $\pm$ 0.01 | 224 $\pm$ 18 | 156 $\pm$ 9 | 273 $\pm$ 15 | 35.0 $\pm$ 3.9 | 0.14 $\pm$ 0.02 | 1983.06 $\pm$ 0.52
KHY1 | 0.21 $\pm$ 0.01 | 335 $\pm$ 15 | 139 $\pm$ 8 | 363 $\pm$ 14 | 22.6 $\pm$ 1.8 | 0.08 $\pm$ 0.01 | 1990.07 $\pm$ 0.11
A1 | 0.37 $\pm$ 0.03 | 347 $\pm$ 5 | 249 $\pm$ 26 | 427 $\pm$ 16 | 35.7 $\pm$ 4.4 | 0.06 $\pm$ 0.00 | 1995.44 $\pm$ 0.13
kHVC | 0.31 $\pm$ 0.02 | 237 $\pm$ 20 | 209 $\pm$ 15 | 316 $\pm$ 18 | 41.4 $\pm$ 5.7 | 0.11 $\pm$ 0.01 | 1999.48 $\pm$ 0.05
kLVC | 0.21 $\pm$ 0.02 | 82 $\pm$ 25 | 141 $\pm$ 17 | 163 $\pm$ 19 | 59.6 $\pm$ 29.2 | 0.40 $\pm$ 0.10 | 1999.80 $\pm$ 0.10
WB | 0.16 $\pm$ 0.02 | 167 $\pm$ 13 | 108 $\pm$ 15 | 199 $\pm$ 14 | 32.8 $\pm$ 6.0 | 0.27 $\pm$ 0.04 | 2003.12 $\pm$ 0.32
TC | 0.10 $\pm$ 0.02 | 138 $\pm$ 4 | 68 $\pm$ 14 | 154 $\pm$ 7 | 26.1 $\pm$ 5.8 | 0.45 $\pm$ 0.04 | 2008.97 $\pm$ 0.02
TD | 0.14 $\pm$ 0.01 | 136 $\pm$ 10 | 90 $\pm$ 8 | 164 $\pm$ 10 | 33.5 $\pm$ 4.5 | 0.40 $\pm$ 0.05 | 2012.93 $\pm$ 0.24
TE | 0.11 $\pm$ 0.02 | 148 $\pm$ 7 | 77 $\pm$ 14 | 167 $\pm$ 9 | 27.4 $\pm$ 5.6 | 0.38 $\pm$ 0.04 | 2016.46 $\pm$ 0.02
Average | 0.21 $\pm$ 0.08 | 207 $\pm$ 74 | 140 $\pm$ 53 | 248 $\pm$ 90 | 35.6 $\pm$ 10.9 | 0.24 $\pm$ 0.15 | $\cdots$

Note. — (a) Average radial velocity for each knot group. (b) Tangential
velocity for each knot group calculated under the assumed distance of 141
$\pm$ 7 pc to TMC, taken from the GAIA DR2 edition (Zucker et al., 2019). (c)
deprojected flow velocity $V_{f}=\sqrt{{\bar{V}_{r}}^{2}+{V_{t}}^{2}}$. (d)
Inclination angle with respect to the line of sight,
$I~{}=\arctan(V_{t}/|\bar{V}_{r}|)$. (e) from Eislöffel & Mundt (1998).

Figure 1: (left)The image on the left shows the [Fe II] $\lambda$1.257 $\mu$m
emission after removing the continuum. The three knots along the jet, PA, PB,
and PC, are marked and labeled. The field of view is $2\farcs 6\times 2\farcs
6$, and the central star, DG Tau, is located at (X,Y)=(0,0). The vertical and
horizontal dash-dotted lines indicate the star position. The emission line was
integrated over $\pm$ 600 km s-1. The contour levels are 10$\sigma$,
20$\sigma$, 30$\sigma$, 41$\sigma$, 53$\sigma$, 66$\sigma$, and 81$\sigma$
above the sky level where $\sigma$ is the standard deviation of the sky area.
The upward direction (Y $\geq$ 0$\arcsec$) indicates the direction of the
blue-shifted jet with a position angle (PA) of 222∘. The red-shifted jet is
moving in the opposite direction (Y $\leq$ 0$\arcsec$) from the blue-shifted
jet, leaving a gap of about 1 arc second at the star position. The gap is the
part obscured by the disk (Pyo et al., 2003). (right)The position-velocity
diagram on the right shows the same emission. The systemic radial velocity is
$V_{\rm LSR}=+$6.4 km s-1 (Kimura, Kawabe, & Saito, 1996). The vertical dash-
dotted line marks V${}_{LSR}=$ 0 km s-1. The peak velocity along the jet is
indicated with the dotted line. The contour levels are 10$\sigma$, 15$\sigma$,
21$\sigma$, 27$\sigma$, 35$\sigma$, 43$\sigma$ and 52$\sigma$ above the
background level where the $\sigma$ is the standard deviation of the
background. It was observed using ALTAIR with NIFS/Gemini on February 13, 2007
(UT). The spatial resolution was $0\farcs 14$, and the spectral resolution was
60 km s-1 ($R~{}\sim$ 5000).

Figure 2: Observed knots along the DG Tau jet from 1984 to 2019. The dotted
lines indicate the proper motions of the 12 knot groups.

Figure 3: Absolute radial velocities ($|V_{r}|$) of the knots along the DG Tau
jet. The crowded area within 2$\arcsec$ is zoomed (left side) to show the
knots in detail. The dotted line shows the deceleration trend ( $-$4.8 km s-1
arcsec-1) of $|V_{r}|$ for the knots located further than 2$\arcsec$ away from
the star.

Figure 4: Histogram displaying the radial velocities ($|V_{r}|$) of the knots
in the DG Tau jet. The average velocity is 199 $\pm$ 73 km s-1 denoted by a
vertical dash-dotted line. Twenty-one knots ($\sim$ 51%) exhibit velocities
within the range of 150 - 250 km s-1.

Figure 5: Change of various parameters with the ejection date (yr) of knots.
The parameters are: (a) Proper motions, (b) Radial ($|\bar{V}_{r}|$;
$\blacksquare$ with dotted line) and tangential ($V_{t}$; $\bullet$ with solid
line) velocities, (c) deprojected flow velocities
($V_{f}=\sqrt{|\bar{V}_{r}|^{2}+V_{t}^{2}}$), and (d) Inclination angle ($I$).
The dash-dotted line shows the average inclination angle of 35$\hbox
to0.0pt{.\hss}^{\circ}$6\.

Figure 6: Variation of knot velocities on the on the $|\bar{V}_{r}|$ vs.
$V_{t}$ plane. The earliest knot in 1937 is depicted in black, while the
latest one in 2016 is shown in gray. Additionally, the connection line
progressively thins out from 1937 to 2016. The dash-dotted line indicates the
average inclination angle of $\bar{I}~{}=$ 35$\hbox to0.0pt{.\hss}^{\circ}$6\.

Figure 7:  Variation of knot launching radii ($r_{\rm o}$; AU) from 1937 until
2016. The launching radii of the stationary MHD jet models are indicated:
stellar winds($r_{\rm o}\lesssim$ 0.05 AU), X-winds (gray-shaded area;
0.05$\lesssim r_{\rm o}\lesssim$0.07), and extended disc winds (0.07 $\lesssim
r_{\rm o}$) (Ferreira, Dougados, & Cabrit, 2006). The range of the X-winds
launching radii is marked from 0.05 AU to 0.07 AU based on Shu et al. (1994,
1995); Shang, Shu, & Glassgold (1998).

Figure 8:  Comparison of DG Tau’s (a) $V$, (b) $\ub$, (c) $\bv$, and (d) $\vr$
magnitudes (left axis)variation with deprojected knot velocities
($V_{f}$)(right axis). Opened circles, filled triangles, filled squares, and
filled circles denote $V$, $\ub$, $\bv$, and $\vr$ magnitudes averaged over 10
rotational periods, 63 days, by Grankin et al. (2007) before 2005. The $V$
magnitudes after 2009 (gray color) are from the AAVSO database (Kafka, 2015).
The circles filled with red and blue colors in the $V$ data represent points
that are brighter and redder or fainter and bluer compared to the immediately
preceding data points (See Figure 12). Crosses depict $V_{f}$ with the error
bars showing standard deviations.

Figure 9: Color-magnitude diagrams (CMD) of DG Tau. (left) U B vs $V$,
(middle) (B V) vs $V$, (right) V R vs $V$. The data was taken from Grankin et
al. (2007). The vertical dotted lines in each panel indicate the intrinsic
color of DG Tau for its spectral type of K5 (Hartigan, Edwards, & Ghandour,
1995) and M0 (Herbst et al., 1994). The luminosity classes of CTTSs are
between giants (III) and dwarfs (V) (Greene & Lada, 1996). The arrows in the
middle panel show the direction of extinction with $R_{\rm v}=$ 3.1 and 6.9
where $R_{\rm v}=~{}A_{\rm V}$ / $E$(B V).  Figure 10: The variations of the
colors ($\Delta$($\ub$), $\Delta$($\bv$), $\Delta$($\vr$)) and magnitudes
($\Delta U$, $\Delta B$, $\Delta V$, $\Delta R$) by cool and hot spots model
(Carpenter et al., 2001; Vrba et al., 1993). Panels (a) and (c) display color
variations, while panels (b) and (d) depict magnitude variations resulting
from cool and hot spots, respectively. The spots’ coverage fractions ($f$)
range from 0.01 to 0.3. The temperatures of the cool and hot spots are
indicated by gray dash-dotted lines(2000 K), blue solid lines (2500 K), yellow
dotted lines(8000 K), and red dashed lines (12000 K). The star’s effective
temperature is $T_{\ast}$ = 4000 K. In panel (b), it is noteworthy that the
$\Delta m$ of cool spots at $T_{\text{spot}}=2000$ K and 2500 K exhibit
minimal differences due to the narrow (500 K) temperature range.

Figure 11: Variation of $\Delta V$ and $\Delta$($\bv$) for cool and hot spot
models (Carpenter et al., 2001; Vrba et al., 1993). The gray and blue filled
squares in the upper right corner ($\Delta V\geq 0$ and $\Delta$($\bv$) $\geq
0$) represent cool spots with $T_{spot}=$ 2000 K (gray dash-dotted line) and
2500 K (blue solid line). The inset box provides a zoomed-in view of this
region. The yellow, red, and dark-red filled squares depict the hot spot model
with $T_{spot}=$ 8000 K (yellow dotted line), 12000 K (red dashed line), and
40000 K (dark-red long-dashed line). The labels of the data points correspond
to the coverage fraction ($f$) values: 0.0001, 0.001, 0.01, 0.1, 0.2, and 0.3.

Figure 12: Relative variations in color, $d(\bv)_{rel}$, and magnitude,
$dV_{rel}$, are observed between successive data points. In the upper panel
(a), changes in brightness and color corresponding to the four quadrants are
classified: brighter & bluer, brighter & redder, fainter & bluer, and fainter
& redder. The labels of the data points indicate the observed date. In the
lower panel (b), the relative variations along the observation dates are
presented. Open circles represent $dV_{rel}$, while filled squares denote
$d(\bv)_{rel}$. The filled squares are further classified based on relative
changes: blue squares indicate fainter & bluer, red squares indicate brighter
& redder, and green squares indicate fainter & redder or brighter & bluer.
Notably, all nine blue squares exist in the period from 1984 to 1995, during
which the overall brightness of DG Tau was increasing.