# pix2code: Generating Code from a Graphical User Interface Screenshot

Tony Beltramelli
UIzard Technologies
Copenhagen, Denmark
<EMAIL_ADDRESS>
###### Abstract

Transforming a graphical user interface screenshot created by a designer into
computer code is a typical task conducted by a developer in order to build
customized software, websites, and mobile applications. In this paper, we show
that deep learning methods can be leveraged to train a model end-to-end to
automatically generate code from a single input image with over 77% of
accuracy for three different platforms (i.e. iOS, Android and web-based
technologies).

##  1 Introduction

The process of implementing client-side software based on a _Graphical User
Interface (GUI)_ mockup created by a designer is the responsibility of
developers. Implementing GUI code is, however, time-consuming and prevent
developers from dedicating the majority of their time implementing the actual
functionality and logic of the software they are building. Moreover, the
computer languages used to implement such GUIs are specific to each target
runtime system; thus resulting in tedious and repetitive work when the
software being built is expected to run on multiple platforms using native
technologies. In this paper, we describe a model trained end-to-end with
stochastic gradient descent to simultaneously learns to model sequences and
spatio-temporal visual features to generate variable-length strings of tokens
from a single GUI image as input.

Our first contribution is _pix2code_ , a novel approach based on Convolutional
and Recurrent Neural Networks allowing the generation of computer tokens from
a single GUI screenshot as input. That is, no engineered feature extraction
pipeline nor expert heuristics was designed to process the input data; our
model learns from the pixel values of the input image alone. Our experiments
demonstrate the effectiveness of our method for generating computer code for
various platforms (i.e. iOS and Android native mobile interfaces, and multi-
platform web-based HTML/CSS interfaces) without the need for any change or
specific tuning to the model. In fact, _pix2code_ can be used as such to
support different target languages simply by being trained on a different
dataset. A video demonstrating our system is available
online111https://uizard.io/research#pix2code.

Our second contribution is the release of our synthesized datasets consisting
of both GUI screenshots and associated source code for three different
platforms. Our datasets and our _pix2code_ implemention are publicly
available222https://github.com/tonybeltramelli/pix2code to foster future
research.

##  2 Related Work

The automatic generation of programs using machine learning techniques is a
relatively new field of research and program synthesis in a human-readable
format have only been addressed very recently. A recent example is DeepCoder
[2], a system able to generate computer programs by leveraging statistical
predictions to augment traditional search techniques. In another work by Gaunt
et al. [5], the generation of source code is enabled by learning the
relationships between input-output examples via differentiable interpreters.
Furthermore, Ling et al. [12] recently demonstrated program synthesis from a
mixed natural language and structured program specification as input. It is
important to note that most of these methods rely on _Domain Specific
Languages (DSLs)_ ; computer languages (e.g. markup languages, programming
languages, modeling languages) that are designed for a specialized domain but
are typically more restrictive than full-featured computer languages. Using
DSLs thus limit the complexity of the programming language that needs to be
modeled and reduce the size of the search space.

Although the generation of computer programs is an active research field as
suggested by these breakthroughs, program generation from visual inputs is
still a nearly unexplored research area. The closest related work is a method
developed by Nguyen et al. [14] to reverse-engineer Android user interfaces
from screenshots. However, their method relies entirely on engineered
heuristics requiring expert knowledge of the domain to be implemented
successfully. Our paper is, to the best of our knowledge, the first work
attempting to address the problem of user interface code generation from
visual inputs by leveraging machine learning to learn latent variables instead
of engineering complex heuristics.

In order to exploit the graphical nature of our input, we can borrow methods
from the computer vision literature. In fact, an important number of research
[21, 4, 10, 22] have addressed the problem of image captioning with impressive
results; showing that deep neural networks are able to learn latent variables
describing objects in an image and their relationships with corresponding
variable-length textual descriptions. All these methods rely on two main
components. First, a _Convolutional Neural Network (CNN)_ performing
unsupervised feature learning mapping the raw input image to a learned
representation. Second, a _Recurrent Neural Network (RNN)_ performing language
modeling on the textual description associated with the input picture. These
approaches have the advantage of being differentiable end-to-end, thus
allowing the use of gradient descent for optimization.

(a) Training

(b) Sampling

Figure 1: Overview of the _pix2code_ model architecture. During training, the
GUI image is encoded by a CNN-based vision model; the context (i.e. a sequence
of one-hot encoded tokens corresponding to DSL code) is encoded by a language
model consisting of a stack of LSTM layers. The two resulting feature vectors
are then concatenated and fed into a second stack of LSTM layers acting as a
decoder. Finally, a _softmax_ layer is used to sample one token at a time; the
output size of the _softmax_ layer corresponding to the DSL vocabulary size.
Given an image and a sequence of tokens, the model (i.e. contained in the gray
box) is differentiable and can thus be optimized end-to-end through gradient
descent to predict the next token in the sequence. During sampling, the input
context is updated for each prediction to contain the last predicted token.
The resulting sequence of DSL tokens is compiled to the desired target
language using traditional compiler design techniques.

##  3 pix2code

The task of generating computer code written in a given programming language
from a GUI screenshot can be compared to the task of generating English
textual descriptions from a scene photography. In both scenarios, we want to
produce a variable-length strings of tokens from pixel values. We can thus
divide our problem into three sub-problems. First, a computer vision problem
of understanding the given scene (i.e. in this case, the GUI image) and
inferring the objects present, their identities, positions, and poses (i.e.
buttons, labels, element containers). Second, a language modeling problem of
understanding text (i.e. in this case, computer code) and generating
syntactically and semantically correct samples. Finally, the last challenge is
to use the solutions to both previous sub-problems by exploiting the latent
variables inferred from scene understanding to generate corresponding textual
descriptions (i.e. computer code rather than English) of the objects
represented by these variables.

###  3.1 Vision Model

CNNs are currently the method of choice to solve a wide range of vision
problems thanks to their topology allowing them to learn rich latent
representations from the images they are trained on [16, 11]. We used a CNN to
perform unsupervised feature learning by mapping an input image to a learned
fixed-length vector; thus acting as an encoder as shown in Figure 1.

The input images are initially re-sized to $256\times 256$ pixels (the aspect
ratio is not preserved) and the pixel values are normalized before to be fed
in the CNN. No further pre-processing is performed. To encode each input image
to a fixed-size output vector, we exclusively used small $3\times 3$ receptive
fields which are convolved with stride $1$ as used by Simonyan and Zisserman
for VGGNet [18]. These operations are applied twice before to down-sample with
max-pooling. The width of the first convolutional layer is $32$, followed by a
layer of width $64$, and finally width $128$. Two fully connected layers of
size $1024$ applying the _rectified linear unit_ activation complete the
vision model.

(a) iOS GUI screenshot

(b) Code describing the GUI written in our DSL

Figure 2: An example of a native iOS GUI written in our markup-like DSL.

###  3.2 Language Model

We designed a simple lightweight DSL to describe GUIs as illustrated in Figure
2. In this work we are only interested in the GUI layout, the different
graphical components, and their relationships; thus the actual textual value
of the labels is ignored. Additionally to reducing the size of the search
space, the DSL simplicity also reduces the size of the vocabulary (i.e. the
total number of tokens supported by the DSL). As a result, our language model
can perform token-level language modeling with a discrete input by using one-
hot encoded vectors; eliminating the need for word embedding techniques such
as word2vec [13] that can result in costly computations.

In most programming languages and markup languages, an element is declared
with an opening token; if children elements or instructions are contained
within a block, a closing token is usually needed for the interpreter or the
compiler. In such a scenario where the number of children elements contained
in a parent element is variable, it is important to model long-term
dependencies to be able to close a block that has been opened. Traditional RNN
architectures suffer from vanishing and exploding gradients preventing them
from being able to model such relationships between data points spread out in
time series (i.e. in this case tokens spread out in a sequence). Hochreiter
and Schmidhuber proposed the _Long Short-Term Memory (LSTM)_ neural
architecture in order to address this very problem [9]. The different LSTM
gate outputs can be computed as follows:

$\displaystyle i_{t}$ $\displaystyle=\phi(W_{ix}x_{t}+W_{iy}h_{t-1}+b_{i})$
(1) $\displaystyle f_{t}$
$\displaystyle=\phi(W_{fx}x_{t}+W_{fy}h_{t-1}+b_{f})$ (2) $\displaystyle
o_{t}$ $\displaystyle=\phi(W_{ox}x_{t}+W_{oy}h_{t-1}+b_{o})$ (3)
$\displaystyle c_{t}$ $\displaystyle=f_{t}\bullet
c_{t-1}+i_{t}\bullet\sigma(W_{cx}x_{t}+W_{cy}h_{t-1}+b_{c})$ (4)
$\displaystyle h_{t}$ $\displaystyle=o_{t}\bullet\sigma(c_{t})$ (5)

With $W$ the matrices of weights, $x_{t}$ the new input vector at time $t$,
$h_{t-1}$ the previously produced output vector, $c_{t-1}$ the previously
produced cell state’s output, $b$ the biases, and $\phi$ and $\sigma$ the
activation functions _sigmoid_ and _hyperbolic tangent_ , respectively. The
cell state $c$ learns to memorize information by using a recursive connection
as done in traditional RNN cells. The input gate $i$ is used to control the
error flow on the inputs of cell state $c$ to avoid input weight conflicts
that occur in traditional RNN because the same weight has to be used for both
storing certain inputs and ignoring others. The output gate $o$ controls the
error flow from the outputs of the cell state $c$ to prevent output weight
conflicts that happen in standard RNN because the same weight has to be used
for both retrieving information and not retrieving others. The LSTM memory
block can thus use $i$ to decide when to write information in $c$ and use $o$
to decide when to read information from $c$. We used the LSTM variant proposed
by Gers and Schmidhuber [6] with a forget gate $f$ to reset memory and help
the network model continuous sequences.

###  3.3 Decoder

Our model is trained in a supervised learning manner by feeding an image $I$
and a contextual sequence $X$ of $T$ tokens $x_{t},t\in\left\\{0\ldots
T-1\right\\}$ as inputs; and the token $x_{T}$ as the target label. As shown
on Figure 1, a CNN-based vision model encodes the input image $I$ into a
vectorial representation $p$. The input token $x_{t}$ is encoded by an LSTM-
based language model into an intermediary representation $q_{t}$ allowing the
model to focus more on certain tokens and less on others [8]. This first
language model is implemented as a stack of two LSTM layers with $128$ cells
each. The vision-encoded vector $p$ and the language-encoded vector $q_{t}$
are concatenated into a single feature vector $r_{t}$ which is then fed into a
second LSTM-based model decoding the representations learned by both the
vision model and the language model. The decoder thus learns to model the
relationship between objects present in the input GUI image and the associated
tokens present in the DSL code. Our decoder is implemented as a stack of two
LSTM layers with $512$ cells each. This architecture can be expressed
mathematically as follows:

$\displaystyle p$ $\displaystyle=CNN(I)$ (6) $\displaystyle q_{t}$
$\displaystyle=LSTM(x_{t})$ (7) $\displaystyle r_{t}$
$\displaystyle=(q,p_{t})$ (8) $\displaystyle y_{t}$
$\displaystyle=softmax(LSTM^{\prime}(r_{t}))$ (9) $\displaystyle x_{t+1}$
$\displaystyle=y_{t}$ (10)

This architecture allows the whole _pix2code_ model to be optimized end-to-end
with gradient descent to predict a token at a time after it has seen both the
image as well as the preceding tokens in the sequence. The discrete nature of
the output (i.e. fixed-sized vocabulary of tokens in the DSL) allows us to
reduce the task to a classification problem. That is, the output layer of our
model has the same number of cells as the vocabulary size; thus generating a
probability distribution of the candidate tokens at each time step allowing
the use of a _softmax_ layer to perform multi-class classification.

###  3.4 Training

The length $T$ of the sequences used for training is important to model long-
term dependencies; for example to be able to close a block of code that has
been opened. After conducting empirical experiments, the DSL input files used
for training were segmented with a sliding window of size $48$; in other
words, we unroll the recurrent neural network for $48$ steps. This was found
to be a satisfactory trade-off between long-term dependencies learning and
computational cost. For every token in the input DSL file, the model is
therefore fed with both an input image and a contextual sequence of $T=48$
tokens. While the context (i.e. sequence of tokens) used for training is
updated at each time step (i.e. each token) by sliding the window, the very
same input image $I$ is reused for samples associated with the same GUI. The
special tokens $<START>$ and $<END>$ are used to respectively prefix and
suffix the DSL files similarly to the method used by Karpathy and Fei-Fei
[10]. Training is performed by computing the partial derivatives of the loss
with respect to the network weights calculated with backpropagation to
minimize the multiclass log loss:

$\displaystyle L(I,X)$ $\displaystyle=-\sum^{T}_{t=1}x_{t+1}\log(y_{t})$ (11)

With $x_{t+1}$ the expected token, and $y_{t}$ the predicted token. The model
is optimized end-to-end hence the loss $L$ is minimized with regard to all the
parameters including all layers in the CNN-based vision model and all layers
in both LSTM-based models. Training with the _RMSProp_ algorithm [20] gave the
best results with a learning rate set to $1e-4$ and by clipping the output
gradient to the range $[-1.0,1.0]$ to cope with numerical instability [8]. To
prevent overfitting, a dropout regularization [19] set to $25\%$ is applied to
the vision model after each max-pooling operation and at $30\%$ after each
fully-connected layer. In the LSTM-based models, dropout is set to $10\%$ and
only applied to the non-recurrent connections [24]. Our model was trained with
mini-batches of $64$ image-sequence pairs.

Table 1: Dataset statistics. Dataset type | Synthesizable | Training set | Test set
---|---|---|---
Instances | Samples | Instances | Samples
iOS UI (Storyboard) | $26\times 10^{5}$ | $1500$ | $93672$ | $250$ | $15984$
Android UI (XML) | $21\times 10^{6}$ | $1500$ | $85756$ | $250$ | $14265$
web-based UI (HTML/CSS) | $31\times 10^{4}$ | $1500$ | $143850$ | $250$ | $24108$

###  3.5 Sampling

To generate DSL code, we feed the GUI image $I$ and a contextual sequence $X$
of $T=48$ tokens where tokens $x_{t}\ldots x_{T-1}$ are initially set empty
and the last token of the sequence $x_{T}$ is set to the special $<START>$
token. The predicted token $y_{t}$ is then used to update the next sequence of
contextual tokens. That is, $x_{t}\ldots x_{T-1}$ are set to $x_{t+1}\ldots
x_{T}$ ($x_{t}$ is thus discarded), with $x_{T}$ set to $y_{t}$. The process
is repeated until the token $<END>$ is generated by the model. The generated
DSL token sequence can then be compiled with traditional compilation methods
to the desired target language.

(a) _pix2code_ training loss

(b) Micro-average ROC curves

Figure 3: Training loss on different datasets and ROC curves calculated during
sampling with the model trained for 10 epochs.

##  4 Experiments

Access to consequent datasets is a typical bottleneck when training deep
neural networks. To the best of our knowledge, no dataset consisting of both
GUI screenshots and source code was available at the time this paper was
written. As a consequence, we synthesized our own data resulting in the three
datasets described in Table 1. The column _Synthesizable_ refers to the
maximum number of unique GUI configuration that can be synthesized using our
stochastic user interface generator. The columns _Instances_ refers to the
number of synthesized (GUI screenshot, GUI code) file pairs. The columns
_Samples_ refers to the number of distinct image-sequence pairs. In fact,
training and sampling are done one token at a time by feeding the model with
an image and a sequence of tokens obtained with a sliding window of fixed size
$T$. The total number of training samples thus depends on the total number of
tokens written in the DSL files and the size of the sliding window. Our
stochastic user interface generator is designed to synthesize GUIs written in
our DSL which is then compiled to the desired target language to be rendered.
Using data synthesis also allows us to demonstrate the capability of our model
to generate computer code for three different platforms.

Table 2: Experiments results reported for the test sets described in Table 1. Dataset type | Error (%)
---|---
greedy search | beam search 3 | beam search 5
iOS UI (Storyboard) | 22.73 | 25.22 | 23.94
Android UI (XML) | 22.34 | 23.58 | 40.93
web-based UI (HTML/CSS) | 12.14 | 11.01 | 22.35

Our model has around $109\times 10^{6}$ parameters to optimize and all
experiments are performed with the same model with no specific tuning; only
the training datasets differ as shown on Figure 3. Code generation is
performed with both greedy search and beam search to find the tokens that
maximize the classification probability. To evaluate the quality of the
generated output, the classification error is computed for each sampled DSL
token and averaged over the whole test dataset. The length difference between
the generated and the expected token sequences is also counted as error. The
results can be seen on Table 2.

Figures 4, 5, and 6 show samples consisting of input GUIs (i.e. ground truth),
and output GUIs generated by a trained _pix2code_ model. It is important to
remember that the actual textual value of the labels is ignored and that both
our data synthesis algorithm and our DSL compiler assign randomly generated
text to the labels. Despite occasional problems to select the right color or
the right style for specific GUI elements and some difficulties modelling GUIs
consisting of long lists of graphical components, our model is generally able
to learn the GUI layout in a satisfying manner and can preserve the
hierarchical structure of the graphical elements.

(a) Groundtruth GUI 1

(b) Generated GUI 1

(c) Groundtruth GUI 2

(d) Generated GUI 2

Figure 4: Experiment samples for the iOS GUI dataset.

##  5 Conclusion and Discussions

In this paper, we presented _pix2code_ , a novel method to generate computer
code given a single GUI image as input. While our work demonstrates the
potential of such a system to automate the process of implementing GUIs, we
only scratched the surface of what is possible. Our model consists of
relatively few parameters and was trained on a relatively small dataset. The
quality of the generated code could be drastically improved by training a
bigger model on significantly more data for an extended number of epochs.
Implementing a now-standard attention mechanism [1, 22] could further improve
the quality of the generated code.

Using one-hot encoding does not provide any useful information about the
relationships between the tokens since the method simply assigns an arbitrary
vectorial representation to each token. Therefore, pre-training the language
model to learn vectorial representations would allow the relationships between
tokens in the DSL to be inferred (i.e. learning word embeddings such as
word2vec [13]) and as a result alleviate semantical error in the generated
code. Furthermore, one-hot encoding does not scale to very big vocabulary and
thus restrict the number of symbols that the DSL can support.

(a) Groundtruth GUI 5

(b) Generated GUI 5

(c) Groundtruth GUI 6

(d) Generated GUI 6

Figure 5: Experiment samples from the web-based GUI dataset.

_Generative Adversarial Networks GANs_ [7] have shown to be extremely powerful
at generating images and sequences [23, 15, 25, 17, 3]. Applying such
techniques to the problem of generating computer code from an input image is
so far an unexplored research area. GANs could potentially be used as a
standalone method to generate code or could be used in combination with our
_pix2code_ model to fine-tune results.

A major drawback of deep neural networks is the need for a lot of training
data for the resulting model to generalize well on new unseen examples. One of
the significant advantages of the method we described in this paper is that
there is no need for human-labelled data. In fact, the network can model the
relationships between graphical components and associated tokens by simply
being trained on image-sequence pairs. Although we used data synthesis in our
paper partly to demonstrate the capability of our method to generate GUI code
for various platforms; data synthesis might not be needed at all if one wants
to focus only on web-based GUIs. In fact, one could imagine crawling the World
Wide Web to collect a dataset of HTML/CSS code associated with screenshots of
rendered websites. Considering the large number of web pages already available
online and the fact that new websites are created every day, the web could
theoretically supply a virtually unlimited amount of training data;
potentially allowing deep learning methods to fully automate the
implementation of web-based GUIs.

(a) Groundtruth GUI 3

(b) Generated GUI 3

(c) Groundtruth GUI 4

(d) Generated GUI 4

Figure 6: Experiment samples from the Android GUI dataset.

## References

  * [1] D. Bahdanau, K. Cho, and Y. Bengio.  Neural machine translation by jointly learning to align and translate.  arXiv preprint arXiv:1409.0473, 2014.
  * [2] M. Balog, A. L. Gaunt, M. Brockschmidt, S. Nowozin, and D. Tarlow.  Deepcoder: Learning to write programs.  arXiv preprint arXiv:1611.01989, 2016.
  * [3] B. Dai, D. Lin, R. Urtasun, and S. Fidler.  Towards diverse and natural image descriptions via a conditional gan.  arXiv preprint arXiv:1703.06029, 2017.
  * [4] J. Donahue, L. Anne Hendricks, S. Guadarrama, M. Rohrbach, S. Venugopalan, K. Saenko, and T. Darrell.  Long-term recurrent convolutional networks for visual recognition and description.  In Proceedings of the IEEE conference on computer vision and pattern recognition, pages 2625–2634, 2015.
  * [5] A. L. Gaunt, M. Brockschmidt, R. Singh, N. Kushman, P. Kohli, J. Taylor, and D. Tarlow.  Terpret: A probabilistic programming language for program induction.  arXiv preprint arXiv:1608.04428, 2016.
  * [6] F. A. Gers, J. Schmidhuber, and F. Cummins.  Learning to forget: Continual prediction with lstm.  Neural computation, 12(10):2451–2471, 2000.
  * [7] I. Goodfellow, J. Pouget-Abadie, M. Mirza, B. Xu, D. Warde-Farley, S. Ozair, A. Courville, and Y. Bengio.  Generative adversarial nets.  In Advances in neural information processing systems, pages 2672–2680, 2014.
  * [8] A. Graves.  Generating sequences with recurrent neural networks.  arXiv preprint arXiv:1308.0850, 2013.
  * [9] S. Hochreiter and J. Schmidhuber.  Long short-term memory.  Neural computation, 9(8):1735–1780, 1997.
  * [10] A. Karpathy and L. Fei-Fei.  Deep visual-semantic alignments for generating image descriptions.  In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pages 3128–3137, 2015.
  * [11] A. Krizhevsky, I. Sutskever, and G. E. Hinton.  Imagenet classification with deep convolutional neural networks.  In Advances in neural information processing systems, pages 1097–1105, 2012.
  * [12] W. Ling, E. Grefenstette, K. M. Hermann, T. Kočiskỳ, A. Senior, F. Wang, and P. Blunsom.  Latent predictor networks for code generation.  arXiv preprint arXiv:1603.06744, 2016.
  * [13] T. Mikolov, I. Sutskever, K. Chen, G. S. Corrado, and J. Dean.  Distributed representations of words and phrases and their compositionality.  In Advances in neural information processing systems, pages 3111–3119, 2013.
  * [14] T. A. Nguyen and C. Csallner.  Reverse engineering mobile application user interfaces with remaui (t).  In Automated Software Engineering (ASE), 2015 30th IEEE/ACM International Conference on, pages 248–259. IEEE, 2015.
  * [15] S. Reed, Z. Akata, X. Yan, L. Logeswaran, B. Schiele, and H. Lee.  Generative adversarial text to image synthesis.  In Proceedings of The 33rd International Conference on Machine Learning, volume 3, 2016.
  * [16] P. Sermanet, D. Eigen, X. Zhang, M. Mathieu, R. Fergus, and Y. LeCun.  Overfeat: Integrated recognition, localization and detection using convolutional networks.  arXiv preprint arXiv:1312.6229, 2013.
  * [17] R. Shetty, M. Rohrbach, L. A. Hendricks, M. Fritz, and B. Schiele.  Speaking the same language: Matching machine to human captions by adversarial training.  arXiv preprint arXiv:1703.10476, 2017.
  * [18] K. Simonyan and A. Zisserman.  Very deep convolutional networks for large-scale image recognition.  arXiv preprint arXiv:1409.1556, 2014.
  * [19] N. Srivastava, G. E. Hinton, A. Krizhevsky, I. Sutskever, and R. Salakhutdinov.  Dropout: a simple way to prevent neural networks from overfitting.  Journal of Machine Learning Research, 15(1):1929–1958, 2014.
  * [20] T. Tieleman and G. Hinton.  Lecture 6.5-rmsprop: Divide the gradient by a running average of its recent magnitude.  COURSERA: Neural networks for machine learning, 4(2), 2012.
  * [21] O. Vinyals, A. Toshev, S. Bengio, and D. Erhan.  Show and tell: A neural image caption generator.  In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pages 3156–3164, 2015.
  * [22] K. Xu, J. Ba, R. Kiros, K. Cho, A. C. Courville, R. Salakhutdinov, R. S. Zemel, and Y. Bengio.  Show, attend and tell: Neural image caption generation with visual attention.  In ICML, volume 14, pages 77–81, 2015.
  * [23] L. Yu, W. Zhang, J. Wang, and Y. Yu.  Seqgan: sequence generative adversarial nets with policy gradient.  arXiv preprint arXiv:1609.05473, 2016.
  * [24] W. Zaremba, I. Sutskever, and O. Vinyals.  Recurrent neural network regularization.  arXiv preprint arXiv:1409.2329, 2014.
  * [25] H. Zhang, T. Xu, H. Li, S. Zhang, X. Huang, X. Wang, and D. Metaxas.  Stackgan: Text to photo-realistic image synthesis with stacked generative adversarial networks.  arXiv preprint arXiv:1612.03242, 2016.


# Asymmetry of Lines in the Spectra of the Sun and Solar-Type Stars

V. A. Sheminova

###### Abstract

We have analysed the asymmetry of lines Fe I and Fe II in spectra of a solar
flux using three FTS atlases and the HARPS atlas and also in spectra of 13
stars using observation data on the HARPS spectrograph. To reduce observation
noise individual line bisectors of each star have been averaged. The obtained
average bisectors in the stellar spectra are more or less similar to the shape
C well known to the Sun. In stars with rotation speeds greater than 5 km/s the
shape of the bisectors is more like /. The curvature and span of the bisectors
increase with the temperature of the star. Our results confirm the known facts
about strong influence of rotation velocity on the span and shape of
bisectors. The average convective speed was determined based on the span of
the average bisector, which shows the largest difference between the velocity
of cold falling and hot rising convective flows of the matter. It’s equal to
$-420$ m/s for the Sun as a star. In solar-type stars, it grows from $-150$ to
$-700$ m/s with an effective temperature of 4800 to 6200 K, respectively. For
stars with greater surface gravity and greater metallicity, the average
convective velocity decreases. It also decreases with star age and correlates
with the speed of micro and macroturbulent movements. The results of solar
flux analysis showed that absolute wavelength scales in the FTS atlases used
coincide with an accuracy of about $-10$ m/s, except for the atlas of Hinkle,
etc., whose scale is shifted and depends on the wavelength. In the range from
450 to 650 nm, the scale shift of this atlas varies from $-100$ to $-330$ m/s,
respectively, and it equals on average of $-240$ m/s. The resulting average
star bisectors contain information about the fields of convective velocities
and may be useful for hydrodynamic modeling of stellar atmospheres in order to
study the characteristic features of surface convection.

Main Astronomical Observatory, National Academy of Sciences of Ukraine,

Akademika Zabolotnoho 27, Kyiv, 03143 Ukraine

e-mail<EMAIL_ADDRESS>
Keywords: line asymmetry, line profiles, solar-type stars, velocity field,
granulation.

##  1 Introduction

In stellar spectra asymmetry of absorption lines may arise due to dark spots,
bright points, and bright plages, as well as due to oscillations, pulsations,
granulation, and line blending. The main cause of asymmetry in solar-type
stars is granulation [14]. It is directly observed in images of the solar
surface as a granulation pattern. For main-sequence stars direct observation
of granulation is impossible, however, signatures of stellar granulation can
be studied by analyzing the asymmetries of the absorption line profiles. By
studying the asymmetry of many lines, we can obtain information about the
characteristics of surface convection in stars. The observation of spectral
lines in stars, as compared with the Sun, has a number of limitations.
Firstly, in cases of stars we observe a disk-integral flux, which leads to the
weakening of the Doppler shifts. Secondly, the absolute line shifts cannot be
measured due to the lack of data on the exact radial velocities of many stars.
Thirdly, the rotation of the stars affects the shape of the line profiles and
alters the asymmetry. Fourthly, spectral lines become weaker in low-resolution
stellar spectra, and profile distortions increase. This should be borne in
mind when studying the asymmetry of lines in stars.

The appearance of granulation on the stellar surface is caused by convection
in the surface layers, which creates wide upward flows of hot matter, so-
called granules, and cold downward flows, or intergranules. Granules make a
blue contribution to the spectral line due to the matter’s ascent velocity.
Cold intergranules are narrow because of the smaller area and they make a red
contribution due to the matter’s descent velocity. The combined effect of all
contributions integrated over the stellar surface creates asymmetric line
profiles [14]. Photospheric velocities that are directly caused by convection
and affect the asymmetry of lines are called convective or granulation
velocities. According to [22], the average velocity of the distribution of
granulation velocities is the average ascent velocity of granules $V=V_{\rm
col}-V_{\rm hot}$, where $V_{\rm col}$ and $V_{\rm hot}$ velocity of granules
are the velocities of cold and hot matter, respectively; the dispersion of
this distribution is the so-called macroturbulent velocity $V_{\rm mac}$.
Simple modeling showed that the average granulation velocity affects the
asymmetry, while the velocity dispersion affects the line width, and always
$V<V_{\rm mac}$. As noted in [18], asymmetry is controlled by three
parameters: (1) average velocity of granules and intergranules, (2) brightness
contrast between granules and intergranules, and (3) the fraction of the
stellar disk area occupied by granules in relation to intergranules. For a
star, (2) and (3) are combined into a flux contrast $(F_{\rm hot}-F_{\rm
col})/(F_{\rm hot}+F_{\rm col})$, where $F_{\rm hot}$ is the flux emitted by
all granules, and $F_{\rm col}$ is the flux emitted by all intergranules on
the stellar disk. The higher the degree of correlation between the velocity
and temperature in the granules and in the intergranules, the stronger the
asymmetry of the observed line. The rotation velocity $v\sin i$ also affects
the asymmetry when it is higher than $V_{\rm mac}$. According to [38], the
asymmetry is to a lesser extent influenced by the abundance of chemical
elements and the activity of stars, and the latter decreases the line
asymmetry in comparison with inactive stars.

The asymmetry of the profiles is quantitatively characterized by the bisector,
which represents the dependence between convective shifts and the relative
flux of radiation at each point of the line profile. The most complete picture
of convective shifts is seen for strong lines. This is due to a significant
difference in physical conditions at different effective formation depths of
each segment of the line where the bisector is measured. Early studies of the
solar spectrum [7, 14, 29, 32] established that the bisector has a typical
shape similar to the letter C. The upper part of the profile near the
continuum has a small blue shift due to significant contributions of radiation
from intergranules in the deep layers of the photosphere. The largest blue
shift occurs closer to the middle of the profile due to the predominant
fraction of radiation from hot rising granules. The core of a strong line has
a smaller blue shift because the radiation contrast between granules and
intergranules decreases in the upper photospheric layers of the core
formation. The asymmetry of the line profile varies depending on the line
depth and atomic parameters, such as chemical composition, excitation and
ionization potential, wavelength, oscillator strength, i.e., the parameters
that determine the effective depth of line formation in the atmosphere.

Stellar bisectors show significant differences both in the shape and the
magnitude of the blue span with variations in effective temperature,
luminosity class, or gravity. According to the results of stellar spectra
studies [3, 11, 12, 13, 18, 19, 21, 22, 38], the bisector does not show the
C-shape in all cold FGK stars of the main sequence; this shape is often
slightly distorted, and sometimes only the upper part of the letter C is
visible. The differences in shape from one star to another depend on the
strength of granulation and the structure of the atmosphere. The span or shift
of the bluest point of the bisector, which characterizes the average
granulation velocity of the star, decreases from F- to K-stars. For giants,
the span is larger than for dwarfs. The brighter the star (or the greater the
luminosity, or the smaller the gravity), the lower the height of the bluest
point of the bisector, and the higher the granulation penetrates into the
atmosphere. The spectra of stars with a very low metal abundance reveal a
significantly greater line asymmetry and a wider range of velocities than
stars of the same class with solar metallicity [3]. This fact is interpreted
as a signature of a low metal abundance and, therefore, a low opacity in
convective atmospheres. There is also another feature of asymmetry: if we
proceed along the main sequence towards higher temperatures, the line
bisectors in the F0 zone change their tilt to the opposite direction. This
boundary separating the two modes is called the granulation boundary [21, 23].
It is assumed that convection in hot F0V stars is qualitatively different:
granules are small and quickly rise upward, while intergranules are large and
descend slowly.

To better understand stellar granulation and its effect on the observed
spectra, hydrodynamic modeling of the solar and stellar atmospheres is
performed, which takes into account atmospheric inhomogeneities associated
with the granulation phenomenon. The first studies on granulation modeling [5,
6, 15] predicted that the energy flux passing through the atmosphere is the
most important parameter that affects the magnitude of blue shifts and,
therefore, the span of the bisector. The correlation between temperature,
velocity, and density is weaker in cold dwarfs than in hotter stars.
Subsequent studies on modeling the convective envelopes of stars, e.g., [1, 4,
9, 39], confirmed and explained many observational features of bisectors.
Despite the great progress in modeling, theoretical calculations were carried
out only for some FGK stars using a limited set of spectral characteristics
and were verified against a limited set of observations. It is necessary to
study a larger number of stars with hotter and colder temperatures. Careful
validation of the models on high-quality observed spectra is required. It is
especially important to verify the models for coincidence with the bisectors
of many observed lines. This should be done so that the granulation properties
predicted by the models can be guaranteed to be fully reliable. Therefore,
obtaining new data on the asymmetry of the observed lines and on their
bisectors for stars with different characteristics remains an important
problem in the physics of stellar atmospheres. Solving this problem requires
high quality stellar spectra and careful selection of absorption lines to
reduce the bisector errors.

The aim of this study is to measure the line bisectors in the spectra of
solar-type stars with different characteristics and to trace the change in
their shape with the variations in the stars’ parameters. In other words, this
study is dedicated to the second signature of stellar granulation, i.e., line
asymmetry. The first signature of granulation is the broadening of the line
profiles by macroturbulent motions, and the third signature of granulation is
the blue shift of lines due to the dominant contributions of hot granules. The
well-known parameter of classical macroturbulence shows the strength of
granulation in one or another star. In our previous study [42], we already
investigated the first signature of granulation in the stars that are analyzed
here. This enables us to perform a comparison of convective and macroturbulent
velocities and confirm the existing understanding of granulation processes and
its effect on spectral lines.

##  2 Initial data and methods

Observation data. Table 1 shows the basic data for the stars whose spectra we
will analyze. These stars are from the Calan-Hertfordshire Extrasolar Planet
Search (CHEPS) sample. The main characteristics of the stars, such as
effective temperature $T_{\rm eff}$, surface gravity $\log g$, and metallicity
[M/H], were taken from [27], the data on mass $m$ and age from [36], and the
data on macroturbulent velocity $V_{\rm mac}$ and rotational velocity $v\sin
i$ from [42]. The spectral types correspond to the data from the SIMBAD
database (http://cdsportal.u-strasbg.fr). All the stars are single and
inactive ($\log R_{\rm HK}\leq-4.5$ dex). The $T_{\rm eff}$ range is small,
from 4800 to 6200 K. Some of the stars are slightly enriched in metals with
[M/H] from 0.06 to 0.34, while others have a small metal deficit from $-0.05$
to $-0.15$.

Table 1: Values of the parameters of the stars ($n$ is the number of lines; $V$ is the average convective velocity in the Mg scale) HD | Type | $T_{\rm eff}$ | $\log g$ | [M/H] | $M/M_{\odot}$ | Age | $V_{\rm mac}$ | $v\sin i$ | $n$ | $V$
---|---|---|---|---|---|---|---|---|---|---
|  | (K) |  |  |  | (Ga) | (km/s) | (km/s) |  | m/s
189627 | F7 V | 6210 | 4.40 |  0.07 | 1.244 | 4.0 | 5.52 | 5.93 | 31 |  $-$606
92003 | F0 V | 6158 | 5.10 |  0.27 | 1.087 | 7.2 | 2.20 | 2.09 | 37 |  $-$573
93849 | G0/1 V | 6153 | 4.21 |  0.08 | 1.268 | 3.5 | 2.92 | 3.05 | 41 |  $-$696
158469 | F8/G2 V | 6105 | 4.19 |  $-$0.14 | 1.223 | 2.0 | 3.61 | 3.10 | 44 |  $-$686
127423 | G0 V | 6020 | 4.26 |  $-$0.09 | 1.107 | 3.1 | 2.90 | 2.53 | 34 |  $-$442
6790 | G0 V | 6012 | 4.40 |  $-$0.06 | 1.089 | 3.5 | 3.16 | 2.94 | 32 |  $-$511
102196 | G2 V | 6012 | 3.90 |  $-$0.05 | 1.395 | 3.0 | 4.26 | 3.56 | 36 |  $-$630
102361 | F8 V | 5978 | 4.12 |  $-$0.15 | 1.250 | 2.0 | 5.62 | 5.03 | 47 |  $-$600
147873 | G1 V | 5972 | 3.90 |  $-$0.09 | 1.493 | 2.6 | 5.95 | 6.51 | 34 |  $-$586
Sun | G2 V | 5777 | 4.44 |  0.00 | 1.000 | 4.6 | 2.11 | 1.84 | 61 |  $-$456
38459 | K1 IV-V | 5233 | 4.43 |  0.06 | 0.882 | 9.0 | 3.20 | 1.85 | 37 |  $-$220
42936 | K0 IV-V | 5126 | 4.44 |  0.19 | 0.881 | 12.0 | 1.74 | 0.97 | 26 |  $-$338
221575 | K2 V | 5037 | 4.49 |  $-$0.11 | 0.823 | 6.0 | 2.79 | 1.89 | 36 |  $-$152
128356 | K2.5 IV | 4875 | 4.58 |  0.34 | 0.824 | 15.5 | 1.74 | 1.01 | 19 |  $-$220

The spectra of the stars were obtained with the HARPS (High-Accuracy Radial
Velocity Planetary Searcher) spectrograph at La Silla in Chile [28]. The
signal-to-noise ratio is above 100. The spectral resolution $R\approx 120000$
is rather low for this problem; however, according to [19], this is
sufficient, since the nominal spectral resolution should be at least 100000.

List of spectral lines. The lines were selected visually using a graphic solar
atlas [44] with a high resolution $R\approx 700000$. To reduce uncertainties
due to noise, bisectors were measured in the $F_{\lambda}{/}F_{\rm c}$ range
from 0.1 to 0.95. Here, $F_{\lambda}$ and $F_{\rm c}$ are the radiation fluxes
in the line and the continuum, respectively. The selected lines were in the
wavelength range from 450 to 650 nm. The number of lines varied from star to
star, because new blends appeared in the line profiles with decreasing $T_{\rm
eff}$. The presence of subtle blends in the profiles was checked by analyzing
the bisectors and comparing them with the mean bisector. As a result of this
selection, the final lists of lines in different stars mainly differed in the
range of very weak and strong lines. The number of lines $n$ is given in Table
1.

Measurement of the central wavelength of a line. To accurately determine the
wavelength of a line in the spectrum of a star is only possible for a
symmetrical profile. In reality, stellar spectral lines are always asymmetric
and distorted by blends and noise, which makes it impossible to accurately and
unambiguously calculate their central wavelength. There are various procedures
for measuring the line wavelength [25]. In most cases, the wavelength of the
line is determined by fitting polynomials of the 2–4 order by the number of
points nearest to the minimum. We have chosen, in our opinion, a more error-
tolerant method using the center of gravity of the line core [10]. The
accuracy of this method is not affected by a strong or weak, wide or narrow
line in the core. Mainly what matters is the interpolation step of the
profile. In our case, it was equal to 0.1 pm.

Calculation of the convective line shifts. For stars, it is difficult to
determine convective line shifts because the true radial velocity of the
stellar center of mass is not known accurately enough. To exclude the
influence of the radial velocity, the zero point of the shifts is established
using absolute solar shifts, as was done in [20], or an arbitrary zero point
is adopted using the shifts of the core wavelengths of the strongest lines. In
[32], to determine the convective line shifts observed in magnetic formations
on the solar surface, the strong line Mg I ${\rm b}_{2}$ 517.27 nm was used
(hereinafter referred to as the “Mg line”) based on the assumption that the
core of this line forms high in the atmosphere, where the brightness contrast
nearly disappears and its convective shift will be almost zero. This line was
also used in [10] a long with a group of strong iron lines to determine the
zero point of the shifts of the observed lines in magnetic formations. In the
study of K-dwarfs [38], the shifts of the cores of the strongest lines of iron
were also used on the basis of the assumption that they are equal to zero,
which was based on the results obtained for the solar spectrum in [2].

In this study, we used the Mg line. This allowed us to eliminate Doppler
shifts due to the radial velocity of the star relative to the observer on the
Earth, gravitational displacement, and other displacements if they were not
taken into account in the calibration of the observations as well as errors in
the method of measuring the central wavelength. Such a zero point of the shift
scale will contain an error that is equal to the intrinsic convective shift of
the Mg line. Such a scale for stellar spectra will henceforth be called the Mg
scale. In this scale, the convective shift of the spectral line is easily
calculated relative to the laboratory wavelength in units of velocity using
the following formula

$V=c(\lambda_{\rm obs}-\lambda_{\rm lab}){/}\lambda_{\rm lab}-c(\lambda_{\rm
obs}^{\rm Mg}-\lambda_{\rm lab}^{\rm Mg}){/}\lambda_{\rm lab}^{\rm Mg}.$ (1)

Here $\lambda_{\rm obs}$ is the measured central wavelength of the observed
line in the spectrum of the star, $\lambda_{\rm lab}$ is the laboratory
wavelength. The superscript Mg indicates the Mg line. We have applied this
formula to all stars and tested it using the Sun’s spectrum.

Laboratory wavelengths. To measure convective shifts, we used the results of
laboratory studies of wavelengths [34]. The catalog of laboratory wavelengths
presented in [34] is currently one of the most accurate and contains a
complete list of iron lines. All the line wavelengths are divided into four
categories depending on their errors (0.04 to 1 pm and larger). Most lines
have an error below 0.05 pm, and approximately half of them are below 0.1 pm
(or 60 m/s for a wavelength of 500.0 nm). For weaker lines
($F_{\lambda}{/}F_{\rm c}>0.7$), the error of laboratory wavelengths is
approximately 2.5 times larger than for stronger lines.

Plotting of the bisectors. To obtain a bisector, we first interpolate the line
profile using a cubic spline with a step of 0.01 pm. We then calculate the
midpoints of the horizontal lines drawn from the measured point on the blue
side of the line profile to the interpolated point on the red side of the
profile. By connecting the resulting series of midpoints, we obtain a curve
that is the bisector of the line. The ordinate axis on the bisector plots is
the radiation flux $F_{\lambda}{/}F_{\rm c}$ normalized to the continuum, and
the abscissa axis is the wavelength shifts of the measured profile points
calculated according to the above formula and converted to velocity units.

Since more and more blends appear in the spectra when proceeding to cooler
stars, the line profiles become more noisy, their bisectors are more
distorted, and the number of blendless lines decreases. For this reason,
reliable results cannot be obtained by analyzing individual bisectors in the
spectra of many stars. There is a practice to remove any noise by averaging
the bisectors. For example, it was shown [3, 13] that it is possible by
averaging bisectors to obtain a statistically reliable result by discarding
points that deviate sharply from the mean bisector. We applied this method as
well. All abrupt outliers were considered erroneous and removed from the
averaging or the top and/or bottom parts of the bisector were cut off if they
contained a large outlier. In such cases, there is inevitably some degree of
subjective judgment, so we acted with caution. The mean bisector was
calculated as the arithmetic mean of the positions of those bisectors that
contribute at each absorption depth with a step of 1% of the intensity of the
continuum. For each star, we averaged the individual bisectors in the Mg scale
and then smoothed the mean bisector.

##  3 Line bisectors in the solar flux spectrum

Individual and mean bisectors. We performed an analysis of solar line
bisectors to add the Sun to our sample of solar-type stars to ensure the
validity of our results. For this purpose, we used the solar atlas [33], that
represents the spectrum of the Moon-reflected solar flux obtained at HARPS in
2010 with a resolution $R\approx 120000$ in the spectral range from 476 to 585
nm and with a small gap in the range of 530–534 nm. We will refer to this
atlas as HARPS. This atlas was calibrated using an ideal laser frequency comb
(LFC) calibrator. The authors of this atlas claim that the wavelength solution
is the most accurate of all currently possible and requires no other
correction. We will use the HARPS atlas for the comparison with the results of
the analysis of stellar spectra obtained with the same spectral resolution.

In addition, three more high-resolution solar flux atlases obtained with FTS
(Fourier-Transform Spectrometers) were used. The HINKL FTS atlas [26] provides
convenient access to the KURUCZ atlas [31] with a resolution $R\approx
340500$, 521360, and 522900 for the wavelength ranges of 401.9–473.8,
473.8–576.5, 576.5–753.9 nm, respectively. The NECKL FTS atlas [35] provides a
high-quality spectrum with $R\approx 400000$ for the center of the disk and
averaged spectrum over the entire disk. The IAG FTS atlas [40] has a very high
resolution $R\approx 1000000$ and an accurate wavelength scale. Its systematic
deviation from the HARPS scale is approximately $-5\pm 5$ m/s and it has no
significant trend in the range of 480–580 nm. We will further use the IAG
atlas as a reference for the comparison with the other atlases of the solar
spectrum.

Figure 1: Individual and mean bisectors of iron lines measured in the absolute
wavelength scale of four spectral atlases of the solar flux HINKL, NECKL, IAG,
and HARPS: the thin lines are the bisectors of the Fe I lines, the dashed
lines are the bisectors of the Fe II lines, the bold lines are the mean
bisectors.

Figure 2: Mean bisectors of the lines in the solar flux spectrum (a) in the
absolute scale and (b) in the Mg-scale according to the observations from four
atlases: IAG (solid line), HINKL (dash-dotted line), NECKL (dotted line), and
HARPS (dashed line).

We have measured bisectors for 61 lines. All the bisectors of these lines, as
well as the mean bisector, are shown in Fig. 1 in the absolute scales of each
atlas. As can be seen, the range of blue shifts is the same for the IAG and
NECKL atlases, while that for HARPS is narrower, and it is significantly
broadened and shifted towards the blue part of the spectrum for the HINKL
atlas. Figure 2a shows the mean bisectors of the four atlases in absolute
scales. The average convective velocity was determined from the span of the
mean bisector. For NECKL and IAG, it is $-418$ and $-423$ m/s, respectively.
For the HARPS atlas, it is slightly lower ($-395$ m/s) due to the lower
spectral resolution. The lower-resolution solar spectrum weakened by the
reflection from the Moon shows smoother line profiles with less pronounced
asymmetry. The authors of the HARPS spectrum provided the solar radiation flux
that was not normalized to the continuum, so we used the local continuum for
each individual line. This may partially affect the shape of the bisector near
the continuum. For the HINKL atlas, the bisector span turned out to be the
largest and, accordingly, the average convective velocity turned out to be
large ($-636$ m/s). The shape of this bisector is also different from the IAG.
The bend in the upper part of the bisector is smaller due to increased blue
shifts of the weakest lines with wavelengths greater than 600 nm. Based on the
comparison results, we came to the conclusion that the wavelength scales in
the IAG, NECKL, and HARPS atlases nearly coincide within the analysis errors
and are quite accurate. The average convective velocity in the photosphere of
the Sun as a star measured from the data of these atlases is approximately
$-420$ m/s. There appears to be an issue with the wavelength scale in the
HINKL atlas, which will be discussed below.

Errors of the absolute scales of solar atlases. We compared the absolute
shifts of the mean bisectors of each atlas with the IAG and, thus, estimated
the error of the absolute scales in the wavelength range from 450 to 650 nm.
The NECKL and HARPS atlases have an insignificant error of approximately $-10$
m/s (Fig. 2a), while the error in the HINKL atlas is $-240$ m/s. This infers
that the absolute scales of the IAG, NECKL, and HARPS atlases can be
considered to be reliable, while the HINKL scale cannot. The analysis of the
shifts of individual bisectors showed that the scale shift in the HINKL atlas
relative to IAG

Figure 3: Absolute convective shifts of the iron lines in the solar flux
spectrum according to the IAG, HINKL, NECKL, and HARPS atlases depending on
the wavelength (the squares show the line shifts from the IAG atlas for
comparison).

Figure 4: Mean bisectors of the lines according to the data of the IAG, HINKL,
NECKL, and HARPS atlases in the absolute scales (solid lines) and Mg scales
(dashed lines).

depends on the line wavelength. Figure 3 demonstrates this dependence for the
HINKL atlas, while the IAG, NECKL, and HARPS atlases show only an
insignificant trend, which is a consequence of this line selection. In the
HINKL atlas, blue line shifts increase with wavelength. The greater the
wavelength, the greater the bisector shift. The difference between the shifts
in HINKL and IAG is approximately $-100$, $-220$, and $-330$ m/s for 450, 550,
and 650 nm, respectively. The growth of the blue shifts with wavelength
contradicts the physical sense. Just the opposite should be the case, since
opacity increases with wavelength, the contrast between the granules and
intergranules decreases, and the blue shift should decrease. As shown in [25],
there is hardly any dependence of blue shifts on wavelength for lines of
different strengths. However, if we select lines of equal strength, the blue
shifts of these lines decrease with wavelength. That being said, weak lines
show less dependence than strong lines. We used lines of different strengths,
so there should be no dependence on the wavelength in our case. This suggests
that the wavelength scale of the spectrum in the HINKL atlas is, apparently,
not accurate enough. It was noted in [20] that the zero point in the HINKL
atlas was reset, apparently in order toalign the average positions of the Fe I
lines with laboratory values [34]. According to the data presented [20], the
shift between HINKL and IAG in the range 600–630 nm is $-336$ m/s. This agrees
with our estimates for that particular range.

Verification of the Mg scale. Using the solar data, we can verify the Mg scale
plotted with respect to the Mg line. Since all the solar atlases have absolute
scales, we can trace possible uncertainties in the Mg scale. Figure 4 shows
the mean bisectors in this scale (dashed line) and the bisectors in the
absolute scale for each atlas (solid line). The bisectors in the Mg scale are
shifted by the corresponding absolute shift of the Mg line in each atlas, the
value of which depends on the calibration accuracy of the atlas. In absolute
scales, this shift is $-204$ m/s (HINKL), 64 m/s (IAG), 60 m/s (NECKL), and 61
m/s (HARPS). The close values obtained for three atlases allow us to conclude
that the Mg line in the solar spectrum has a small red shift of $\approx 60$
m/s. When comparing the mean bisectors of four atlases in the Mg scale (Fig.
2b), the difference between HINKL and IAG is significantly reduced. Almost all
the mean bisectors are in satisfactory agreement within the analysis error.
Therefore, we can conclude that the use of the Mg line for constructing the
wavelength scale allows reducing possible errors in the central wavelengths of
the lines but introduces an additional error equal to the intrinsic shift of
the Mg line core.

Figure 5: (a) Profile of the Mg line core according to the IAG atlas (dashed
line) and the profile synthesized using the FALC model [16] (solid line). (b)
Profile of effective formation heights calculated from depression functions
(solid line) as well as a profile of heights where the optical depth in the
line is equal to one (dashed line).

It is interesting to note that this intrinsic Mg line shift turned out to be
red. In general, the reddening effect of shifts of very strong lines was
previously known both for solar lines [43] and stellar lines [3]. It is
possible that this small red shift is caused in part by convective currents
penetrating into the region between the upper photosphere and lower
chromosphere. Doppler images obtained in the core of this line show that the
granulation structure still exists, but it differs markedly from the
granulation pattern seen in the wings of this line [41]. According to [8, 30],
the structure of granulation is preserved up to heights of approximately
500–700 km, while its characteristics vary with height. The temperature
inversion in granules and intergranules almost always occurs, on average, at
heights of 200–300 km, while the inversion of vertical velocities occurs in
most cases at heights $>500$ km. At heights of approximately 700 km, 20% of
the granulation structures are still observed. These are the heights between
the upper photosphere and lower chromosphere where the core of the Mg line
should form. According to [37], the formation height of the core of this line
calculated for the center of the solar disk is approximately 600 km. The
maximum sensitivity to temperature variations is manifested at a height of
approximately 550 km, and that to vertical velocity variations is at
approximately 800 km. We calculated the formation height of the Mg line core
in the solar flux spectrum with the aid of the SPANSAT software [17] using the
depression contribution functions [24]. Figure 5 shows the observed and
calculated profiles of the Mg line core and the profiles of effective heights.
As can be seen, in the range $F_{\lambda}{/}F_{\rm c}=0.1$–0.2 , the core
forms at heights of 600–800 km, i.e., in the region where granulation
structures are weakened but still visible on Doppler images. It can be assumed
that the altered structure of granulation in the lower chromosphere is one of
the causes for the reddening of shifts in the cores of strong lines. It is
possible that the red contributions to the line from the downward flows
slightly prevail over the blue contributions from the upward flows, and the
shift of the core of the velocity-sensitive Mg line may be red.

##  4 Line bisectors in stellar spectra

Bisector noises and errors. Individual bisectors in the spectra of the stars
are distorted by noise to a greater or lesser extent. Note that a slight
tortuosity of the bisectors is visible even at very high resolution in the IAG
solar flux atlas (Fig. 1). The tortuous character is even more pronounced in
the stars with $v\sin i\geq 3$ km/s. In principle, radiation at any given
point of the line profile comes from a wide range of depths in the photosphere
and total disk. This should lead to averaging and smoothing of all possible
oscillations and to a smoother shape of the bisectors. However, we see large
observational noise in the line profiles and their bisectors, which does not
allow obtaining reliable results for individual bisectors. Small noise of
rotationally broadened profiles leads to even larger errors in the bisectors.
In addition, if we proceed from the fact that the tortuosity is more
pronounced in stars with $v\sin i>5$ km/s, it can be assumed that another
cause for this may be the redistribution of Doppler shifts caused by rotation
in the surface regions near the limb, as was noted in [22]. In colder, slowly
rotating stars, one can also see the chaotic appearance of the bisectors. In
addition to noise, there is uncertainty with blends, which also introduces
additional scatter and affects the reliability of the results. Averaging the
bisectors makes it possible to remove noise; for this reason, all the results
of the analysis of stellar spectra were obtained based on the averaged
bisectors.

Figure 6: (a) Individual bisectors are shown on the Mg scale for the Fe I
(thin solid lines) and Fe II (dashed lines) lines and the mean bisector (thick
solid lines) for the star HD 102361; (b) the same bisectors taking into
account the correction for wavelength dispersion.

Figure 7: Convective line shifts are shown on the Mg scale in the spectrum of
the star HD 102361 depending on the wavelength (crosses) and shifts of the
same lines taking into account the correction for wavelength dispersion
(squares).

For stars, we will consider all bisectors on the Mg scale. This means that
their position has already been corrected for the gravitational displacement
and radial velocity of the star, and it corresponds to the convective shift of
the line core with an error equal to the convective shift of the Mg line in
this star. The magnitude of this convective shift is unknown, but it should be
presumably small within the accuracy of this analysis for solar-type stars.

Figure 8: Individual bisectors of the Fe I (thin solid lines) and Fe II
(dashed lines) lines as well as the averaged bisectors (bold solid lines) in
the Mg scale for solar-type stars with the indicated main parameters $T_{\rm
eff}{/}\log g/{\rm[N/H]}{/}v\sin i$.

Figure 9: Changes in the shape of the mean bisectors with the effective
temperature of the star. The horizontal arrangement of the bisectors on the
shift scale is arbitrary but ordered by the effective temperature. Star
parameters $T_{\rm eff}{/}\log g/{\rm[N/H]}{/}v\sin i$, as well as the type of
star, are noted under each bisector.

Line bisectors in the spectrum of the star HD 102361 F8V. Our preliminary
analysis of individual bisectors showed a very wide range of bisector shifts
in the spectrum of HD 102361, which is sharply different from other stars. In
addition, the magnitude of the shifts depends on the line wavelength (Fig.
6a). Blue shifts reach approximately $-2000$ m/s in the 538–643 nm range, and
red shifts reach almost 500 m/s in the 441–460 nm range, i.e., the line shifts
become bluer with increasing wavelength. This is clearly seen in Fig. 7, which
shows the dependence of the shifts of the individual lines on the wavelength.
This dependence resembles the dependence found in the HINKL atlas (see Fig.
3). In this regard, we introduced a dispersion correction to the wavelength
scale for this star. Figure 7 shows the corrected line shifts, and Fig. 6b
shows the positions of the bisectors of these lines in the corrected shift
scale for the star HD 102361.

Changes in the bisectors with star temperature. Figure 8 shows the bisectors
of the lines of all the stars and the Sun as a star (HARPS atlas) in order of
decreasing effective temperature from left to right and from top to bottom on
the same scale. We see no significant differences in the bisectors of the Fe I
and Fe II lines, although it is known that the bisectors of strong Fe II lines
for the Sun have a more pronounced C shape than the bisectors of Fe I lines of
the same strength. Apparently, this is a consequence of the lower resolution,
higher noise, and a small number of Fe II lines. If we consider the bisector
of the strong line from the top near the continuum to the bottom near the
core, we can get an idea of the variation in the total velocity of upward and
downward flows from bottom to top in the photosphere of a particular star. Any
point on the bisector reflects the average velocity of convective flows at
photospheric heights from which radiation is emitted at the corresponding
point of the line profile. To more accurately tie this velocity to the height,
it is necessary to calculate the profile of the line formation heights.
However, even one bisector can tell a lot.

Figure 8 shows that, in general, the individual bisectors of the weakest lines
are most blue-shifted on the shift scale, since these lines form in deep
layers, where the convective velocities of ascending granules are large. The
strong lines are less displaced because their cores form in the high layers of
the photosphere, where granulation is significantly weaker. The placement of
the bisector shifts on the scale from the weakest to the strongest lines shows
a certain range of convective velocities inherent in a particular star. The
width of this range increases with effective temperature and with decreasing
surface gravity and metallicity. We can say that this range of shifts is one
of the signatures of the efficiency of convection in the stellar photosphere.

The mean bisectors, as seen in Fig. 9, have a C shape for most stars, same as
in the case of the Sun, but with different bulge in the middle of the
bisector, the size of which depends on the main parameters of the star. Among
all the bisectors, the bisectors for the stars HD 189627 F7V, HD 102361 F8V,
and HD 147873 G1V with a rotation velocity $v\sin i\geq 5$ km/s stand out the
most. The shape of these three bisectors resembles a greatly elongated top of
the letter C or even a slash symbol (/). This shape is caused by the effects
of rotation on the line profiles. As is known from [22], the lower part of the
bisector compresses or contracts with an increase in the rotation velocity and
deviates more and more towards the blue part of the spectrum; for this reason,
the shifts of the line cores in such stars become bluer.

Mean bisectors differ in length. In the spectra of hotter stars, absorption
lines are significantly weakened as compared to cooler stars; their bisectors
become shorter and their cores more displaced since they form in deeper
layers. As a rule, the hotter the star, the faster it rotates. The length of
the mean bisector depends not only on temperature but also on the star’s
rotation velocity and metallicity. This is especially noticeable for stars
with large $v\sin i$, e.g., the mean bisector in the star HD 189627 is much
shorter than the others and the shift of its lower part is bluer.

Figure 10: Average convective velocities derived from the bisector span
depending on the effective temperature, gravity, metallicity, age of the star,
rotation velocity, and micro- and macroturbulence velocities (the dashed lines
show the linear approximation).

An important characteristic of the bisector is its span or shift of the bluest
point. It can be seen from Fig. 9 that the span depends on the temperature.
The higher the effective temperature, the greater the curvature or span of the
bisector and the greater the average convective velocity. In addition, the
span depends on the surface gravity, which tends to weaken the motion of the
ascending granules and, due to this, the span decreases in stars with higher
gravity. In metal-rich stars, the opacity of the photosphere increases, the
motion of the ascending granules slows down, and the span becomes smaller. For
example, in the stars HD 93849 and HD 158469 with close $T_{\rm eff}$, $\log
g$, and $v\sin i$ values, the mean bisectors differ due to different
metallicities. In this regard, we cannot expect a clear dependence of the span
of the bisectors on temperature in our sample of stars, since the other
parameters vary from star to star, albeit not much. In general, we can say
that the span of the bisector depends on the structure of the atmosphere,
which is determined by the main parameters of the star.

The height of the span of the mean bisector is also a characteristic feature.
As can be seen from Fig. 9a, the hotter the star, the smaller the span height.
This means that effective granulation in hot stars extends into the higher
layers of the photosphere. In colder stars (Fig. 9c), the span height is large
and this indicates that the effective ascent of matter in the granules occurs
in deeper layers than in the case of hot stars. Hence, it follows that the
higher the temperature, the higher the convective flows of hot matter rise
into the atmosphere.

Convective velocities in stellar photospheres. We measured the shift of the
bluest point on the mean bisector, i.e., the span of the bisector, and
obtained the average convective velocity in the photosphere for each star. The
results are presented in Table 1 and Fig. 10 depending on the parameters of
the star. It is clearly seen that the closest correlation exists between the
average convective velocity or the ascent velocity of granules and the
effective temperature of the star. The rest of the dependences shown in Fig.
10 have a greater scatter of values; however, a certain tendency is observed
everywhere in accordance with the physical sense. We would like to note the
existence of a correlation between the convective velocity and the micro- and
macroturbulence parameters. It is not without reason that the macroturbulence
velocity is often called the first signature of granulation in a star’s
atmosphere. The higher the ascent velocity of granules, the more efficient the
granulation and the stronger the micro- and macroturbulence in stars, i.e.,
the stronger the dispersion of convective velocities. The influence of the
star’s rotation on the line bisectors is also confirmed. The higher the
rotation velocity, the higher the ascent velocity of the granules. The
correlation between the mean convective velocity and the age of the star
should also be noted. It is obvious that granulation efficiency should
decrease with age. Thus, none of the dependences presented in Fig. 10
contradict the physical sense and they confirm the reliability of
observational data on stellar asymmetry.

##  5 Conclusions

In this study, we have analyzed the asymmetry of lines in the spectra of
solar-type stars obtained with the HARPS spectrograph with a resolution of
approximately 120000 and a signal-to-noise ratio above 100. Bisectors, which
are chains of convective blue shifts in the photospheric layers where the
corresponding points of the spectral line profile are formed, were measured
for the selected lines. The main findings are as follows.

In solar-type stars, the bisectors of very weak lines show blue shifts in the
deepest layers of the photosphere, and their shape resembles the upper part of
the letter C. The bisectors of strong lines carry information about the
convective velocities of almost all the photospheric layers, and they have a
shape similar to the letter C for most stars, which becomes more like the /
symbol if the rotation velocity of the star $v\sin i\geq 5$ km/s.

With an increase in the effective temperature, the curvature of the bisector
systematically increases. This is confirmed by the clear dependence of the
magnitude of the bisector span on the effective temperature. The mean
convective velocities derived from the magnitude of the bisector span increase
from $-150$ to $-700$ m/s in stars with an effective temperature of 4800 to
6200 K, respectively. With increasing surface gravity and metallicity,
convective velocities become lower. As the star ages, convective velocities
decrease, as expected. The photospheric macroturbulent velocity, which is the
dispersion of convective velocities, directly depends on the magnitude of
these velocities.

The line bisectors in the solar flux spectrum presented in the HARPS atlas fit
well into the general picture of the bisectors of solar-type stars. The
comparative analysis of the bisectors measured in the absolute scales of four
solar flux atlases showed that the wavelength scale in the FTS atlases IAG,
NECKL, and the HARPS atlas is set with an accuracy of $\pm 10$ m/s in the
range of 450–650 nm. The wavelength scale in the HINKL atlas is shifted
relative to the IAG atlas by an average of $-240$ m/s. In addition, the HINKL
scale shift changes with wavelength from $-100$ to $-330$ m/s in the range of
450–650 nm, respectively. In this regard, it is undesirable to use the HINKL
atlas to determine the convective blue line shifts.

The above conclusions confirm the known facts regarding the nature of the line
bisectors in the spectra of solar-type stars and show that, in general terms,
the stellar line bisectors are similar to the solar line bisectors but with
their own characteristic features. When we compare the shape of the bisectors
and the magnitude of the blue shifts of the lines of different strengths in
the stellar spectrum with solar data, it is possible to obtain preliminary
estimates of the average convective velocity, height of penetrating
convection, rotation velocity, and macroturbulent velocity in the stellar
photosphere. By the magnitude of the bisector span, it is possible to estimate
the layers of the maximum effect of granulation on the profiles of spectral
lines and the energy of granulation in the photosphere.

Observation noise, blends, and insufficiently high spectral resolution distort
the shape of the bisectors. This affects the bisectors of rapidly rotating
stars to a greater extent. Due to these reasons, the individual bisectors are
not reliable enough, so all the conclusions were drawn on the basis of the
mean bisectors. The mean convective velocities derived from the bisector span
are only the lower limit since insufficiently high resolution reduces this
span. In addition, convective velocities in the scale of shifts relative to
the Mg line may contain the Mg line shift itself if it turns out to be
nonzero. The values of the obtained average convective velocities do not
exceed the accuracy of laboratory wavelengths, which is 30–60 m/s at best. To
improve the analysis results, it is necessary to have more accurate laboratory
wavelengths and stellar spectra with higher resolution and signal-to-noise
ratio.

ACKNOWLEDGMENTS

I am sincerely grateful to Ya. Pavlenko and A. Ivanyuk for providing the
observed spectra of stars and would also like to thank the reviewer for
important remarks.

## References

  * [1] Allende Prieto C., Asplund M., Garcia Lopez R. J., Lambert D. L. Signatures of Convection in the Spectrum of Procyon: Fundamental Parameters and Iron Abundance. ApJ 2002. 567. P. 544–565.
  * [2] Allende Prieto C., Garcia Lopez R. J. Fe i line shifts in the optical spectrum of the Sun. A&A Suppl. 1998. 129. P. 41–44.
  * [3] Allende Prieto C., Garcia Lopez R.J., Lambert D. L., Gustafsson B. Spectroscopic Observations of Convective Patterns in the Atmospheres of Metal-poor Stars. ApJ 1999. 526. P. 991–1000.
  * [4] Allende Prieto C., Koesterke L., Ludwig H. -G., Freytag B., Caffau E. Convective line shifts for the GAIA RVS from the CIFIST 3D model atmosphere grid. A&A 2013. 550. id.A103, 13 pp.
  * [5] Asplund M., Nordlund A., Trampedach R., Allende Prieto C., Stein R. F. Line formation in solar granulation. I. Fe line shapes, shifts and asymmetries. A&A 2000. 359. P. 729–742.
  * [6] Atroshchenko I. N., Gadun A. S. Three-dimensional hydrodynamic models of solar granulation and their application to a spectral analysis problem. A&A 1994. 291. P. 635–656.
  * [7] Balthasar, H. Asymmetries and Wavelengths of Solar Spectral Lines and the Solar Rotation Determined from Fourier-Transform Spectra. Sol. Phys. 1984. 93. P. 219–241.
  * [8] Baran O. A., Stodilka M. I. Convection structure in the solar photosphere at granulation and mesogranulation scales. Kinematics and Physics of Celestial Bodies 2015. 31. P.65–72.
  * [9] Beeck B., Cameron R. H., Reiners A., Schussler M. Three-dimensional simulations of near-surface convection in main-sequence stars. II. Properties of granulation and spectral lines. A&A 2013. 558. id.A49, 18 pp.
  * [10] Brandt P. N., Gadun A. S., Sheminova V. A. Absolute shifts of Fe I and Fe II lines in solar active regions (disk center). Kinem. Phys. Cel. Bod. 1997\. 13. N5, P. 65–74.
  * [11] Dravins D. Stellar granulation. II. Stellar photospheric line asymmetries. A&A 1987. 172. P. 211–224.
  * [12] Dravins D. Stellar Surface Convection, Line Asymmetries, and Wavelength Shifts ASP Conference Series, IAU Colloquium 170. Eds. J. B. Hearnshaw and C. D. 1999. 185. P. 268–276.
  * [13] Dravins D. Ultimate information content in solar and stellar spectra. Photospheric line asymmetries and wavelength shifts. A&A 2008. 492. P. 199–213.
  * [14] Dravins D., Lindegren L., Nordlund A. Solar granulation – Influence of convection on spectral line asymmetries and wavelength shifts. A&A 1981. 96. P. 345–364.
  * [15] Dravins D., Nordlund A. Stellar granulation. V. Synthetic spectral lines in disk-integrated starlight. A&A 1990. 228. P. 203–217.
  * [16] Fontenla J. M., Avrett E. H., Loeser R. Energy Balance in the Solar Transition Region. III. Helium Emission in Hydrostatic, Constant-Abundance Models with Diffusion. ApJ 1993. 406. P. 319–345.
  * [17] Gadun A. S., Sheminova V. A. SPANSAT: the Program for LTE Calculations of Absorption Line Profiles in Stellar Atmospheres. Preprint No. ITF-88-87P (Institute for Theoretical Physics of the Ukrainian SSR Academy of Sciences, Kiev, 1988).
  * [18] Gray D. F. Observations of spectral line asymmetries and convective velocities in F, G and K stars. ApJ 1982. 255. P. 200–209.
  * [19] Gray D. F. Shapes of Spectral Line Bisectors for Cool Stars. Publications of the Astronomical Society of the Pacific 2005. 117. P. 711–720.
  * [20] Gray D. F. Solar-flux Line-broadening Analysis. ApJ 2018. 857. Id. 139. 8 p.
  * [21] Gray D.F., Nagel T. The Granulation Boundary in the H-R Diagram. ApJ 1989. 341. P. 421–426.
  * [22] Gray D. F., Toner C. G. Inferred properties of stellar granulation. ApJ 1985. 97. P. 543–550.
  * [23] Gray D. F., Toner C. G. The remarkable spectral line asymmetries of F and G Ib supergiant stars. Publications of the Astronomical Society of the Pacific 1986. 98. P. 499–503.
  * [24] Gurtovenko E. A., Sheminova V. A. Formation depths of Fraunhofer lines // eprint arXiv:1505.00975
  * [25] Hamilton D., Lester J. B. A Technique for the Study of Stellar Convection: The Visible Solar Flux Spectrum. The Publications of the Astronomical Society of the Pacific 1999. 763. P.1132–1143.
  * [26] Hinkle K., Wallace L. The Spectrum of Arcturus from the Infrared through the Ultraviolet in Astronomical Society of the Pacific Conference. Series. 336. Cosmic Abundances as Records of Stellar Evolution and Nucleosynthesis, eds. T. G. Barnes, F. N. Bash. 2005. 321 p.
  * [27] Ivanyuk O. M., Jenkins J. S., Pavlenko Ya. V., et. al. The metal-rich abundance pattern – spectroscopic properties and abundances for 107 main-sequence stars. MNRAS. 2017. 468. P. 4151–4169.
  * [28] Jenkins J. S., Jones H. R. A., Gozdziewski K. First results from the Calan-Hertfordshire Extrasolar Planet Search: exoplanets and the discovery of an eccentric brown dwarf in the desert. MNRAS. 2009. 398. P. 911–917.
  * [29] Kostik R. I., Orlova T. V. On the asymmetry of selected Fraunhofer lines. Sol. Phys. 1977. 53. P. 353–358.
  * [30] Kostyk R. I., Shchukina N. G. Fine Structure of Convective Motions in the Solar Photosphere: Observations and Theory. Astronomy Reports 2004. 48. P. 769–780.
  * [31] Kurucz R. L., Furenlid I., Brault J., Testerman L. Solar flux atlas from 296 to 1300 nm. National Solar Observatory Atlas, Sunspot, New Mexico, National Solar Observatory, 1984.
  * [32] Livingston W. C. Magnetic fields and convection – New observations. Solar and stellar magnetic fields: Origins and coronal effects; Proceedings of the Symposium, Zurich, Switzerland. 1983. P. 149–152.
  * [33] Molaro P., Esposito M., Monai S., Lo Curto G., et al. A frequency comb calibrated solar atlas. A&A 2013. 560. id.A61, 9 pp.
  * [34] Nave G., Johansson S., Learner R. C. M., Thorne A. P., Brault J. W. A New Multiplet Table for Fe i. ApJS 1994. 94. p.221.
  * [35] Neckel H. Announcement Spectal Atlas of Solar Absolute Disk-averaged and Disk-Center Intensity from 3290 to 12510 A (Brault and Neckel, 1987). Sol. Phys. 1999. 184. P. 421–422.
  * [36] Pavlenko. Y. V., Kaminsky B. M., Jenkins, J. S., Ivanyuk O. M., Jones, H. R. A., Lyubchik Y. P. Masses, oxygen, and carbon abundances in CHEPS dwarf stars. A&A 2019. 621. id.A112, 13 pp.
  * [37] Quintero Noda C., Uitenbroek H., Carlsson M., Orozco Suarez D., Katsukawa Y., et al. Study of the polarization produced by the Zeeman effect in the solar Mg I b lines. MNRAS. 2018. 481. P. 5675–5686.
  * [38] Ramirez I., Allende Prieto C., Lambert D. L. Granulation in K-type dwarf stars. I. Spectroscopic observations. A&A 2008. 492. P. 841–855.
  * [39] Ramirez I., Allende Prieto C., Koesterke L., Lambert D. L., Asplund M. Granulation in K-type dwarf stars. II. Hydrodynamic simulations and 3D spectrum synthesis. A&A 2009. 501. P. 1087–1101.
  * [40] Reiners A., Mrotzek N., Lemke U., Hinrichs J., Reinsch K. The IAG solar flux atlas: Accurate wavelengths and absolute convective blueshift in standard solar spectra. A&A 2016. 587. id.A65, 8 pp.
  * [41] Rutten R. J., Leenaarts J., Rouppe van der Voort L. H. M., de Wijn A. G., Carlsson, M., Hansteen V. Quiet-Sun imaging asymmetries in Na I D1 compared with other strong Fraunhofer lines. A&A 2011. 531. id.A17, 16 pp.
  * [42] Sheminova V. A. Turbulence and Rotation in Solar-Type Stars. Kinematics and Physics of Celestial Bodies 2019. 35. P.129–142.
  * [43] Sheminova V. A., Gadun A. S. Convective shifts of iron lines in the solar photosphere. Kinematika i Fizika Nebesnykh Tel 2002. 18. N1, P.18–32.
  * [44] Wallace L., Hinkle K. H., Livingston, W. C., Davis S. P. An Optical and Near-infrared (2958–9250 A) Solar Flux Atlas. ApJS 2011. 1. id. 6, 8 pp.