#  Equation Parsing : Mapping Sentences to Grounded Equations

Subhro Roy Shyam Upadhyay Dan Roth
University of Illinois, Urbana Champaign
{sroy9, upadhya3<EMAIL_ADDRESS>

###### Abstract

Identifying mathematical relations expressed in text is essential to
understanding a broad range of natural language text from election reports, to
financial news, to sport commentaries to mathematical word problems. This
paper focuses on identifying and understanding mathematical relations
described within a single sentence. We introduce the problem of Equation
Parsing – given a sentence, identify noun phrases which represent variables,
and generate the mathematical equation expressing the relation described in
the sentence. We introduce the notion of projective equation parsing and
provide an efficient algorithm to parse text to projective equations. Our
system makes use of a high precision lexicon of mathematical expressions and a
pipeline of structured predictors, and generates correct equations in $70\%$
of the cases. In $60\%$ of the time, it also identifies the correct noun
phrase $\rightarrow$ variables mapping, significantly outperforming baselines.
We also release a new annotated dataset for task evaluation.

##  1 Introduction

Understanding text often involves reasoning with respect to quantities
mentioned in it. Understanding the news article statement in Example 1
requires identifying relevant entities and the mathematical relations
expressed among them in text, and determining how to compose them. Similarly,
solving a math word problem with a sentence like Example 2, requires realizing
that it deals with a single number, knowing the meaning of “difference” and
composing the right equation – “25” needs to be subtracted from a number only
after it is multiplied by $3$.

Example 1 Emanuel’s campaign contributions total three times those of his
opponents put together.
---
Example 2 Twice a number equals 25 less than triple the same number.
Example 3 Flying with the wind , a bird was able to make 150 kilometers per
hour.
Example 4 The sum of two numbers is 80.
Example 5 There are 54 5-dollar and 10-dollar notes.

As a first step towards understanding such relations, we introduce the
Equation Parsing task - given a sentence expressing a mathematical relation,
the goal is to generate an equation representing the relation, and to map the
variables in the equation to their corresponding noun phrases. To keep the
problem tractable, in this paper we restrict the final output equation form to
have at most two (possibly coreferent) variables, and assume that each
quantity mentioned in the sentence can be used at most once in the final
equation.111We empirically found that around 97% of sentences describing a
relation have this property. In example 1, the gold output of an equation
parse should be $V_{1}=3\times V_{2}$, with $V_{1}=\text{``Emanuel's campaign
contributions''}$ and $V_{2}=\text{``those of his opponents put together''}$.

The task can be seen as a form of semantic parsing [Goldwasser and Roth, 2011,
Kwiatkowski et al., 2013] where instead of mapping a sentence to a logical
form, we want to map it to an equation. However, there are some key
differences that make this problem very challenging in ways that differ from
the “standard” semantic parsing. In Equation Parsing, not all the components
of the sentence are mapped to the final equation. There is a need to identify
noun phrases that correspond to variables in the relations and determine that
some are irrelevant and can be dropped. Moreover, in difference from semantic
parsing into logical forms, in Equation Parsing multiple phrases in the text
could correspond to the same variable, and identical phrases in the text could
correspond to multiple variables.

We call the problem of mapping noun phrases to variables the problem of
grounding variables. Grounding is challenging for various reasons, key among
them are that: (i) The text often does not mention “variables” explicitly,
e.g., the sentence in example 3 describes a mathematical relation between the
speed of bird and the speed of wind, without mentioning “speed” explicitly.
(ii) Sometimes, multiple noun phrases could refer to the same variable. For
instance, in example 2, both “a number” and “the same number” refer to the
same variable. On the other hand, the same noun phrase might refer to multiple
variables, as in example 4, where the noun phrase “two numbers” refer to two
variables.

In addition, the task involves deciding which of the quantities identified in
the sentence are relevant to the final equation generation. In example 5, both
“5” and “10” are not relevant for the final equation “$V_{1}+V_{2}=54$”.
Finally, the equation needs to be constructed from a list of relevant
quantities and grounded variables. Overall, the output space becomes
exponential in the number of quantities mentioned in the sentence.

Determining the final equation that corresponds to the text is an inference
step over a very large space. To address this, we define the concept of
“projectivity” - a condition where the final equation can be generated by
combining adjacent numbers or variables, and show that most sentences
expressing mathematical relations exhibit the projectivity property. Finally,
we restrict our inference procedure to only search over equations which have
this property.

Our approach builds on a pipeline of structured predictors that identify
irrelevant quantities, recognize coreferent variables, and, finally, generate
equations. We also leverage a high precision lexicon of mathematical
expressions and develop a greedy lexicon matching strategy to guide inference.
We discuss and exemplify the advantages of this approach and, in particular,
explain where the “standard” NLP pipeline fails to support equation parsing,
and necessitates the new approach proposed here. Another contribution of this
work is the development of a new annotated data set for the task of equation
parsing. We evaluate our method on this dataset and show that our method
predicts the correct equation in $70\%$ of the cases and that in $60\%$ of the
time we also ground all variables correctly.

The next section presents a discussion of related work. Next we formally
describe the task of equation parsing. The following sections describe our
equation representation and the concept of projectivity, followed by the
description of our algorithm to generate the equations and variable groundings
from text. We conclude with experimental results.

##  2 Related Work

The work most related to this paper is [Madaan et al., 2016], which focuses on
extracting relation triples where one of the arguments is a number. In
contrast, our work deals with multiple variables and complex equations
involving them. There has been a lot of recent work in automatic math word
problem solving [Kushman et al., 2014, Roy et al., 2015, Hosseini et al.,
2014, Roy and Roth, 2015]. These solvers cannot handle sentences individually.
They require the input to be a complete math word problem, and even then, they
only focus on retrieving a set of answer values without mentioning what each
answer value corresponds to. Our work is also conceptually related to work on
semantic parsing – mapping natural language text to a formal meaning
representation [Wong and Mooney, 2007, Clarke et al., 2010, Cai and Yates,
2013, Kwiatkowski et al., 2013, Goldwasser and Roth, 2011]. However, as
mentioned earlier, there are some significant differences in the task
definition that necessitate the development of a new approach.

##  3 The Equation Parsing Task

Equation parsing takes as input a sentence $x$ describing a single
mathematical equation, comprising one or two variables and other quantities
mentioned in $x$. Let $N$ be the set of noun phrases in the sentence $x$. The
output of the task is the mathematical equation described in $x$, along with a
mapping of each variable in the equation to its corresponding noun phrase in
$N$. We refer to this mapping as the “grounding” of the variable; the noun
phrase represents what the variable stands for in the equation. Table 1 gives
an example of an input and output for the equation parsing of the text in
example 2. Since an equation can be written in various forms, we use the form
which most agrees with text, as our target output. So, for example 1, we will
choose $V_{1}=3\times V_{2}$ and not $V_{2}=V_{1}\div 3$. In cases where
several equation forms seem to be equally likely to be the target equation, we
randomly choose one of them, and keep this choice consistent across the
dataset.

The Equation Parsing Task
---
Input |  Twice a number equals 25 less than triple the same number.
Output |  $2\times V_{1}=(3\times V_{1})-25$ (Equation)
|  $V_{1}=\text{``a number''}$ (Grounding)
Table 1: Input and output for Equation Parsing

###  3.1 Equation Parse Representation

Figure 1: A sentence with its trigger list and equation tree. $-_{r}$
indicates subtraction with order $rl$.

In this section, we introduce an equation parse for a sentence. An equation
parse of a sentence $x$ is a pair $(T,E)$, where $T$ represents a set of
triggers extracted from $x$, and $E$ represents an equation tree formed with
the set $T$ as leaves. We now describe these terms in detail.

Trigger Given a sentence $x$ mentioning a mathematical relation, a trigger can
either be a quantity trigger expressed in $x$, or variable trigger which is a
noun phrase in $x$ corresponding to a variable. A quantity trigger is a tuple
$(q,s)$, where $q$ is the numeric value of the quantity mentioned in text, and
$s$ is the span of text from the sentence $x$ which refers to the quantity. A
variable trigger is a tuple $(l,s)$, where $l$ represents the label of the
variable, and $s$ represents the noun phrase representing the variable. For
example, for the sentence in Fig 1, the spans “Twice”, “25”, and “triple”
generate quantity triggers, whereas “a number” and “the same number” generate
variable triggers, with label $V_{1}$.

Trigger List The trigger list $T$ for a sentence $x$ contains one trigger for
each variable mention and each numeric value used in the final equation
expressed by the sentence $x$. The trigger list might consist of multiple
triggers having the same label, or extracted from the same span of text. In
the example sentence in Fig 1, the trigger list comprises two triggers having
the same label $V_{1}$. The final trigger list for the example in Fig 1 is
{($2$, “2”), ($V_{1}$, “a number”), ($25$, “25”), ($3$, “triple”), ($V_{1}$,
“the same number”)}. Note that there can be multiple valid trigger lists. In
our example, we could have chosen both variable triggers to point to the same
mention “a number”. Quantity triggers in the trigger list form the quantity
trigger list, and the variable triggers in trigger list form the variable
trigger list.

Notation |  Definition
---|---
Quantity Trigger |  Mention of a quantity in text
Variable Trigger |  Noun phrase coupled with variable label
Trigger |  Quantity or variable trigger
Quantity Trigger List |  List of quantity triggers, one for each number mention in equation
Variable Trigger List |  List of variable triggers, one for each variable mention in equation
Trigger List |  Union of quantity and variable trigger list
Equation Tree |  Binary tree representation of equation
$lc(n)$, $rc(n)$ |  Left and right child of node $n$
$\operatorname{\textsc{Expr}}(n)$ |  Expression represented by node $n$
$\operatorname{\odot}(n)$ |  Operation at node $n$
$\operatorname{\textsc{Order}}(n)$ |  Order of operation at node $n$
$\operatorname{\mathrm{Location}}(n)$ |  Character offset of trigger representing leaf node $n$
$\operatorname{\mathrm{Span-Start}}(n)$, $\operatorname{\mathrm{Span-End}}(n)$ |  Start and end character offsets of span covered by node $n$
Table 2: Summary of notations used in the paper

Equation Tree An equation tree of a sentence $x$ is a binary tree whose leaves
constitute the trigger list of $x$, and internal nodes (except the root) are
labeled with one of the following operations – addition, subtraction,
multiplication, division. In addition, for nodes which are labeled with
subtraction or division, we maintain a separate variable to determine order of
its children. The root of the tree is always labeled with the operation equal.

An equation tree is a natural representation for an equation. Each node $n$ in
an equation tree represents an expression $\operatorname{\textsc{Expr}}(n)$,
and the label of the parent node determines how the expressions of its
children are to be composed to construct its own expression. Let us denote the
label for a non-leaf node $n$ to be $\operatorname{\odot}(n)$, where
$\operatorname{\odot}(n)\in\\{+,-,\times,\div,=\\}$ and the order of a node
$n$’s children by $\operatorname{\textsc{Order}}(n)$ (defined only for
subtraction and division nodes), which takes values $lr$ (Left-Right) or $rl$
(Right-Left). For a leaf node $n$, the expression
$\operatorname{\textsc{Expr}}(n)$ represents the variable label, if $n$ is a
variable trigger, and the numeric value of the quantity, if it is a quantity
trigger. Finally, we use $lc(n)$ and $rc(n)$ to represent the left and right
child of node $n$, respectively. The equation represented by the tree can be
generated as follows. For all non-leaf nodes $n$, we have

$\operatorname{\textsc{Expr}}(n)=\\\
\begin{cases}\operatorname{\textsc{Expr}}(lc(n))\operatorname{\odot}(n)\operatorname{\textsc{Expr}}(rc(n))\\\
\qquad\mbox{if }\operatorname{\odot}(n)\in\\{+,\times,=\\}\\\
\operatorname{\textsc{Expr}}(lc(n))\operatorname{\odot}(n)\operatorname{\textsc{Expr}}(rc(n))\\\
\qquad\mbox{if
}\operatorname{\odot}(n)\in\\{-,\div\\}\wedge\operatorname{\textsc{Order}}(n)=lr\\\
\operatorname{\textsc{Expr}}(rc(n))\operatorname{\odot}(n)\operatorname{\textsc{Expr}}(lc(n))\\\
\qquad\mbox{if
}\operatorname{\odot}(n)\in\\{-,\div\\}\wedge\operatorname{\textsc{Order}}(n)=rl\\\
\end{cases}$ (1)

Given an equation tree $\operatorname{\mathcal{T}}$ of a sentence, the
equation represented by it is the expression generated by the root of
$\operatorname{\mathcal{T}}$ (following Equation 1). Referring to the equation
tree in Fig 1, the node marked “$-_{r}$” represents $(3\times V_{1})-25$, and
the root represents the full equation $2\times V_{1}=(3\times V_{1})-25$.

##  4 Projectivity

For each leaf $n$ of an equation tree $T$, we define a function
$\operatorname{\mathrm{Location}}(\cdot)$, to indicate the position of the
corresponding trigger in text. We also define for each node $n$ of equation
tree $T$, functions $\operatorname{\mathrm{Span-Start}}(n)$ and
$\operatorname{\mathrm{Span-End}}(n)$ to denote the minimum span of text
containing the leaves of the subtree rooted at $n$. We define them as follows:

$\operatorname{\mathrm{Span-Start}}(n)=\\\
\begin{cases}\operatorname{\mathrm{Location}}(n)\qquad\mbox{if $n$ is a
leaf}\\\ \min(\operatorname{\mathrm{Span-
Start}}(lc(n)),\operatorname{\mathrm{Span-Start}}(rc(n)))\\\
\qquad\qquad\qquad\qquad\mbox{otherwise}\end{cases}$ (2)
$\operatorname{\mathrm{Span-End}}(n)=\\\
\begin{cases}\operatorname{\mathrm{Location}}(n)\qquad\mbox{if $n$ is a
leaf}\\\ \max(\operatorname{\mathrm{Span-
End}}(lc(n)),\operatorname{\mathrm{Span-End}}(rc(n)))\\\
\qquad\qquad\qquad\qquad\mbox{otherwise}\end{cases}$ (3)

An equation tree $T$ is called projective iff for every node $n$ of $T$,
either $\operatorname{\mathrm{Span-End}}(lc(n))\leq\operatorname{\mathrm{Span-
Start}}(rc(n))$ or $\operatorname{\mathrm{Span-
End}}(rc(n))\leq\operatorname{\mathrm{Span-Start}}(lc(n))$. In other words,
the span of the left child and the right child cannot intersect in a
projective equation tree222This is more general than the definition of
projective trees used in dependency parsing [McDonald et al., 2005]..

The key observation, as our corpus analysis indicates, is that for most
sentences, there exists a trigger list, such that the equation tree
representing the relation in the sentence is projective. However this might
involve mapping two mentions of the same variable to different noun phrases.
Figure 1 shows an example of a projective equation tree, which requires
different mentions of $V_{1}$ to be mapped to different noun phrases. If we
had mapped both mentions of $V_{1}$ to same noun phrase “a number”, the
resulting equation tree would not have been projective. We collected $385$
sentences which represent an equation with one or two mentions of variables,
and each number in the sentence used at most once in the equation. We found
that only one sentence among these could not generate a projective equation
tree. (See Section 6.1 for details on dataset creation). Therefore, we develop
an algorithmic approach for predicting projective equation trees, and show
empirically that it compares favourably with ones which do not make the
projective assumption.

##  5 Predicting Equation Parse

Equation parsing of a sentence involves predicting three components – Quantity
Trigger List, Variable Trigger List and Equation Tree. We develop three
structured prediction modules to predict each of the above components.

All our prediction modules take a similar form: given input $x$ and output
$y$, we learn a scoring function $f_{w}(x,y)$, which scores how likely is the
output $y$ given input $x$. The scoring function $f_{w}(x,y)$ is linear,
$f_{w}(y)=w^{T}\phi(x,y)$, where $\phi(x,y)$ is a feature vector extracted
from $x$ and $y$. The inference problem, that is, the prediction $y^{*}$ for
an input $x$ is then: $y^{*}=\arg\max_{y\in\mathcal{Y}}f_{w}(y)$, where
$\mathcal{Y}$ is the set of all allowed values of $y$.

###  5.1 Predicting Quantity Trigger List

Given input text and the quantities mentioned in it, the role of this step is
to identify , for each quantity in the text, whether it should be part of the
final equation. For instance, in example 5 in Section 1, both “5” and “10” are
not relevant for the final equation “$V_{1}+V_{2}=54$”. Similarly, in example
4, the number “two” is irrelevant for the equation “$V_{1}+V_{2}=80$”.

We define for each quantity $q$ in the sentence, a boolean value
$\operatorname{\mathrm{Relevance}}(q)$, which is set to $true$ if $q$ is
relevant for the final equation, and to $false$ otherwise. For the structured
classification, the input $x$ is the sentence along with a set of recognized
quantities mentioned in it, and the output $y$ is the relevance values for all
quantities in the sentence. We empirically found that predicting all relevance
values jointly performs better than having a binary classifier predict each
one separately. The feature function $\phi(x,y)$ used for the classification
generates neighborhood features (from neighborhood of $q$) and quantity
features (properties of the quantity mention). Details added to the appendix.

###  5.2 Predicting Variable Trigger List

The goal of this step is to predict the variable trigger list for the
equation. Our structured classifier takes as input the sentence $x$, and the
output $y$ is either one or two noun-phrases, representing variables in the
final equation. As we pointed out earlier, multiple groundings might be valid
for any given variable, hence there can be multiple valid variable trigger
lists. For every sentence $x$, we construct a set $Y$ of valid outputs. Each
element in $Y$ corresponds to a valid variable trigger list. Finally, we aim
to output only one of the elements of $Y$.

We modified the standard structured prediction algorithm to consider “superset
supervision” and take into account multiple gold structures for an input $x$.
We assume access to $N$ training examples of the form :
$(x_{1},Y_{1}),(x_{2},Y_{2}),\ldots,(x_{N},Y_{N})$, where each $Y_{i}$ is a
set of valid outputs for the sentence $x_{i}$. Since we want to output only
one variable trigger list, we want to score at least one $y$ from $Y_{i}$
higher than all other possible outputs, for each $x_{i}$. We use a modified
latent structured SVM to learn the weight vector $w$. The algorithm treats the
best choice among all of $Y_{i}$ as a latent variable. At each iteration, for
all $x_{i}$, the algorithm chooses the best choice $y_{i}^{*}$ from the set
$Y_{i}$, according to the weight vector $w$. Then, $w$ is updated by learning
on all $(x_{i},y_{i}^{*})$ by a standard structured SVM algorithm. The details
of the algorithm are in Algorithm 1.

Algorithm 1 Structural SVM with Superset Supervision

0: Training data $T=\\{(x_{1},Y_{1}),(x_{2},Y_{2}),\ldots,(x_{N},Y_{N})\\}$

0: Trained weight vector $w$

1: $w\leftarrow w_{0}$

2: repeat

3: $T^{\prime}\leftarrow\emptyset$

4: for all $(x_{i},Y_{i})\in T$ do

5: $y_{i}^{*}\leftarrow\arg\max_{y\in Y_{i}}w^{T}\phi(x_{i},y)$

6: $T^{\prime}\leftarrow T^{\prime}\cup\\{(x_{i},y_{i}^{*})\\}$

7: end for

8: Update $w$ by running standard Structural SVM algorithm on $T^{\prime}$

9: until convergence

10: return $w$

The distinction from standard latent structural SVM is in line $5$ of
Algorithm 1. In order to get the best choice $y_{i}^{*}$ for input $x_{i}$, we
search only inside $Y_{i}$, instead of all of $\mathcal{Y}$. A similar
formulation can be found in ?). The features $\phi(x,y)$ used for variable
trigger prediction include variable features (properties of noun phrase
indicating variable) and neighborhood features (lexical features from
neighborhood of variable mention). Details added to the appendix.

If the output of the classifier is a pair of noun phrases, we use a rule based
variable coreference detector, to determine whether both noun phrases should
have the same variable label or not. The rules for variable coreference are as
follows :

  1. 1.

If both noun phrases are the same, and they do not have the token “two” or
“2”, they have the same label.

  2. 2.

If the noun phrases are different, and the noun phrase appearing later in the
sentence contains tokens “itself”, “the same number”, they have the same
label.

  3. 3.

In all other cases, they have different labels.

Finally, each noun phrase contributes one variable trigger to the variable
trigger list.

###  5.3 Predicting Equation Tree

It is natural to assume that the syntactic parse of the sentence could be very
useful in addressing all the predictions we are making in the equation parsing
tasks. However, it turns out that this is not the case – large portions of the
syntactic parse will not be part of the equation parse, hence we need the
aforementioned modules to address this. Nevertheless, in the next task of
predicting the equation tree, we attempted to constraint the output space
using guidance from the syntactic tree; we found, though, that even enforcing
this weak level of output expectation is not productive. This was due to the
poor performance of current syntactic parsers on the equation data (eg., in
$32\%$ of sentences, the Stanford parser made a mistake which does not allow
recovering the correct equation).

The tree prediction module receives the trigger list predicted by the previous
two modules, and the goal is to create an equation tree using the trigger list
as the leaves of that tree. The input $x$ is the sentence and the trigger
list, and the output $y$ is the equation tree representing the relation
described in the sentence. We assume that the output will be a projective
equation tree. For features $\phi(x,y)$, we extract for each non-leaf node $n$
of the equation tree $y$, neighborhood features (from neighborhood of node
spans of $n$’s children), connecting text features (from text between the
spans of $n$’s children) and number features (properties of number in case of
leaf nodes). Details are included in the appendix.

The projectivity assumption implies that the final equation tree can be
generated by combining only adjacent nodes, once the set of leaves is sorted
based on $\operatorname{\mathrm{Span-Start}}(\cdot)$ values. This allows us to
use CKY algorithm for inference. A natural approach to further reduce the
output space is to conform to the projective structure of the syntactic parse
of the sentence. However, we found this to adversely affect performance, due
to the poor performance of syntactic parser on equation data.

Lexicon To bootstrap the equation parsing process, we developed a high
precision lexicon to translate mathematical expressions to operations and
orders, like “sum of A and B” translates to “A+B”, “A minus B” translates to
“A-B”, etc. (where A and B denote placeholder numbers or expressions). At each
step of CKY, while constructing a node $n$ of the equation tree, we check for
a lexicon text expression corresponding to node $n$. If found, we allow only
the corresponding operation (and order) for node $n$, and do not explore other
operations or orders. We show empirically that reducing the space using this
greedy lexicon matching help improve performance. We found that using the
lexicon rules as features instead of hard constraints do not help as much.
Note that our lexicon comprises only generic math concepts, and around $50\%$
of the sentences in our dataset do not contain any pattern from the lexicon.

Finally, given input sentence, we first predict the quantity trigger and the
variable trigger lists. Given the complete trigger list, we predict the
equation tree relating the components of the trigger list.

###  5.4 Alternatives

A natural approach could be to jointly learn to predict all three components,
to capture the dependencies among them. To investigate this, we developed a
structured SVM which predicts all components jointly, using the union of the
features of each component. We use approximate inference, first enumerating
possible trigger lists, and then equation trees, and find the best scoring
structure. However, this method did not outperform the pipeline method. The
worse performance of joint learning is due to: (1) search space being too
large for the joint model to do well given our dataset size of 385, and (2)
our independent classifiers being good enough, thus supporting better joint
inference. This tradeoff is strongly supported in the literature [Punyakanok
et al., 2005, Sutton and McCallum, 2007].

Another option is to enforce constraints between trigger list predictions,
such as, variable triggers should not overlap with the quantity triggers.
However, we noticed that often noun phrases returned by the Stanford parser
were noisy, and would include neighboring numbers within the extracted noun
phrases. This prevented us from enforcing such constraints.

##  6 Experimental Results

We now describe the data set, and the annotation procedure used. We then
evaluate the system’s performance on predicting trigger list, equation tree,
and the complete equation parse.

###  6.1 Dataset

We created a new dataset consisting of $385$ sentences extracted from algebra
word problems and financial news headlines. For algebra word problems, we used
the MIT dataset [Kushman et al., 2014], and two high school mathematics
textbooks, Elementary Algebra (College of Redwoods) and Beginning and
Intermediate Algebra (Tyler Wallace). Financial news headlines were extracted
from The Latest News feed of MarketWatch, over the month of February, 2015.
All sentences with information describing a mathematical relation among at
most two (possibly coreferent) variables, were chosen. Next, we pruned
sentences which require multiple uses of a number to create the equation. This
only removed a few time related sentences like “In 10 years, John will be
twice as old as his son.”. We empirically found that around 97% of sentences
describing a relation fall under the scope of our dataset.

The annotators were shown each sentence paired with the normalized equation
representing the relation in the sentence. For each variable in the equation,
the annotators were asked to mark spans of text which best describe what the
variable represents. The annotation guidelines are provided in the appendix.
We wanted to consider only noun phrase constituents for variable grounding.
Therefore, for each annotated span, we extracted the noun phrase with maximum
overlap with the span, and used it to represent the variables. Finally, a
tuple with each variable being mapped to one of the noun phrases representing
it, forms a valid output grounding (variable trigger list). We computed inter-
annotator agreement on the final annotations where only noun phrases represent
variables. The agreement (kappa) was 0.668, indicating good agreement. The
average number of mention annotations per sentence was 1.74.

###  6.2 Equation Parsing Modules

In this section, we evaluate the performance of the individual modules of the
equation parsing process. We report Accuracy - the fraction of correct
predictions. Table 3 shows the $5$-fold cross validation accuracy of the
various modules. In each case, we also report accuracy by removing each
feature group, one at a time. In addition, for equation tree prediction, we
also show the effect of lexicon, projectivity, conforming to syntactic parse
constraints, and using lexicon as features instead of hard constraints. For
all our experiments, we use the Stanford Parser [Socher et al., 2013], the
Illinois POS tagger [Roth and Zelenko, 1998] and the Illinois-SL structured
prediction package [Chang et al., 2015].

Quantity Trigger List Prediction | Accuracy
---|---
All features | 95.3
No Neighborhood features | 42.5
No Quantity features | 93.2
Variable Trigger List Prediction | Accuracy
All features | 75.5
No Variable features | 58.6
No Neighborhood features | 70.3
Equation Tree Prediction | Accuracy
All features | 78.9
No Neighborhood features | 64.3
No Connecting Text features | 70.2
No Number features | 77.6
No Lexicon | 72.7
No Projectivity | 72.8
Conform with Syntactic Parse | 70.2
Lexicon as Features | 74.5
Table 3: Performance of system components Source |  Equation Accuracy |  Equation + Grounding Accuracy
---|---|---
Our System |  71.3 |  61.2
Joint Learning |  60.9 |  50.0
SPF |  3.1 |  N/A
Table 4: Performance on equation parsing

###  6.3 Equation Parsing Results

In this section, we evaluate the performance of our system on the overall
equation parsing task. We report Equation Accuracy - the fraction of sentences
for which the system got the equation correct, and Equation+Grounding Accuracy
- the fraction of sentences for which the system got both the equation and the
grounding of variables correct. Table 4 shows the overall performance of our
system, on a $5$-fold cross validation. We compare against Joint Learning - a
system which jointly learns to predict all relevant components of an equation
parse (Section 5.4). We also compare with SPF [Artzi and Zettlemoyer, 2013], a
publicly available semantic parser, which can learn from sentence-logical form
pairs. We train SPF with sentence-equation pairs and a seed lexicon for
mathematical terms (similar to ours), and report equation accuracy. Our
structured predictors pipeline approach is shown to be superior to both Joint
Learning and SPF.

SPF gets only a few sentences correct. We attribute this to the inability of
SPF to handle overlapping mentions (like in Example 4), as well as its
approach of parsing the whole sentence to the final output form. The
developers of SPF also confirmed 333Private communication that it is not
suitable for equation parsing and that these results are expected. Since
equation parsing is a more involved process, a slight adaptation of SPF does
not seem possible, necessitating a more involved process , of the type we
propose. Our approach, in contrast to SPF, can handle overlapping mentions,
selects triggers from text, and parses the trigger list to form equations.

###  6.4 Error Analysis

For variable trigger list prediction, around $25\%$ of the errors were due to
the predictor choosing a span which is contained within the correct span,
e.g., when the target noun phrase is “The cost of a child’s ticket”, our
predictor chose only “child’s ticket”. Although this choice might be
sufficient for downstream tasks, we consider it to be incorrect in our current
evaluation. Another $25\%$ of the errors were due to selection of entities
which do not participate in the relation. For example, in “A rancher raises 5
times as many cows as horses.”, our predictor chose “A rancher” and “cows” as
variables, whereas the relation exists between “cows” and “horses”. For the
prediction of the equation tree, we found that $35\%$ of the errors were due
to rare math concepts expressed in text. For example, “7 dollars short of the
price” represents $7$ dollars should be subtracted from the price. These
errors can be handled by carefully augmenting the lexicon. Another $15\%$ of
the errors were due to lack of world knowledge, requiring understanding of
time, speed, and distance.

##  7 Conclusion

This paper investigates methods that identify and understand mathematical
relations expressed in text. We introduce the equation parsing task, which
involves generating an equation from a sentence and identifying what the
variables represent. We define the notion of projectivity, and construct a
high precision lexicon, and use these to reduce the equation search space. Our
experimental results are quite satisfying and raise a few interesting issues.
In particular, it suggests that predicting equation parses using a pipeline of
structured predictors performs better than jointly trained alternatives. As
discussed, it also points out the limitation of the current NLP tools in
supporting these tasks. Our current formulation has one key limitation; we
only deal with expressions that are described within a sentence. Our future
work will focus on lifting this restriction, in order to allow relations
expressed across multiple sentences and multiple relations expressed in the
same sentence. Code and dataset are available at
http://cogcomp.cs.illinois.edu/page/publication_view/800.

## Acknowledgements

This work is funded by DARPA under agreement number FA8750-13-2-0008, and a
grant from the Allen Institute for Artificial Intelligence (allenai.org).

## Appendix

##  Appendix A Features

###  A.1 Quantity Trigger List Prediction

The feature function $\phi(x,y)$ used for the classification generates the
following features :

  1. 1.

Neighborhood features : For each quantity $q$ in the input sentence, we add
unigrams and bigrams generated from a window around $q$, part of speech tags
of neighborhood tokens of $q$. We conjoin these features with
$\operatorname{\mathrm{Relevance}}(q)$.

  2. 2.

Quantity Features : For each quantity $q$, we add unigrams and bigrams of the
phrase representing the quantity. Also, we add a feature indicating whether
the number is associated with number one or two, and whether it is the only
number present in the sentence. These features are also conjoined with
$\operatorname{\mathrm{Relevance}}(q)$.

###  A.2 Variable Trigger List Prediction

The features $\phi(x,y)$ used for variable trigger prediction are as follows:

  1. 1.

Variable features : Unigrams and bigrams generated from the noun phrase
representing variables, part of speech tags of tokens in noun phrase
representing variables.

  2. 2.

Neighborhood Features : Unigrams and POS tags from neighborhood of variables.

All the above features are conjoined with two labels, one denoting whether $y$
has two variables or one, and the second denoting whether $y$ has two
variables represented by the same noun phrase.

###  A.3 Equation Tree Prediction

For features $\phi(x,y)$, we extract for each non-leaf node $n$ of the
equation tree $y$, the following:

  1. 1.

Neighborhood Features : Unigrams, bigrams and POS tags from neighborhood of
$\operatorname{\mathrm{Span-Start}}(lc(n))$, $\operatorname{\mathrm{Span-
Start}}(rc(n))$, $\operatorname{\mathrm{Span-End}}(lc(n))$ and
$\operatorname{\mathrm{Span-End}}(rc(n))$, conjoined with
$\operatorname{\odot}(n)$ and $\operatorname{\textsc{Order}}(n)$.

  2. 2.

Connecting Text Features : Unigrams, bigrams and POS tags between
$\min(\operatorname{\mathrm{Span-End}}(lc(n)),\operatorname{\mathrm{Span-
End}}(rc(n)))$ and $\max(\operatorname{\mathrm{Span-
Start}}(lc(n)),\operatorname{\mathrm{Span-Start}}(rc(n)))$, conjoined with
$\operatorname{\odot}(n)$ and $\operatorname{\textsc{Order}}(n)$.

  3. 3.

Number Features : In case we are combining two leaf nodes representing
quantity triggers, we add a feature signifying whether one number is larger
than the other.

##  Appendix B Annotation Guidelines

The annotators were shown each sentence paired with the normalized equation
representing the relation in the sentence. For each variable in the equation,
the annotators were asked to mark spans of text which best describe what the
variable represents. They were asked to annotate associated entities if exact
variable description was not present. For instance, in example 3 (Section 1),
the relation holds between the speed of bird and the speed of wind. However,
“speed” is not explicitly mentioned in the sentence. In such cases, the
annotators were asked to annotate the associated entities “the wind” and “a
bird” as representing variables.

The guidelines also directed annotators to choose the longest possible
mention, in case they feel the mention boundary is ambiguous. As a result, in
the sentence, “City Rentals rent an intermediate-size car for 18.95 dollars
plus 0.21 per mile.”, the phrase “City Rentals rent an intermediate-size car”
was annotated as representing variable. We allow multiple mentions to be
annotated for the same variable. In example 2 (Section 1), both “a number” and
“the same number” were annotated as representing the same variable.

##  Appendix C Lexicon

We construct a high precision list of rules, to parse sentences describing
mathematical concepts, for example, “difference of”, “greater than”, etc. For
each non-leaf node $n$ of a projective equation tree, we define the following
terms :

  1. 1.

$\operatorname{\mathrm{MidSpan}}(n)$ : The string from
$\min(\operatorname{\mathrm{Span-End}}(lc(n)),\operatorname{\mathrm{Span-
End}}(rc(n)))$ to $\max(\operatorname{\mathrm{Span-
Start}}(lc(n)),\operatorname{\mathrm{Span-Start}}(rc(n)))$.

  2. 2.

$\operatorname{\mathrm{LeftSpan}}(n)$ : The string ending at
$\min(\operatorname{\mathrm{Span-Start}}(lc(n)),\operatorname{\mathrm{Span-
Start}}(rc(n)))$ and starting from the nearest trigger position on the left.

  3. 3.

$\operatorname{\mathrm{RightSpan}}(n)$ : The string starting at
$\max(\operatorname{\mathrm{Span-End}}(lc(n)),\operatorname{\mathrm{Span-
End}}(rc(n)))$ and ending at the nearest trigger position on the right.

  4. 4.

$\operatorname{\mathrm{LeftToken}}(n)$ : Defined only for leaves, indicates
the span of text for the trigger of $n$.

The rules in our lexicon are described using the above terms. They are as
follows, ordered from low precedence to high precedence.

  1. 1.

If $\operatorname{\mathrm{LeftSpan}}(n)$ contains “sum of” and
$\operatorname{\mathrm{MidSpan}}(n)$ contains “and” or is the empty string,
$\operatorname{\odot}(n)$ should be $+$.

  2. 2.

If $\operatorname{\mathrm{MidSpan}}(n)$ contains one of “added to”, “plus”,
“more than” “taller than”, “greater than”, “larger than”, “faster than”,
“longer than”, “increased”, $\operatorname{\odot}(n)$ should be $+$.

  3. 3.

If $\operatorname{\mathrm{MidSpan}}(n)$ contains one of “more than” “taller
than”, “greater than”, “larger than”, “faster than”, “longer than”, and
$\operatorname{\mathrm{RightSpan}}(n)$ contains “by”,
$\operatorname{\odot}(n)$ should be $-$, and
$\operatorname{\textsc{Order}}(n)$ should be $lr$.

  4. 4.

If $\operatorname{\mathrm{LeftSpan}}(n)$ contains “difference of” and
$\operatorname{\mathrm{MidSpan}}(n)$ contains “and” or is the empty string,
$\operatorname{\odot}(n)$ should be $-$ and $\operatorname{\textsc{Order}}(n)$
should be $lr$.

  5. 5.

If $\operatorname{\mathrm{LeftSpan}}(n)$ contains one of “exceeds”, “minus”,
“decreased”, $\operatorname{\odot}(n)$ should be $-$, and
$\operatorname{\textsc{Order}}(n)$ should be $lr$.

  6. 6.

If $\operatorname{\mathrm{MidSpan}}(n)$ contains one of “subtracted” “shorter
than”, “less than”, “slower than”, “smaller than”, $\operatorname{\odot}(n)$
should be $-$, and $\operatorname{\textsc{Order}}(n)$ should be $rl$.

  7. 7.

If $\operatorname{\mathrm{MidSpan}}(n)$ contains “multiplied by”,
$\operatorname{\odot}(n)$ is $\times$.

  8. 8.

If $\operatorname{\mathrm{LeftSpan}}(n)$ contains “product of” and
$\operatorname{\mathrm{MidSpan}}(n)$ contains “and”, $\operatorname{\odot}(n)$
should be $\times$.

  9. 9.

If $\operatorname{\mathrm{LeftSpan}}(n)$ contains “ratio of”,
$\operatorname{\odot}(n)$ should be $\div$, and
$\operatorname{\textsc{Order}}(n)$ should be $lr$.

  10. 10.

If $\operatorname{\mathrm{LeftToken}}(n)$ contains one of “thrice”, “triple”,
“twice”, “double”, “half”, or if $\operatorname{\mathrm{MidSpan}}(n)$ contains
“times”, $\operatorname{\odot}(n)$ is $\times$.

  11. 11.

If $\operatorname{\mathrm{LeftToken}}(n)$ contains one of “thrice”, “triple”,
“twice”, “double”, “half”, or if $\operatorname{\mathrm{MidSpan}}(n)$ contains
“times”, and $\operatorname{\mathrm{MidSpan}}(n)$ contains “as”, and
$\operatorname{\mathrm{RightSpan}}(n)$ contains “as”, operation at
$\operatorname{\odot}(n)$ is $\div$, and $\operatorname{\textsc{Order}}(n)$ is
$rl$.

## References

  * [Artzi and Zettlemoyer, 2013]  Yoav Artzi and Luke Zettlemoyer.  2013\.  UW SPF: The University of Washington Semantic Parsing Framework.
  * [Björkelund and Kuhn, 2014]  Anders Björkelund and Jonas Kuhn.  2014\.  Learning structured perceptrons for coreference resolution with latent antecedents and non-local features.  In Proceedings of the 52nd Annual Meeting of the Association for Computational Linguistics (Volume 1: Long Papers).
  * [Cai and Yates, 2013]  Qingqing Cai and Alexander Yates.  2013\.  Semantic Parsing Freebase: Towards Open-domain Semantic Parsing.  In Proceedings of the Second Joint Conference on Lexical and Computational Semantics (*SEM).
  * [Chang et al., 2015]  Kai-Wei Chang, Shyam Upadhyay, Ming-Wei Chang, Vivek Srikumar, and Dan Roth.  2015\.  Illinoissl: A JAVA library for structured prediction.  In Arxiv Preprint, volume abs/1509.07179.
  * [Clarke et al., 2010]  J. Clarke, D. Goldwasser, M. Chang, and D. Roth.  2010\.  Driving semantic parsing from the world’s response.  In Proc. of the Conference on Computational Natural Language Learning (CoNLL), 7.
  * [Goldwasser and Roth, 2011]  D. Goldwasser and D. Roth.  2011\.  Learning from natural instructions.  In Proc. of the International Joint Conference on Artificial Intelligence (IJCAI).
  * [Hosseini et al., 2014]  Mohammad Javad Hosseini, Hannaneh Hajishirzi, Oren Etzioni, and Nate Kushman.  2014\.  Learning to solve arithmetic word problems with verb categorization.  In Proc. of the Conference on Empirical Methods in Natural Language Processing (EMNLP) 2014.
  * [Kushman et al., 2014]  N. Kushman, L. Zettlemoyer, R. Barzilay, and Y. Artzi.  2014\.  Learning to automatically solve algebra word problems.  In ACL.
  * [Kwiatkowski et al., 2013]  Tom Kwiatkowski, Eunsol Choi, Yoav Artzi, and Luke Zettlemoyer.  2013\.  Scaling semantic parsers with on-the-fly ontology matching.  In Proceedings of the 2013 Conference on Empirical Methods in Natural Language Processing.
  * [Madaan et al., 2016]  A. Madaan, A. Mittal, Mausam, G. Ramakrishnan, and S. Sarawagi.  2016\.  Numerical relation extraction with minimal supervision.  In Proc. of the Conference on Artificial Intelligence (AAAI).
  * [McDonald et al., 2005]  Ryan McDonald, Fernando Pereira, Kiril Ribarov, and Jan Hajič.  2005\.  Non-projective dependency parsing using spanning tree algorithms.  In Proceedings of the Conference on Human Language Technology and Empirical Methods in Natural Language Processing.
  * [Punyakanok et al., 2005]  V. Punyakanok, D. Roth, W. Yih, and D. Zimak.  2005\.  Learning and inference over constrained output.  In Proc. of the International Joint Conference on Artificial Intelligence (IJCAI), pages 1124–1129.
  * [Roth and Zelenko, 1998]  D. Roth and D. Zelenko.  1998\.  Part of speech tagging using a network of linear separators.  In Coling-Acl, The 17th International Conference on Computational Linguistics, pages 1136–1142.
  * [Roy and Roth, 2015]  S. Roy and D. Roth.  2015\.  Solving general arithmetic word problems.  In Proc. of the Conference on Empirical Methods in Natural Language Processing (EMNLP).
  * [Roy et al., 2015]  S. Roy, T. Vieira, and D. Roth.  2015\.  Reasoning about quantities in natural language.  Transactions of the Association for Computational Linguistics, 3\.
  * [Socher et al., 2013]  Richard Socher, John Bauer, Christopher D. Manning, and Andrew Y. Ng.  2013\.  Parsing With Compositional Vector Grammars.  In ACL.
  * [Sutton and McCallum, 2007]  C. Sutton and A. McCallum.  2007\.  Piecewise pseudolikelihood for efficient training of conditional random fields.  In Zoubin Ghahramani, editor, Proceedings of the International Conference on Machine Learning (ICML), pages 863–870. Omnipress.
  * [Wong and Mooney, 2007]  Y.-W. Wong and R. Mooney.  2007\.  Learning synchronous grammars for semantic parsing with lambda calculus.  In Proceedings of the Annual Meeting of the Association for Computational Linguistics (ACL), pages 960–967, Prague, Czech Republic, June. Association for Computational Linguistics.


# NaCl & KCl in Io’s Atmosphere

Erin Redwing  University of California - Berkeley, Department of Earth &
Planetary Science, 307 McCone Hall, Berkeley, CA, USA  Imke de Pater
University of California - Berkeley, Department of Astronomy, 501 Campbell
Hall, Berkeley, CA, USA  Statia Luszcz-Cook  University of Columbia, Astronomy
Department, New York, USA  Katherine de Kleer  California Institute of
Technology, 1200 East California Boulevard, Pasadena, CA, USA  Arielle Moullet
SOFIA/USRA, NASA Ames Building N232, Moffett Field, CA 94035, USA  Patricio M
Rojo  Universidad de Chile, Departamento de Astronomia, Casilla 36-D,
Santiago, Chile

(Accepted September 17, 2022)

###### Abstract

We present the first comprehensive study of NaCl and KCl gases in Io’s
atmosphere in order to investigate their characteristics, and to infer
properties of Io’s volcanoes and subsurface magma chambers. In this work, we
compile all past spectral line observations of NaCl and KCl in Io’s atmosphere
from the Atacama Large Millimeter/submillimeter Array (ALMA) and use
atmospheric models to constrain the physical properties of the gases on
several dates between 2012 and 2018. NaCl and KCl appear to be largely
spatially confined and for observations with high spectral resolution, the
temperatures are high ($\sim$500-1000 K), implying a volcanic origin. The
ratio of NaCl:KCl was found to be $\sim$5-6 in June 2015 and $\sim$3.5-10 in
June 2016, which is consistent with predictions based on observations of Io’s
extended atmosphere, and less than half the Na/K ratio in chondrites. Assuming
these gases are volcanic in origin, these ratios imply a magma temperature of
$\sim$1300 K, such that the magma will preferentially outgas KCl over NaCl.

Radio astronomy (1338) — Galilean satellites (627) — Planetary atmospheres
(1244)

##  1 Introduction

Jupiter’s moon Io, the innermost of the Galilean moons, is the most
volcanically active body in our solar system due to intense tidal forces from
its 4:2:1 orbital resonance with Europa and Ganymede. Strong internal heating
from tidal forcing on Io results in extreme volcanic activity – there are over
400 known volcanoes on Io’s surface, approximately 150 of which had thermal
activity detected during the Galileo mission (Lopes et al., 2004). Io’s
atmosphere is primarily sulfur dioxide (SO2), thin and tenuous (Lellouch et
al., 1990), and is replenished both by plume outgassing from its many active
volcanoes, as well as sublimation of surface frost deposits (Lellouch et al.,
2007; Tsang et al., 2016; de Pater et al., 2020). As such, Io’s atmosphere is
not uniform across its surface, but rather is highly spatially localized
around active volcanoes and over surface frost deposits at latitudes
$<|30-40|\degree$ (e.g., (Feaga et al., 2009); (de Pater et al., 2021)).

Plume outgassing on Io is primarily SO2, with trace gases SO, NaCl, and KCl
(Lellouch et al., 2007; de Pater et al., 2021). NaCl was first observed in
Io’s atmosphere by Lellouch et al. (2003), while KCl was first detected by
Moullet et al. (2013). Long before NaCl was detected in Io’s atmosphere, its
presence was predicted due to the strong detection of sodium in Io’s neutral
clouds (Brown, 1974), coupled to the ubiquity of alkali and chlorine
outgassing in terrestrial silicate volcanism (Pennisi et al., 1988; Symonds et
al., 1992, 1994). The origin of NaCl and KCl in Io’s atmosphere is likely
volcanic outgassing, though sputtering of materials on Io’s surface may also
be a source, which is the case for the atmospheres of the Moon and Mercury
(Hunten et al., 1988; Killen & Ip, 1999). Sputtering as a primary source of
these materials, however, would be limited to areas where the atmospheric
column density is $\leq 10\textsuperscript{16}$ cm-2, such as at high
latitudes or when Io is in Jupiter’s shadow and its SO2 atmosphere has
collapsed (Lanzerotti et al., 1982; de Pater et al., 2021). In the case where
these gases are indeed volcanic in origin, their ratio of abundances can be
used to indirectly infer the temperature and composition of the magma chambers
from which they originate (Fegley & Zolotov, 2000), something which is at
present poorly constrained.

No comprehensive studies correlating observations with models of NaCl and KCl
in Io’s atmosphere have been carried out – photochemical models suggest that
volcanically-outgassed NaCl and KCl would be removed from Io’s atmosphere in
$\sim$3 hours (Moses et al., 2002), while observations of NaCl over a four-
month period from 2016-2017 indicate that NaCl is remarkably stable over long
time periods (Roth et al., 2020). In this work, we compile all past spectral
line observations of NaCl and KCl in Io’s atmosphere from the Atacama Large
Millimeter/submillimeter Array (ALMA) and use atmospheric models to constrain
the physical properties of the gases (and Io’s putative subsurface magma
chambers). Each observation was reduced and analyzed using identical spectral
line modeling methods so that they may be assessed individually as well as
jointly, in order to piece together an understanding of the nature of these
gases in Io’s atmosphere. This paper has several main components: Section 2
details the observations and data reduction, Section 3 describes the spectral
line modeling used to analyze the data, Section 4 presents results from this
analysis including the physical properties inferred from our models, Section 5
discusses the common themes, interesting findings, anomalies, and remaining
questions, and Section 6 summarizes the main conclusions of the paper.

Figure 1: All ALMA images used in this study, top row is NaCl lines, bottom
row is KCl lines. Middle columns are dates where NaCl and KCl were observed
concurrently. In these images (as well as all images in this paper), Io North
is up, bottom left white circle is the HPBW of the synthesized beam (spatial
resolution), center white circle is the outline of the disk of Io, taken from
continuum maps of Io observed simultaneously, and black circles are the
apertures analyzed in this study. Identification of each aperture (i.e.
apertures 1, 2, etc.) is summarized in Figs. 12 and 13). The rms error for
each image, calculated from emission off Io’s disk, is noted in the bottom
right corner of each image

The radius of the apertures was determined by the smaller axis of the HPBW for
each image (see Table 2).

Table 1: Spectral Line Species and Frequencies Observed Species | Dates Observed | Frequency | Wavelength | Line Strength | E (low)
---|---|---|---|---|---
|  | (GHz) | (mm) | (cm-1/mol/cm2) | (cm-1)
KCl | Oct 17 2012, Oct 19 2012, Mar 20 2018, Sept 2 2018, Sept 11 2018 | 344.820 | 0.869 | 2.235E-19 | 253.489
NaCl | June 9 2015 | 338.021 | 0.887 | 2.747E-19 | 141.082
KCl | June 9 2015 | 337.208 | 0.889 | 2.207E-19 | 242.241
KCl | June 9 2016 | 260.916 | 1.149 | 1.647E-19 | 143.748
NaCl | June 9 2016, July 26 2016 | 260.223 | 1.152 | 1.668E-19 | 82.510

##  2 Observations & Data Reduction

Figure 2: Example continuum image from July 26, 2016, where the lower left
white circle is the HPBW and the white circle in the center of the image is
determined to be the edge of the disk of Io, which is then used to determine
Io’s size and location in the corresponding spectral line images.

Data were obtained on eight dates from 2012-2018 with the Atacama Large
Millimeter/submillimeter Array (ALMA). All data are publicly available on the
ALMA archive - in some cases, NaCl and/or KCl were the primary observational
goals, and in some they were lines added to observations whose goals were
primarily to observe other atmospheric gases (such as SO2). Observations were
made in Bands 6 and 7, which have central frequencies of 233 GHz (1.3 mm) and
343.5 GHz (0.87 mm) respectively. Several transitions of NaCl and KCl were
observed, as detailed in Table 1. In two datasets, both NaCl and KCl were
detected (June 9 2015, June 9 2016), in one both were targeted but only NaCl
was detected (July 26 2016), and in two only a single line of KCl was targeted
and detected (Oct 19 2012, March 20 2018), resulting in three total detections
of NaCl and four detections of KCl. KCl was also targeted on October 17 2012,
and September 2 and 11 2018, but not detected. SO2 was observed on all dates
where NaCl and KCl were observed.

The spatial resolution varied greatly from observation to observation – in
most cases, the half-power beam width (HPBW) is large and asymmetrical (see
Figure 1, bottom left corner in each image; Table 2). In the observation with
the highest spatial resolution, $\sim$4 beams across the disk of Io were
obtained. For each observation, all scans on Io have been combined to create a
single spectral-line data cube per line per observation. In three datasets
(July 26 2016, September 2 and 11 2018), Io went from eclipse into sunlight
during the time of observation, and in one (March 20 2018) Io went from
sunlight to eclipse – these scans have been combined in this analysis, since
the eclipse had no apparent effect on the observed NaCl nor KCl emission when
imaged separately. Several of the observations used in this study have been
previously published, as shown in Table 2.

Table 2: Observation Details Date & Time (UT) | Sub-Earth | Sub-Earth | Io diameter | HPBW | Resolution | ALMA Program,
---|---|---|---|---|---|---
(month/day/yr hr:m:s) | Longitude | Latitude | (arcsec) | (Bmax, Bmin, PA) | (kHz) | Publication
| (deg (W)) | (deg (N)) |  |  |  |
Oct 17 2012 09:42:31-10:33:00 | 62.488 | 3.115 | 1.156 | 0.784”, 0.406”, -29.296∘ | 61.0 | 2011.0.00779.S (Moullet, 2015)
Oct 19 2012 09:39:29-10:28:08 | 193.741 | 3.120 | 1.165 | 0.846”, 0.466”, -27.883∘ | 61.0 | 2011.0.00779.S (Moullet, 2015)
June 9 2015 21:30:33-23:20:54 | 281.054 | -0.153 | 0.867 | 0.351”, 0.288”, -29.627∘ | 61.0 | 2012.1.00853.S
June 9 2016 01:14:54-02:17:07 | 24.587 | -1.550 | 0.931 | 0.585”, 0.358”, -59.54∘ | 245.6 | 2015.1.00995.S
July 26 2016 19:13:09-20:16:13 | 13.882 | -1.671 | 0.828 | 0.237”, 0.206”, -88.52∘ | 245.6 | 2015.1.00995.S
Mar 20 2018 10:54:43-11:13:53 | 343.264 | -3.405 | 1.058 | 0.328”, 0.284”, -83.470∘ | 144.0 | 2017.1.00670.S (de Pater et al., 2020)
Sept 2 2018 21:46:21-22:40:08 | 19.472 | -2.957 | 0.885 | 0.30”, 0.30”, 0∘ | 144.0 | 2017.1.00670.S (de Pater et al., 2020)
Sept 11 2018 17:36:12-18:41:56 | 13.381 | -2.948 | 0.867 | 0.22”, 0.22”, 0∘ | 144.0 | 2017.1.00670.S (de Pater et al., 2020)

CASA version 5.4.0-68 was used for all steps of the data reduction and
imaging. The raw data were calibrated and flagged via the ALMA pipeline, after
which Io was split off into a separate calibrated measurement set (MS). An
ephemeris file with Io’s position and velocity at a 1-minute time interval was
attached to each MS to ensure Io’s tracking at high precision, which is
necessary to distinguish gas velocity effects and get the right shape of the
velocity profile in Io’s frame of motion. Io’s continuum was modeled as a
limb-darkened disk – the limb-darkening coefficient and brightness temperature
were adjusted to fit each individual observation, with limb-darkening
generally being asymmetrical about the poles and equator (a value of q $\sim$
0.3 North-South and q $\sim$ 0.1 East-West, where brightness falls off at
higher emission angles ($\theta$) as a function of
$\cos\textsuperscript{q}(\theta)$)). Peak brightness temperatures ranged from
80-100 K. This disk was used to self-calibrate the data, and to clean (i.e.,
deconvolve) the continuum map. Before imaging the line emission in each MS,
the continuum emission was subtracted using CASA’s UVCONTSUB (see example
continuum image Figure 2), leaving only spectral line emission of each species
(NaCl or KCl). Continuum images were used to determine the edge of the disk of
Io for each spectral line image (see central white circles in Figure 1). Line-
averaged images of Io were created by averaging emission over 0.4 km/s
centered at each spectral line’s rest frequency, except for June 9 2016 –
these images are centered at -0.1 km/s from the line center for NaCl and -0.5
km/s from the line center for KCl, because these were the centers of the
observed line profiles. The final line center maps are shown in Figure 1. In
order to model the spectral line profiles, image data cubes were created at a
frequency resolution of 61 or 122 kHz (see Table 2), which allows one to
construct a spectral line profile at any position on the disk. All images were
created using uniform weighting to optimize for spatial resolution and cleaned
using the Clark or Briggs algorithm in CASA’s TCLEAN routine. Line spectra
were created from the data cubes by determining the flux density for each
aperture (see apertures in Figure 1) at each frequency - the line profile is
constructed from the total flux density within each aperture at each frequency
of the data cube. Examples of our spectral line profiles for NaCl aperture 1
from June 9 2015 are shown in Figure 3.

Figure 3: Plots demonstrating the effect of varying column density (a),
velocity (b), and temperature (c) on output spectral line profiles compared to
observed line profile of NaCl aperture 1 on June 9 2015.

##  3 Spectral Line Modeling

Models of the spectral line data cubes were created using a radiative transfer
code which integrates the equation of radiative transfer over a model of Io’s
atmosphere consisting of overlying plane-parallel layers. This code was
originally developed to model CO lines in Neptune’s atmosphere (Luszcz-Cook &
de Pater, 2013) and was subsequently adapted to Io’s atmosphere (de Pater et
al., 2020) \- the code used for this analysis is the same as what is used in
de Pater et al. (2020). The models generated by this radiative transfer code
assume that Io’s atmospheric gases are in local thermodynamic equilibrium and
hydrostatic equilibrium. Pixel size, beam size, spectral resolution, and
frequency in each model were matched to the corresponding observations, while
temperature, column density, and velocity of the atmospheric gases were
varied, and the chi-square ($\chi\textsuperscript{2}$) goodness-of-fit was
calculated to find which model spectral line output best fits the observed
spectral line profile and infer reasonable constraints on the free parameters.
To demonstrate how the free parameters (temperature, column density, and
velocity) affect the model spectral line output, in Figure 3 we show the
effect that each individual parameter has on the output line profile. When
column abundance is increased and all other parameters are held constant, the
peak of the line profile increases (see Figure 3a), due to an increased number
of molecules in the system which emit photons at the central line frequency.
Changing the gas velocity shifts the central frequency of the line profile
(shifting to higher frequencies for negative velocity and lower frequencies
for positive velocities, see Figure 3b). When the gas temperature is
increased, the result is an increased line width due to Doppler broadening,
which is the result of increased thermal motion, according to the following
equation:

$\triangle\nu=\frac{\nu\textsubscript{0}}{c}\sqrt{\frac{2kT}{m}}$ (1)

Figure 4: Example plot of $\Delta\chi^{2}$ model fits to NaCl (260.2 GHz)
observed on July 26, 2016 for temperatures ranging from 0 - 2000 K, column
densities ranging from 1012-1014 cm-2, and a velocity of 120 m/s. Top left
inset image indicates the aperture modeled, top right inset shows the line
profile with the best fit line model. These plots were made for each aperture
analyzed in this work, and are shown in the Appendix Figures 12 and 13.

where $\triangle\nu$ is the width of the line profile at half-maximum,
$\nu\textsubscript{0}$ is the central frequency of the line, $c$ is the speed
of light, $k$ is Boltzmann’s constant, $T$ is the temperature of the gas, and
$m$ is the mass of the molecule.

As such, temperature is the factor which determines the width of the line
profile – increasing temperature leads to a larger velocity distribution of
particles ($\triangle\nu\sim\sqrt{T}$), and therefore a broader line profile
(see Figure 3c). Increasing temperature also affects the peak of the line
profile since it increases the overall energy in the system. However, since
higher temperatures lead to broader velocity distributions of particles, at
some point increasing temperature will result in a decrease of the peak of the
line profile because there are fewer molecules to emit energy at the line
center since the molecules are moving around and emitting at nearby
frequencies. As a result, when all other parameters are held constant,
increasing temperature results in a consistently increasing line width, while
the line peak will either increase or decrease depending on the number of
emitting molecules in the system. While the effect of each individual
parameter cannot be completely disentangled, their behaviors are unique enough
to infer constraints on their values for each spectral line observation -
density determines the height of the peak of the line profile, velocity shifts
the line profile, and temperature broadens the line profile.

Figure 5: Plots of $\Delta\chi^{2}=4.61$ contours for each aperture, NaCl
apertures on the left and KCl apertures on the right. Detailed plots of each
aperture are shown in Figure 12 and 13.

A grid of spectral line models was run for each observed line profile, where
the temperature was varied from 100-2000 K in 50 K steps, and the column
number density from 1012-1014 cm-2 for NaCl and 1011-1013 cm-2 for KCl in

Table 3: Velocities for Each Aperture Model Line | Velocity
---|---
October 19, 2012, KCl, aperture 1 | -430 m/s
June 9, 2015, NaCl, aperture 1 | 35 m/s
June 9, 2015, KCl, aperture 1 | 25 m/s
June 9, 2015, NaCl, aperture 2 | -180 m/s
June 9, 2015, NaCl, aperture 3 | -250 m/s
June 9, 2016, NaCl, aperture 1 | -325 m/s
June 9, 2016, KCl, aperture 1 | 200 m/s
June 9, 2016, NaCl, aperture 2 | 120 m/s
June 9, 2016, KCl, aperture 2 | 150 m/s
June 9, 2016, NaCl, aperture 3 | 600 m/s

an equivalent number of logarithmically scaled steps. At each step through the
model grid, a chi-square value was calculated to determine how well the output
model line fits the observed spectral line profile:

$\chi^{2}=\sum_{i=1}^{N}\frac{(O_{i}-E_{i})^{2}}{\sigma_{i}^{2}}$ (2)

where $N$ is the number of data points in the line profile, $O_{i}$ is the
observed flux at each data point, $E_{i}$ is the model flux at each data
point, and $\sigma$ is the root mean square (RMS) uncertainty.

After chi-square values were calculated for each point in the model grid, the
minimum chi-square value was found and subtracted from each chi-square value
in the grid:

$\Delta\chi^{2}=\chi^{2}-\chi_{min}^{2}$ (3)

This created an output grid of $\Delta\chi^{2}$ values which essentially
formed contours of constraints on temperature and density values for each
observation (see example Figure 4). This method can be used to determine the
range of model parameter values which fit the data to a given confidence level
(i.e., a value of $\Delta\chi^{2}=4.61$ gives a 90% confidence level for a two
parameter model fit (Avni, 1976; Wall & Jenkins, 2012)).

Figure 6: a) NaCl emission (260.2 GHz) from July 26, 2016, b) SO2 emission
(262.2 GHz) in eclipse taken concurrently to a), c) KCl emission (344.8 GHz)
from March 20, 2018, d) SO2 emission (346.6 GHz) in eclipse taken concurrently
to c). All maps were integrated over 0.4 km/s at the line center.

Since the only effect of varying gas velocity in our model is shifting the
line profile to higher or lower frequencies, this parameter was chosen before
running the model grid by matching the peak of the model line profile to the
peak of the observed line profile to ensure that the model line profile was
centered on the observed line profile. These values are summarized in Table 3
for each aperture, which can be identified in detail in Figures 12 and 13.
Even large changes in velocity shift the line profile very minimally - see
Figure 3b which shows a shift of +/-100 m/s on either side of the line
profile. For most apertures, the spectral resolution is too low to allow for
precise velocity measurements (see spectral resolutions in 2.

##  4 Results

###  4.1 Constraints on NaCl and KCl Temperature and Column Density

The primary finding of this project are constraints on model temperature and
number density of NaCl and KCl gases for several ALMA spectral line
observations throughout the years, which are shown in Figure 5. For all of
these observations, NaCl and KCl were highly spatially localized around
certain positions, so rather than model the physical parameters for these
gases integrated over the disk, we instead chose smaller apertures at the
emission maxima in each image (see Figure 1). Aperture size for each
observation was determined by the minor axis of the beam size in each image
(see Bmin in Table 2).

The contours in Figure 5 represent the boundary of model fits for which the
delta chi-square value was less than or equal to 4.61 (90% confidence). The
size of the $\Delta\chi^{2}=4.61$ contour for each aperture also shows the
relative quality of the data – data which had good spectral resolution and
high signal to noise ratios resulted in smaller $\Delta\chi^{2}=4.61$
contours, meaning that a smaller range of values fit the line profile well.
This is because observations with higher spectral resolution resulted in line
profiles which had more data points to fill out the line profile in greater
detail, while observations with lower spectral resolution resulted in line
profiles whose shape was less well constrained. As such, plotting contours of
all $\Delta\chi^{2}=4.61$ fits shows the results of all model fits while also
representing the quality of the data used to determine the results.

Figure 7: Flux density vs. time for four different SO2 transitions on July 26,
2016 when Io was observed moving from eclipse into sunlight. The flux
densities have been determined from the center line emission maps of each
$\sim$3-min scan (i.e., maps integrated over 0.4 km/s) by integrating the flux
density on each map over the disk of Io. The gray dotted line represents the
beginning of eclipse egress (when Io first moves into sunlight), after which
SO2 atmosphere reforms.

Model fits for column densities are reasonably well-constrained across all
observations, likely because density is inferred through the line strength
which is usually well-captured even in observations with low spatial or
spectral resolution. However, model fits for temperature are generally poorly
fit except for observations with high spectral resolution, since these
observations have more data points which fill out the shape of the line
profile more precisely. For the observations where temperature is well
constrained, temperatures are relatively high $\sim$500-1000 K, suggesting a
volcanic outgassing source.

For the dates where both NaCl and KCl were observed concurrently (June 2015
and June 2016), the ratio of NaCl:KCl gas abundance was calculated for a
variety of model fits within the $\Delta\chi^{2}=4.61$ solution contour. This
ratio has been determined for Io’s extended atmosphere (Na:K $\sim$ 7-13
(Brown, 2001)) and for chondrites (Na:K $\sim$ 13-17, (Fegley & Zolotov,
2000)), which are assumed to be a proxy for Io’s interior ratio. However, this
ratio had not been determined for Io’s atmosphere. We found that at all
temperatures and densities within our $\Delta\chi^{2}=4.61$ contour fits, the
NaCl:KCl ratio was $\sim$ 5-6 for the June 2015 data and $\sim$ 3.5-10 for the
June 2016 data. This ratio was calculated using the possible range of NaCl and
KCl column density values within the $\Delta\chi^{2}=4.61$ contour fits,
assuming NaCl and KCl gases are the same temperature. These ratios are lower
than what has been found for Io’s extended atmosphere and chondrites. Possible
reasons for this difference are discussed in Section 5.

###  4.2 NaCl/KCl and Sulfurous Compounds Are Not Colocated

For each dataset in our analysis, at least one SO2 transition was observed at
the same time as NaCl/KCl, and two of the datasets involved the observation of
SO2 in eclipse (see Figure 6). Images of the SO2 center line emission were
created in the same way as our NaCl/KCl maps, i.e., the data are integrated
over 0.4 km/s, centered at the center of the observed line profile. Image data
cubes are based on the channel widths listed in Table 2. While in sunlight,
SO2 gas in Io’s atmosphere is sourced both from sublimation of SO2 surface
frost as well as volcanic outgassing (Spencer et al., 2005; Lellouch et al.,
2007, 2015). However, once Io enters Jupiter’s shadow (a process which occurs
once per Io day, i.e., roughly every 42 hours), its temperature drops below
the vapor pressure point of SO2, resulting in a collapse of its SO2
sublimation atmosphere (Saur & Strobel, 2004; Tsang et al., 2016). Io’s
atmosphere reforms once it re-enters sunlight on a timescale of minutes (de
Pater et al., 2020). Figure 7 shows this process for SO2 observations from
July 26, 2016. As such, when SO2 is observed while Io is in eclipse, one can
assume that most observed emission is due to volcanic outgassing (de Pater et
al., 2020).

Figure 8 shows images from March 20, 2018 and July 26, 2016 when KCl and NaCl
were imaged in eclipse and sunlight - it is also notable that NaCl and KCl
showed no significant difference in emission between sunlight and eclipse,
suggesting that Io’s sulfurous atmospheric collapse and reformation has no
effect on its alkali atmospheric gases. While there are subtle differences
between the sunlight and eclipse images, these variations are much lower than
the rms error for each image. As such, when NaCl and KCl were imaged in
sunlight and eclipse, these scans have been combined to boost the SNR.
Assuming that NaCl and KCl are volcanic in origin, it is interesting to
compare the locations of these gases with the location of concurrently
observed volcanic SO2 gas in the eclipse datasets. These two datasets (July
26, 2016 and March 20, 2018) observed just before eclipse egress and ingress,
respectively, are almost at the same viewing geometry of Io (a difference of
5-10∘ in rotation).

In the NaCl and KCl images for both dates, emission from these two trace
species appear in similar areas – along the right (East) limb and at high
eastern latitudes. Interestingly, it appears that the locations of peak
volcanic SO2 emission in these observations are not the same as the locations
of peak NaCl and KCl emission observed concurrently. The SO2 emission in
eclipse also correlates to the SO2 emission in the far line wings of the
combined sunlight and eclipse images (see Figure 9) – line wing emission is
due to SO2 gas which is either at a high temperature or a high velocity, both
suggesting volcanic emission (de Pater et al., 2020).

The presence of SO2 gas at the same locations in eclipse as well as in the far
line wings strongly suggests that this is volcanic emission, and in both cases
does not spatially correlate with NaCl or KCl observed concurrently (see
Figure 9), which we argue are also volcanically sourced. The lack of spatial
correlation between these gases may suggest that the volcanic processes which
produce SO2 outgassing are not the same as those responsible for outgassing of
NaCl and KCl, or that there may be some heterogeneity in the composition of
Io’s subsurface magma chambers (de Pater et al., 2020). Possible explanations
are discussed in Section 5.

Figure 8: Top row: KCl emission from March 20, 2018 imaged over all scans (a),
and separated into sunlight (b) and eclipse (c) scans. Bottom row: NaCl
emission from July 26, 2016 imaged over all scans (d), and separated into
sunlight (e) and eclipse (f) scans. All images from the same dates are imaged
on the same color scale for easy comparison, and rms error is noted in bottom
corners of each image.

###  4.3 Correlating Surface Features (Volcanoes) to NaCl/KCl Emission

If the observed NaCl and KCl indeed result from volcanic outgassing, then it
would only make sense that their locations would be correlated to known
volcanoes on Io’s surface. Unfortunately, it is difficult to concretely
identify the volcanic source for the observed emissions: 1) the spatial
resolution of most of the observations is quite poor (at the highest spatial
resolution, $\sim$4 beams across the disk of Io), and 2) Io has hundreds of
volcanoes on its surface, many in close proximity. Emission locations should
be taken with a grain of salt due to the large beam size in many of the
observations, but it is useful to note areas where NaCl and KCl hotspot
emission has been seen over several observations - a surface map indicating
Io’s volcanic features, overlaid with the KCl and NaCl detections described in
this paper, is shown in Figure 10.

Figure 9: Each row shows NaCl/KCl emission on far left for each date of
observation in this study, followed by a series of images of SO2 emission
taken concurrently. For all images, all scans (sunlight and/or eclipse) are
combined. SO2 images from left to right show SO2 emission across the emission
line, where the leftmost image is emission on the blue-shifted side of the
line wing, the center image is the emission at the center of the line, and the
rightmost image is emission on the red-shifted side of the line wing. See de
Pater et al. (2020) for more detailed analysis.  Figure 10: Approximate
locations of NaCl and KCl emission apertures from all observations. Purple
circles are approximate KCl locations, green circles are approximate NaCl
locations, and red circles are approximate locations where NaCl and KCl were
observed concurrently. Base map from (Williams et al., 2011).  Figure 11: Top
row: a) View of Io on October 17, 2012, when observations taken of Io showed
no KCl emission, b) View of Io on October 19, 2012 when KCl was detected, c)
Same as (b) with observed KCl emission overlaid. Bottom row: Views of Io on d)
September 2, 2018, and e) September 11, 2018, both of which showed no KCl
emission, f) View of Io on March 20, 2018 when KCl was detected, g) Same as
(f) with observed KCl emission overlaid.

It is also interesting to look at how non-detections are correlated to Io’s
orientation. Several dates analyzed in this work showed non-detections of KCl.
The October 19 2012 KCl detection is an interesting case – for this dataset,
Io was observed two nights prior (October 17 2012) and did not show KCl
emission, despite having the same observational set-up and very similar views
of Io’s surface and similar levels of noise (see Figure 11). Comparing the KCl
emission maps to the map of volcanoes on Io for the times of observation show
that from the first observation to the second, the primary change is that
volcano Zamama had rotated into view. Zamama’s location is near that where we
see the observed KCl emission on October 19 (Figure 11). While the spatial
correlation of emission to Zamama may be spurious due to the spatial
resolution and SNR of the images, the fact that KCl is not seen on October 17
but clearly seen on October 19 after Zamama has rotated into view is striking.

Zamama is a well-studied volcanic center on Io. Analysis of SSI and NIMS data
from Galileo indicated silicate magma volcanism at a temperature of $\geq$1100
K (Davies et al., 1997). Surface flows observed on Zamama indicate that it is
a Promethean-style volcano, which are characterized by long-term lava flows
which are likely fed from shallow magma reservoirs (Davies & Ennis, 2011). SSI
images revealed intermittent plume activity at Zamama (Geissler & McMillan,
2008).

No KCl emission was seen on September 2 and 11 2018, even though the emission
frequencies were covered by the observations. A slightly different view of
Io’s surface was observed on March 20 2018, and KCl emission was seen on the
eastern limb of Io over an area above Ulgen and Babbar Patera, which were
barely on the edge of the limb in the September observations. This volcanic
region was first identified in Voyager data (Pearl & Sinton, 1982; McEwen et
al., 1985), whose initial analysis combined the two volcanoes into one hotspot
due to their close proximity. Observations on March 8 2015 from the Large
Binocular Telescope observed volcanic activity in this region (de Kleer et
al., 2021).

NaCl and KCl were both observed on June 9 2015, and both emission hotspots are
very strong and highly co-located with each other. The temperature range for
NaCl was well-constrained to relatively high temperatures ($\sim$450-700 K),
suggesting that these gases are volcanic in origin. Their emission appears to
be directly located over Isum, though it should be noted that Zamama is also
nearby. Isum has no known plume, and SO2 was not observed over Isum at the
time of observation (de Pater et al., 2021).

For the eclipse observations in Figure 6, the same side of Io is seen in both
sets of images. The emission at high latitudes correlates to the area around
Dazhbog Patera, though Amaterasu, Fuchi, Manua, and Kimich Ahau are all active
volcanoes nearby. Plumes have been observed at Dazhbog Patera (McEwen, 2002),
and recent lava flows characterize the geological makeup of the surface in the
area (Williams et al., 2011). Along the eastern limb, emission is seen in both
sets of images over the area around Babbar Patera and Ulgen Patera.

##  5 Discussion

Several interesting questions emerge from our findings. The first is the
apparent lack of correlation between volcanic SO2 gas and NaCl/KCl emission.
If both of these gases are from volcanic sources, then why do some volcanoes
appear to outgas sulfurous compounds while some outgas alkali compounds? The
first possible explanation is that for some volcanoes, SO2 gas emission is the
result of hot lavas vaporizing SO2 surface frost deposits, rather than
outgassing directly from the volcanic vent. In this scenario, SO2 emission in
eclipse would be constrained to the locations of SO2 surface frost deposits,
which are primarily found in Io’s equatorial band. This is consistent with our
SO2 eclipse and linewing observations (see Figures 7 and 9), which primarily
show SO2 emission at mid-latitudes.

Another possible explanation is that NaCl and KCl emission is confined to high
temperature volcanoes, while SO2 emission arises from magma chambers whose
temperatures do not reach the boiling point of NaCl and KCl (1373 K for NaCl
and 1173 K for KCl at Io surface pressures (Fegley & Zolotov, 2000)). If this
were the case, SO2 should be observed at all locations where NaCl and KCl are
observed, while NaCl and KCl would be limited to only high temperature
eruptions. Interestingly, NaCl and KCl are often observed at high latitudes,
indicative of being sourced by deep mantle heating (Segatz et al., 1988; de
Kleer & de Pater, 2016; de Pater et al., 2021). Perhaps at these latitudes,
the temperature of the atmosphere is cold enough that SO2 condenses out very
quickly and is not detectable in our observations, although SO2 emissions have
been detected over energetic high latitude eruptions (e.g., Tvashtar, (Jessup
& Spencer, 2012)). It seems that there is not a clear explanation that
satisfies all observations and open questions - observations with higher
spatial resolution would help to clarify this conundrum.

Assuming the observed NaCl and KCl are indeed products of volcanic outgassing,
then constraints on their abundance can be used to infer the temperature and
composition of the magma chambers themselves (Fegley & Zolotov, 2000). This
relationship between observed atmospheric NaCl:KCl ratio and magma chamber
conditions presents an exciting opportunity to constrain physical properties
of Io’s magma chambers indirectly through observing plume outgassing emission.
Magma thermochemical equilibrium calculations by Fegley & Zolotov (2000)
suggest that the ratios found in our study correlate to a magma temperature
around 1300 K.

Additionally, it is interesting to note that Io’s atmospheric NaCl:KCl ratio
(NaCl:KCl $\sim$ 5-6, 3.5-10) is marginally lower than the ratio found for its
extended atmosphere (Na:K $\sim$ 7-13, (Brown, 2001)) and significantly lower
than the ratio found in chondrites (Na:K $\sim$ 13-17, (Fegley & Zolotov,
2000)). The explanation for the difference between the ratio derived in this
work and that of Io’s extended atmosphere can be explained by the analysis
done by Brown (2001), who notes that his ratio is likely an upper limit on the
Na:K ratio for Io’s surface and near-surface atmosphere, due to the difference
in photoionization times for sodium and potassium (400 and 270 hours,
respectively (Smyth & Marconi, 1995)). Additionally, Brown (2001) detected a
high-speed jet (which they presumed to consist of material more recently
ejected from Io’s surface) with a ratio $\sim$ 5-7, which is comparable to our
ratios.

As for the difference in our ratio compared to the chondritic ratio, it is
possible that this is the result of KCl’s lower condensation temperature (1173
K compared to 1373 for NaCl (Fegley & Zolotov, 2000)), which would result in
KCl being preferentially outgassed in magmas whose temperatures are between
these numbers, lowering the NaCl:KCl ratio in Io’s atmosphere as compared to
the ratio in the magma (de Pater et al., 2021).

##  6 Conclusions

We have presented the first comprehensive study of NaCl and KCl gases in Io’s
atmosphere. The primary findings of this paper include:

  * •

Temperature and column abundance constraints on NaCl and KCl spectral line
emission from several dates over the period of 2012-2018

  * •

NaCl and KCl were highly spatially confined and temperatures (where well-
constrained) were high ($\sim$500-1000 K), suggesting that these compounds are
volcanic in origin

  * •

NaCl:KCl ratios of $\sim$5-6 in June 2015 and $\sim$3.5-10 in June 2016, which
is consistent with (or perhaps marginally lower than) extended atmosphere
calculations from Brown (2001) and a factor of two lower than the chondritic
Na:K ratio, implying a magma temperature of $\sim$1300 K

  * •

Volcanic SO2 emission and volcanic NaCl/KCl emission are not co-located, which
could be explained by heterogeneity in Io’s subsurface magma chambers or a
difference in processes which preferentially outgas SO2 vs. NaCl/KCl

  * •

Several volcanoes (including Zamama and Isum Patera) were identified as
possible sources of NaCl/KCl plumes

###  6.1 Future Work

While the correlations between NaCl and KCl to known volcanoes on Io is
intriguing, observations with higher spatial resolution are needed to
correlate the emissions with a higher accuracy to a location of known
volcanoes. Observations of these compounds at known active volcanoes could
take place on a target-of-opportunity basis. Additionally, an atmospheric
model including volcanic plumes in a sublimation atmosphere (such as those in
(Zhang et al., 2003), 2004, (McDoniel et al., 2015)) would be a great
improvement over the modeling done for this paper, as the model used for this
work assumes LTE and hydrostatic equilibrium, which is not ideal for active
plume outgassing at extremely high temperatures.

###  6.2 Acknowledgements

This paper makes use of the following ALMA data: ADS/JAO.ALMA#2011.0.00779.S,
ADS/JAO.ALMA#2012.1.00853.S, ADS/JAO.ALMA#2015.1.00995.S, and
ADS/JAO.ALMA#2017.1.00670.S. ALMA is a partnership of ESO (representing its
member states), NSF (USA), and NINS (Japan), together with NRC (Canada), MOST
and ASIAA (Taiwan), and KASI (Republic of Korea), in cooperation with the
Republic of Chile. The Joint ALMA Observatory is operated by ESO, AUI/NRAO,
and NAOJ. The National Radio Astronomy Observatory is a facility of the
National Science Foundation operated under cooperative agreement by Associated
Universities, Inc. All data can be downloaded from the ALMA Archive. This
research was supported by the National Science Foundation. Facility: ALMA.

## References

  * Avni (1976) Avni, Y. 1976, The Astrophysical Journal, 210, 642
  * Brown (2001) Brown, M. E. 2001, Icarus, 151, 190
  * Brown (1974) Brown, R. A. 1974, in Symposium-International Astronomical Union, Vol. 65, Cambridge University Press, 527–531
  * Davies & Ennis (2011) Davies, A. G., & Ennis, M. E. 2011, Icarus, 215, 401
  * Davies et al. (1997) Davies, A. G., McEwen, A. S., Lopes-Gautier, R. M., et al. 1997, Geophysical research letters, 24, 2447
  * de Kleer & de Pater (2016) de Kleer, K., & de Pater, I. 2016, Icarus, 280, 405
  * de Kleer et al. (2021) de Kleer, K., Skrutskie, M., Leisenring, J., et al. 2021, The Planetary Science Journal, 2, 227
  * de Pater et al. (2021) de Pater, I., Keane, J. T., de Kleer, K., & Davies, A. G. 2021, Annual Review of Earth and Planetary Sciences, 49
  * de Pater et al. (2020) de Pater, I., Luszcz-Cook, S., Rojo, P., et al. 2020, The Planetary Science Journal, 1, 60
  * Feaga et al. (2009) Feaga, L. M., McGrath, M., & Feldman, P. D. 2009, Icarus, 201, 570
  * Fegley & Zolotov (2000) Fegley, B., & Zolotov, M. Y. 2000, Icarus, 148, 193
  * Geissler & McMillan (2008) Geissler, P., & McMillan, M. 2008, Icarus, 197, 505
  * Hunten et al. (1988) Hunten, D., Morgan, T., & Shemansky, D. 1988, Mercury, 562
  * Jessup & Spencer (2012) Jessup, K. L., & Spencer, J. R. 2012, Icarus, 218, 378
  * Killen & Ip (1999) Killen, R. M., & Ip, W.-H. 1999, Reviews of Geophysics, 37, 361
  * Lanzerotti et al. (1982) Lanzerotti, L., Brown, W., Augustyniak, W., Johnson, R., & Armstrong, T. 1982, The Astrophysical Journal, 259, 920
  * Lellouch et al. (2015) Lellouch, E., Ali-Dib, M., Jessup, K.-L., et al. 2015, Icarus, 253, 99
  * Lellouch et al. (1990) Lellouch, E., Belton, M., de Pater, I., Gulkis, S., & Encrenaz, T. 1990, Nature, 346, 639
  * Lellouch et al. (2007) Lellouch, E., McGrath, M. A., & Jessup, K. L. 2007, in Io After Galileo (Springer), 231–264
  * Lellouch et al. (2003) Lellouch, E., Paubert, G., Moses, J., Schneider, N. M., & Strobel, D. 2003, Nature, 421, 45
  * Lopes et al. (2004) Lopes, R. M., Kamp, L. W., Smythe, W. D., et al. 2004, Icarus, 169, 140
  * Luszcz-Cook & de Pater (2013) Luszcz-Cook, S. H., & de Pater, I. 2013, Icarus, 222, 379
  * McDoniel et al. (2015) McDoniel, W. J., Goldstein, D. B., Varghese, P. L., & Trafton, L. M. 2015, Icarus, 257, 251
  * McEwen (2002) McEwen, A. 2002, in AGU Spring Meeting Abstracts, Vol. 2002, P22A–02
  * McEwen et al. (1985) McEwen, A. S., Matson, D. L., Johnson, T. V., & Soderblom, L. A. 1985, Journal of Geophysical Research: Solid Earth, 90, 12345
  * Moses et al. (2002) Moses, J. I., Zolotov, M. Y., & Fegley Jr, B. 2002, Icarus, 156, 107
  * Moullet (2015) Moullet, A. 2015, Revolution in Astronomy with ALMA: The Third Year, 499, 295
  * Moullet et al. (2013) Moullet, A., Lellouch, E., Moreno, R., et al. 2013, The Astrophysical Journal, 776, 32
  * Pearl & Sinton (1982) Pearl, J., & Sinton, W. 1982, Hot spots on Io, The Satellites of Jupiter D. Morrison, 724–755, Univ. of Arizona Press, Tucson
  * Pennisi et al. (1988) Pennisi, M., Le Cloarec, M., Lambert, G., & Le Roulley, J. 1988, Earth and planetary science letters, 88, 284
  * Roth et al. (2020) Roth, L., Boissier, J., Moullet, A., et al. 2020, Icarus, 350, 113925
  * Saur & Strobel (2004) Saur, J., & Strobel, D. F. 2004, Icarus, 171, 411
  * Segatz et al. (1988) Segatz, M., Spohn, T., Ross, M., & Schubert, G. 1988, Icarus, 75, 187
  * Smyth & Marconi (1995) Smyth, W. H., & Marconi, M. 1995, The Astrophysical Journal, 441, 839
  * Spencer et al. (2005) Spencer, J. R., Lellouch, E., Richter, M. J., et al. 2005, Icarus, 176, 283
  * Symonds et al. (1992) Symonds, R. B., Reed, M. H., & Rose, W. I. 1992, Geochimica et Cosmochimica Acta, 56, 633
  * Symonds et al. (1994) Symonds, R. B., Rose, W. I., Bluth, G. J. S., & Gerlach, T. M. 1994, in Volatiles in Magma, ed. M. R. Carroll & J. R. Holloway (Mineralogical Society of America), 1–66
  * Tsang et al. (2016) Tsang, C. C., Spencer, J. R., Lellouch, E., Lopez-Valverde, M. A., & Richter, M. J. 2016, Journal of Geophysical Research: Planets, 121, 1400
  * Wall & Jenkins (2012) Wall, J. V., & Jenkins, C. R. 2012, Practical statistics for astronomers (Cambridge University Press)
  * Williams et al. (2011) Williams, D. A., Keszthelyi, L. P., Crown, D. A., et al. 2011, Geologic map of Io (US Department of the Interior, US Geological Survey)
  * Zhang et al. (2003) Zhang, J., Goldstein, D., Varghese, P., et al. 2003, Icarus, 163, 182
  * Zhang et al. (2004) —. 2004, Icarus, 172, 479

###  6.3 Appendix

Figure 12: Delta chi-square contour grids for all NaCl apertures analyzed in
this study. Overlain on each plot is an image of the aperture and a smaller
plot with the spectral line profile for the accompanying contour grid plot,
along with the best fit output model line profile, labelled with the velocity
at which the line profile is centered.  Figure 13: Delta chi-square contour
grids for all KCl apertures analyzed in this study. Overlain on each plot is
an image of the aperture and a smaller plot with the spectral line profile for
the accompanying contour grid plot, along with the best fit output model line
profile, labelled with the velocity at which the line profile is centered.