# Gauss’s proof of Descartes’s rule of signs

Melvyn B. Nathanson  Department of Mathematics
Lehman College (CUNY)
Bronx, NY 10468<EMAIL_ADDRESS>
###### Abstract.

For nonzero polynomials $f(x)$ with real coefficients, Descartes’s rule of
signs gives an upper bound for the number of positive zeros. A variation of an
argument of Gauss is used to prove Descartes’s theorem.

###### Key words and phrases:

Polynomials, Descartes’s rule of signs, Budan-Fourier theorem, multiplicity
vector, multiplicity of zeros, location of zeros, zero patterns, real
algebraic geometry.

###### 2000 Mathematics Subject Classification:

11B83, 11C08, 11B75, 12D10

##  1\. Variation of sign

The _sign_ of a real number $\lambda$ is the function

$\operatorname{\text{sign}}(\lambda)=\begin{cases}1&\text{ if $\lambda>0$}\\\
0&\text{ if $\lambda=0$}\\\ -1&\text{ if $\lambda<0$.}\end{cases}$

Let

$A=\left(a_{0},a_{1},a_{2},\ldots,a_{n}\right)$

be a sequence of real numbers. The associated _sign sequence_ is

$\operatorname{\text{sign}}(A)=\left(\operatorname{\text{sign}}(a_{0}),\operatorname{\text{sign}}(a_{1}),\operatorname{\text{sign}}(a_{2}),\ldots,\operatorname{\text{sign}}(a_{n})\right).$

Let $k+1$ be the number of nonzero terms in this sequence. There is a strictly
increasing sequence $0\leq j_{0}<j_{1}<\cdots<j_{k}\leq n$ such that
$\left(a_{j_{0}},a_{j_{1}},\ldots,a_{j_{k}}\right)$ is the subsequence of
nonzero terms in $A$. We have
$\operatorname{\text{sign}}\left(a_{j_{i}}\right)=\pm 1$ for all
$i\in\\{0,1,\ldots,k\\}$. The number of _sign variations_ or _sign changes_ in
the sequence $A$, denoted $V(A)$, is the number of pairs
$(a_{j_{i-1}},a_{j_{i}})$ such that
$\operatorname{\text{sign}}(a_{j_{i-1}})=-\operatorname{\text{sign}}(a_{j_{i}})$.
The number of sign changes in the sequence $(a_{0},a_{1},a_{2},\ldots,a_{n})$
equals the number of sign changes in the associated sign sequence
$\operatorname{\text{sign}}(A)$.

For example, if $A=(1,-1,0,1,-1)$, $A_{1}=(1,-1,0)$, and $A_{2}=(0,1,-1)$,
then

$V(A)=3,\quad V(A_{1})=1,\operatorname{\qquad\text{and}\qquad}V(A_{2})=1.$

Note that

$V(A)\neq V(A_{1})+V(A_{2}).$

Consider $n=4$ and $\ell=2$ in the following lemma.

######  Lemma 1.

Let $A=\left(a_{0},a_{1},a_{2},\ldots,a_{n}\right)$ be a sequence of real
numbers. For $\ell\in\\{1,2,\ldots,n-1\\}$, let

$A_{1}=\left(a_{0},a_{1},\ldots,a_{\ell}\right)$

and

$A_{2}=\left(a_{\ell},a_{\ell+1},\ldots,a_{n}\right).$

If $a_{\ell}\neq 0$, then

$V(A)=V(A_{1})+V(A_{2}).$

If $a_{i}=0$ for all $i\in\\{0,1,\dots,\ell\\}$ or if $a_{j}=0$ for all
$j\in\\{\ell,\ell+1,\ldots,n\\}$, then

$V(A)=V(A_{1})+V(A_{2}).$

If $a_{\ell}=0$ and if $a_{i}\neq 0$ for some $i<\ell$ and $a_{j}\neq 0$ for
some $j>\ell$ and if $\ell_{1}$ is the largest integer such that
$\ell_{1}<\ell$ and $a_{\ell_{1}}\neq 0$, and $\ell_{2}$ is the smallest
integer such that $\ell_{2}>\ell$ and $a_{\ell_{2}}\neq 0$, then

$V(A)=\begin{cases}V(A_{1})+V(A_{2})&\text{if
$\operatorname{\text{sign}}(a_{\ell_{1}})=\operatorname{\text{sign}}(a_{\ell_{2}})$}\\\
V(A_{1})+V(A_{2})+1&\text{if
$\operatorname{\text{sign}}(a_{\ell_{1}})\neq\operatorname{\text{sign}}(a_{\ell_{2}})$.}\end{cases}$

###### Proof.

This is a straightforward verification. ∎

######  Lemma 2.

Consider a sequence
$\varepsilon=(\varepsilon_{0},\varepsilon_{1},\ldots,\varepsilon_{k})$ such
that $\varepsilon_{i}\in\\{1,0,-1\\}$ for all $i\in\\{0,1,\ldots,n\\}$ and
$\varepsilon_{0}\varepsilon_{k}\neq 0$.

  1. (i)

If $\varepsilon_{0}=\varepsilon_{k}$, then $V(\varepsilon)$ is even.

  2. (ii)

If $\varepsilon_{0}=-\varepsilon_{k}$, then $V(\varepsilon)$ is odd.

  3. (iii)

If $\varepsilon_{i-1}\varepsilon_{i+1}=-1$ for some $i\in\\{1,\ldots,k-1\\}$,
then $V(\varepsilon_{i-1},\varepsilon_{i},\varepsilon_{i+1})=1$ for all
$\varepsilon_{i}\in\\{0,1,-1\\}$.

###### Proof.

Statements (i) and (ii) follow from the simple but fundamental observation
that changing signs an even number of times does not change the sign but
changing signs an odd number of times does change the sign.

To prove statement (iii), we observe that if
$\varepsilon_{i-1}=-\varepsilon_{i+1}=1$, then
$(\varepsilon_{i-1},\varepsilon_{i},\varepsilon_{i+1})=(1,1,-1)$ or $(1,0,-1)$
or $(1,-1,-1)$ and each of these sequences has one sign change. Similarly, if
$\varepsilon_{i-1}=-\varepsilon_{i+1}=-1$, then
$(\varepsilon_{i-1},\varepsilon_{i},\varepsilon_{i+1})=(-1,1,1)$ or $(-1,0,1)$
or $(-1,-1,1)$ and each of these sequences has one sign change. This completes
the proof. ∎

A $C^{n}$ function is a function with $n$ continuous derivatives. Let $f(x)$
be a $C^{n}$ function. For $\lambda$ in the domain of $f(x)$, we define the
_derivative sequence_

$D_{f}^{(n)}(\lambda)=\left(f(\lambda),f^{\prime}(\lambda),f^{\prime\prime}(\lambda),\ldots,f^{(n)}(\lambda)\right).$

The variation of sign function

$V^{(n)}_{f}(\lambda)=V\left(D_{f}^{(n)}(\lambda)\right)=V\left(f(\lambda),f^{\prime}(\lambda),f^{\prime\prime}(\lambda),\ldots,f^{(n)}(\lambda)\right)$

counts the number of sign changes in the derivative sequence.

In this paper we consider only polynomials with real coefficients. Let $f(x)$
be a polynomial of degree $n$. For all $\lambda\in\mathbf{R}$, there is the
Taylor expansion

$f(x)=\sum_{j=0}^{n}a_{j}(x-\lambda)^{j}$

with Taylor coefficients

$a_{j}=\frac{f^{(j)}(\lambda)}{j!}.$

It follows that

$\operatorname{\text{sign}}(a_{j})=\operatorname{\text{sign}}\left(f^{(j)}(\lambda)\right)$

for all $j\in\\{0,1,\ldots,n\\}$ and so

$V_{f}^{(n)}(\lambda)=V\left(a_{0},a_{1},\ldots,a_{n}\right)$

also counts the number of sign changes in the sequence of Taylor coefficients
of $f(x)$.

For example, the polynomial

$f(x)=10+8x-3x^{2}-5x^{3}+2x^{5}+7x^{8}+x^{9}$

has coefficient sequence

$A=(10,8,-3,-5,0,2,0,0,7,1).$

The corresponding sign sequence is

$\operatorname{\text{sign}}(A)=(1,1,-1,-1,0,1,0,0,1,1).$

There are two sign changes in these sequences and so $V^{(n)}_{f}(0)=2$.

Let $f(x)$ be a $C^{n}$ function and let

$g(x)=f(x+a).$

For all $j\in\\{0,1,\ldots,n\\}$ we have $g^{(j)}(x)=f^{(j)}(x+a)$ and so

$g^{(j)}(0)=f^{(j)}(a).$

Therefore,

$\displaystyle V_{g}^{(n)}(0)$
$\displaystyle=V\left(g(0),g^{\prime}(0),g^{\prime\prime}(0),\ldots,g^{(n)}(0)\right)$
(1)
$\displaystyle=V\left(f(a),f^{\prime}(a),f^{\prime\prime}(a),\ldots,f^{(n)}(a)\right)$
$\displaystyle=V_{f}^{(n)}(a).$

##  2\. Counting zeros

Let $f(x)$ be a nonzero polynomial with real coefficients. Let
$\mu_{f}(\lambda)$ be the multiplicity of the real or complex number $\lambda$
as a root of the polynomial $f(x)$. We define the _zero counting function_

$Z_{f}(a,b)=\sum_{a<\lambda<b}\mu_{f}(\lambda).$

This is the number (counting multiplicity) of real numbers $\lambda$ such that

$f(\lambda)=0\operatorname{\qquad\text{and}\qquad}a<\lambda<b.$

In particular, $Z_{f}(0,\infty)$ counts the number of positive roots of $f(x)$
and $Z_{f}(-\infty,0)$ counts the number of negative roots of $f(x)$.

We define similarly

$Z_{f}(a,b]=\sum_{a<\lambda\leq
b}\mu_{f}(\lambda).\operatorname{\quad\text{and}\quad}Z_{f}[a,b)=\sum_{a\leq\lambda<b}\mu_{f}(\lambda).$

Let $f(x)$ be a nonzero polynomial and let $g(x)=f(-x)$. For $\lambda>0$ we
have $f(-\lambda)=0$ if and only if $g(\lambda)=0$, and so

$Z_{f}(-\infty,0)=Z_{g}(0,\infty).$

For $a,b\in\mathbf{R}$ with $a<b$, we have

(2) $Z_{f}(a,b]=Z_{f}(a,\infty)-Z_{f}(b,\infty).$

Let $g(x)=f(x+a)$. Equivalently, $f(x)=g(x-a)$. We have $f(\lambda)=0$ if and
only if $g(\lambda-a)=0$. Because $\lambda>a$ if and only if $\lambda-a>0$, it
follows that

$Z_{f}(a,\infty)=Z_{g}(0,\infty).$

Similarly, if $h(x)=f(x+b)$, then

$Z_{f}(b,\infty)=Z_{h}(0,\infty).$

Relation (2) implies that if $f(b)\neq 0$, then

(3) $Z_{f}(a,b]=Z_{g}(0,\infty)-Z_{h}(0,\infty).$

We may use Descartes’s rule of signs (Theorem 1) to compute zero counting
functions of the form $Z_{f}(a,b]$.

If $\mu=\mu_{f}(0)$, then $f(x)=x^{\mu}\ell(x)$, where $\ell(x)$ is a
polynomial of degree $n-\mu$ such that $\ell(0)\neq 0$. For $\lambda\neq 0$,
we have $f(\lambda)=0$ if and only if $\ell(\lambda)=0$, and so the nonzero
roots of $f(x)$ are the nonzero roots of $\ell(x)$. Moreover,
$\mu_{f}(\lambda)=\mu_{\ell}(\lambda)$ for all $\lambda\neq 0$. Thus, to count
the number of nonzero roots of polynomials, it suffices to consider only
polynomials $f(x)$ with $f(0)\neq 0$, that is, polynomials with nonzero
constant terms.

######  Theorem 1 (Descartes).

Let $f(x)$ be a polynomial with $f(0)\neq 0$. There exists a nonnegative even
integer $\nu$ such that

$Z_{f}(0,\infty)=V^{(n)}_{f}(0)-\nu.$

######  Corollary 1.

If $V^{(n)}_{f}(0)=1$, then $f$ has exactly one positive root $\lambda$ and
$\lambda$ is a simple root.

###### Proof.

Because $\nu$ is nonnegative and even, if $\nu\neq 0$, then $\nu\geq 2$ and

$0\leq Z_{f}(0,\infty)=V^{(n)}_{f}(0)-\nu=1-\nu\leq-1$

which is absurd. Therefore, $\nu=0$ and
$Z_{f}(0,\infty)=V^{(n)}_{f}(0)-\nu=1$. This completes the proof. ∎

Let $a,b\in\mathbf{R}$ with $a<b$. Let $f(x)$ be a polynomial of degree $n$
with $f(b)\neq 0$. Consider the polynomials $g(x)=f(x+a)$ and $h(x)=f(x+b)$.
Descartes’s rule of signs and relation (3) give nonnegative even integers
$\nu_{g}$ and $\nu_{h}$ such that

$Z_{f}(a,\infty)=Z_{g}(0,\infty)=V^{(n)}_{g}(0)-\nu_{g}=V_{f}^{(n)}(a)-\nu_{g}$
$Z_{f}(b,\infty)=Z_{h}(0,\infty)=V^{(n)}_{h}(0)-\nu_{h}=V_{f}^{(n)}(b)-\nu_{h}$

From equation (3) we obtain

$Z_{f}(a,b]=V_{f}^{(n)}(a)-V_{f}^{(n)}(b)-\nu$

where

$\nu=\nu_{g}-\nu_{h}.$

This argument does not prove the nonnegativity of the integer $\nu$. The
nonnegativity of $\nu$ would immediately imply the following result, which is
the Budan-Fourier theorem.

######  Theorem 2 (Budan-Fourier).

Let $f(x)$ be a polynomial of degree $n$ and let $a,b\in\mathbf{R}$ with
$a<b$. There exists a nonnegative even integer $\nu$ such that

$Z_{f}(a,b)=V^{(n)}_{f}(a)-V^{(n)}_{f}(b)-\nu.$

The Budan-Fourier theorem implies Descartes’s rule of signs.

For the history and alternate proofs of Descartes’s theorem, see [1, 2, 3, 4,
6, 7]. A recent proof of the Budan-Fourier theorem is Nathanson [8].

##  3\. Gauss’s proof of Descartes’s theorem

The following result is the essential part of Gauss’s beautiful proof [5] of
Descartes’s theorem.

######  Lemma 3.

Let $h(x)$ be a nonzero polynomial with $h(0)\neq 0$ and let $\lambda>0$. If

$\ell(x)=(x-\lambda)h(x)$

then

$V^{(n)}_{\ell}(0)=V^{(n)}_{h}(0)+1+\nu$

for some nonnegative even integer $\nu$.

###### Proof.

Let

$h(x)=\sum_{j=0}^{m}c_{j}x^{j}$

be a polynomial of degree $m$ with $h(0)=c_{0}\neq 0$ and let

$V^{(m)}_{h}(0)=V(c_{0},c_{1},\ldots,c_{m})=v$

be the number of sign changes in the coefficient sequence of $h(x)$. Note that

$v\in\\{0,1,\ldots,m\\}.$

We shall construct an integer sequence $\left(j_{i}\right)_{i=0}^{v}$ with

(4) $0=j_{0}<j_{1}<j_{2}<\cdots<j_{v}\leq m$

such that

(5)
$V(c_{j_{i-1}},\ldots,c_{j_{i}-1})=0\operatorname{\qquad\text{and}\qquad}V(c_{j_{i-1}},\ldots,c_{j_{i}})=1$

for all $i=1,2,\ldots,v$, and

(6) $V(c_{j_{v}},\ldots,c_{m})=0.$

The construction is by induction. Let $j_{0}=0$. Let $i\in\\{1,\ldots,v\\}$.
If $j_{i-1}\in\\{0,1,\ldots,m\\}$ satisfies

$V(c_{0},\ldots,c_{j_{i-1}})=i-1\leq v-1<V^{(m)}_{h}(0)$

then there is a smallest integer $j_{i}>j_{i-1}$ such that

$\operatorname{\text{sign}}(c_{j_{i}})=-\operatorname{\text{sign}}(c_{j_{i-1}})$

and so

$V(c_{0},\ldots,c_{j_{i-1}},c_{j_{i}})=i.$

The sequence $(j_{i})_{i=0}^{v}$ satisfies conditions (4) –
(LABEL:Descartes-3). Moreover, if $j_{v}<m$, then

(7) $\operatorname{\text{sign}}(c_{j_{v}})=\operatorname{\text{sign}}(c_{m}).$

Consider the polynomial

$\displaystyle\ell(x)$
$\displaystyle=(x-\lambda)h(x)=\sum_{j=0}^{m}c_{j}x^{j+1}-\sum_{j=0}^{m}\lambda
c_{j}x^{j}$ $\displaystyle=-\lambda c_{0}+\sum_{j=1}^{m}\left(c_{j-1}-\lambda
c_{j}\right)x^{j}+c_{m}x^{m+1}$ $\displaystyle=\sum_{j=0}^{m+1}b_{j}x^{j}$

with

$b_{j}=\begin{cases}-\lambda c_{0}&\text{for $j=0$}\\\ c_{j-1}-\lambda
c_{j}&\text{for $j=1,\ldots,m$}\\\ c_{m}&\text{for $j=m+1$.}\end{cases}$

For all $i\in\\{1,\ldots,v\\}$ we have

$b_{j_{i}}=c_{j_{i}-1}-\lambda c_{j_{i}}.$

It follows from the construction of the sequence $(j_{i})_{i=0}^{v}$ that
either

$c_{j_{i}-1}=0$

or

$\operatorname{\text{sign}}(c_{j_{i}-1})=\operatorname{\text{sign}}(c_{j_{i-1}})=-\operatorname{\text{sign}}(c_{j_{i}}).$

Because $\lambda>0$, we have

$\operatorname{\text{sign}}(b_{j_{i}})=\operatorname{\text{sign}}(c_{j_{i}-1}-\lambda
c_{j_{i}})=-\operatorname{\text{sign}}(c_{j_{i}})$

and so

$\operatorname{\text{sign}}(b_{j_{i}})=-\operatorname{\text{sign}}(c_{j_{i}})=\operatorname{\text{sign}}(c_{j_{i-1}})=-\operatorname{\text{sign}}(b_{j_{i-1}}).$

We have $j_{v}\leq m$. It follows from (7) that

$\operatorname{\text{sign}}(b_{j_{v}})=-\operatorname{\text{sign}}(c_{j_{v}})=-\operatorname{\text{sign}}(c_{m})=-\operatorname{\text{sign}}(b_{m+1}).$

By Lemma 2, there exist nonnegative even integers
$\nu_{0},\nu_{1},\ldots,\nu_{v}$ such that

$V(b_{j_{i-1}},\ldots,b_{j_{i}})=1+\nu_{i-1}$

for all $i\in\\{1,\ldots,v\\}$ and

$V(b_{j_{v}},\ldots,b_{m+1})=1+\nu_{v}.$

Because $b_{j_{i}}\neq 0$ for all $i\in\\{1,\ldots,v\\}$, the addition formula
in Lemma 1 gives

$\displaystyle V^{(m+1)}_{\ell}(0)$
$\displaystyle=V(b_{0},\ldots,b_{m+1})=\sum_{i=1}^{v}V(b_{j_{i-1}},\ldots,b_{j_{i}})+V(b_{j_{v}},\ldots,b_{m+1})$
$\displaystyle=v+1+\sum_{i=0}^{v}\nu_{i}$ $\displaystyle=V^{(m)}_{h}(0)+1+\nu$

where $\nu=\sum_{i=0}^{v}\nu_{i}$ is a nonnegative even integer. This
completes the proof. ∎

######  Lemma 4.

Let $h(x)$ be a nonzero polynomial with $h(0)\neq 0$. Let
$(\lambda_{i})_{i=1}^{k}$ be a sequence of not necessarily distinct positive
numbers. If

$f(x)=h(x)\prod_{i=1}^{k}(x-\lambda_{i})$

then

$V^{(n)}_{f}(0)=k+V^{(n)}_{h}(0)+\nu$

for some nonnegative even integer $\nu$.

###### Proof.

This follows from Lemma 3 by induction on $k$. ∎

We now prove Descartes’s rule of signs (Theorem 1).

###### Proof.

Let $f(x)$ be a monic polynomial of degree $n$ with $f(0)\neq 0$ and let

$Z_{f}(0,\infty)=k.$

Let $\Re(z)$ denote the real part of the complex number $z$.

Let $(\lambda_{i})_{i=1}^{n}$ be the sequence of $n$ not necessarily distinct
real and complex roots of $f(x)$. Note that $\lambda_{i}\neq 0$ for all
$i\in\\{1,\ldots,n\\}$. We order the roots so that, for nonnegative integers
$k,\ell,m$ with

$k+\ell+2m=n$

we have

  1. (i)

$(\lambda_{i})_{i=1}^{k}$ is the sequence of positive roots of $f(x)$,

  2. (ii)

$(\lambda_{i})_{i=k+1}^{k+\ell}$ is the sequence of negative roots of $f(x)$,

  3. (iii)

$(\lambda_{i})_{i=k+\ell+1}^{k+\ell+2m}$ is the sequence of nonreal complex
roots of $f(x)$ with

$\overline{\lambda_{k+\ell+r}}=\lambda_{k+\ell+r+m}.\qquad\text{for
$r\in\\{1,\ldots,m\\}$.}$

For every nonreal complex number $\lambda_{i}$, we have

$(x-\lambda_{i})(x-\overline{\lambda_{i}})=x^{2}-2\Re(\lambda_{i})x+|\lambda_{i}|^{2}.$

The constant term of this quadratic polynomial is positive and so the constant
term of the polynomial

$\displaystyle\prod_{i=k+\ell+1}^{k+\ell+2m}(x-\lambda_{i})$
$\displaystyle=\prod_{i=k+\ell+1}^{k+\ell+m}(x-\lambda_{i})(x-\overline{\lambda_{i}})$
$\displaystyle=\prod_{i=k+\ell+1}^{k+\ell+m}\left(x^{2}-2\Re(\lambda_{i})x+|\lambda_{i}|^{2}\right)$

is positive:

$\prod_{i=k+\ell+1}^{k+\ell+m}|\lambda_{i}|^{2}>0$

Because $\lambda_{i}<0$ for all $i\in\\{k+1,\ldots,k+\ell\\}$, the constant
term of the polynomial

$\prod_{i=k+1}^{k+\ell}(x-\lambda_{i})$

is also positive:

$\prod_{i=k+1}^{k+\ell}|\lambda_{i}|>0.$

Therefore, the constant term of the monic polynomial

$h(x)=\prod_{i=k+1}^{n}(x-\lambda_{i})$

is positive:

$h(0)=\prod_{i=k+1}^{k+\ell}|\lambda_{i}|\prod_{i=k+\ell+1}^{k+\ell+m}|\lambda_{i}|^{2}>0.$

The leading coefficient and the constant term of $h(x)$ are positive. By Lemma
2, the number of sign changes in the coefficient sequence of $h(x)$ is the
nonnegative even integer $V^{(n-k)}_{h}(0)=\nu_{h}$.

We have

$f(x)=h(x)\prod_{i=1}^{k}(x-\lambda_{i}).$

Because $(\lambda_{i})_{i=1}^{k}$ is a sequence of positive real numbers,
Lemma 4 implies that there is a nonnegative even integer $\nu_{f}$ such that

$V^{(n)}_{f}(0)=V^{(n)}_{h}(0)+k+nu_{f}=k+\nu_{h}+\nu_{f}=k+\nu$

for $\nu=\nu_{h}+\nu_{f}$, and so

$Z_{f}(0,\infty)=k=V^{(n)}_{f}(0)-\nu.$

This completes the proof. ∎

## References

  * [1] S. Basu, R. Pollack, M.-F. Roy, _Algorithms in Real Algebraic Geometry_ , Springer, New York, 2006.
  * [2] Michael Bensimhoun, Historical account and ultra-simple proofs of Descartes’ rule of signs, De Gua, Fourier and Budan’s rules, arXiv:1309.6664v5, 2016.
  * [3] N. B. Conkwright, An elementary proof of the Budan-Fourier theorem, Amer. Math. Monthly 50 (1943), 603–605.
  * [4] L. E. Dickson, _First Course in the Theory of Equations_ , John Wiley & Sons, New York, 1922; reprinted by Wentworth Press in 2019.
  * [5] C. F. Gauss, Beweis eines algebraischen Lehrsatzes, J. reine angew. Math. 3 (1828), 5–8.
  * [6] N. Jacobson, _Basic Algebra 1_ , W. H. Freeman and Company, San Francisco, 1974.
  * [7] A. Hurwitz, Über den Saz von Budan-Fourier, Math. Ann. 71 (1912), 584–591.
  * [8] M. B. Nathanson, Multiplicity vectors and the Budan-Fourier theorem, preprint, 2022.


# The Minimum Variability Time Scale and its Relation to Pulse Profiles of
Fermi GRBs

G. A. MacLachlan1, A. Shenoy1, E. Sonbas2,3, K. S. Dhuga1, A. Eskandarian1, L.
C. Maximon1, and W. C. Parke1
1Department of Physics, The George Washington University, Washington, D.C.
20052, USA.
2University of Adiyaman, Department of Physics, 02040, Adiyaman, Turkey.
3NASA Goddard Space Flight Center, Greenbelt, MD 20771, USA.
E-mail<EMAIL_ADDRESS>(GAM)

###### Abstract

We present a direct link between the minimum variability time scales extracted
through a wavelet decomposition and the rise times of the shortest pulses
extracted via fits of 34 Fermi GBM GRB light curves comprised of 379 pulses.
Pulses used in this study were fitted with log-normal functions whereas the
wavelet technique used employs a multiresolution analysis that does not rely
on identifying distinct pulses. By applying a corrective filter to published
data fitted with pulses we demonstrate agreement between these two independent
techniques and offer a method for distinguishing signal from noise.

###### keywords:

Gamma-ray bursts

##  1 Introduction

One approach for probing light curves which has received attention (Nemiroff,
2000; Norris et al., 2005; Hakkila & Nemiroff, 2009; Nemiroff, 2012) is to
express them as a series of displaced pulses, each with a parametric form.
There is an appeal to this approach because fitting routines are well-
understood and interpretations of rise time, decay time, full width at half
max, etc, are possible. On the other hand, one must make certain assumptions
when using the pulse-fitting procedure such as the choice of the functional
form to use for an individual pulse and the number of parameters to be
included in the fitting function. Moreover, light with high variability at low
power may show variations which are not statistically significant.

A complementary approach using a wavelet-based analysis of a set of both long
and short GRB light curves is discussed by MacLachlan et al. (2012) in which a
time scale, $\tau_{\beta}$, is identified that marks the transition from white
noise to a power law in the power density spectrum (a $f^{-\alpha}$ behavior).
It is argued that over time scales smaller than $\tau_{\beta}$ the light
curves appear stochastic and signal power is distributed uniformly. At time
scales larger than $\tau_{\beta}$, identifiable structures (such as pulses)
with signal power are no longer distributed uniformly over the periods of
light variation. For this reason $\tau_{\beta}$ is referred to as the _minimum
variability time scale_.

The analysis presented in (MacLachlan et al., 2012) is a non-parametric
approach to probing light curves for time scales. It makes no assumptions
about the nature of the structures in a given light curve that give rise to
the $f^{-\alpha}$ character. The technique, however, offers no firm connection
between $\tau_{\beta}$ and the constituent structures although it seems
reasonable to associate $\tau_{\beta}$ with the scale of the smallest emitting
structures present.

Results from an application of a log-normal pulse-fitting procedure to GRB
light curves have been published by Bhat et al. (2012). In this paper we make
a meta-analysis of the timing results presented by Bhat et al. (2012) compared
with the techniques of MacLachlan et al. (2012) for a set of 34 GRBs used in
both studies.

##  2 Analysis

We begin by considering the relation between $\tau_{\beta}$ and the pulse
parameters given in Table 3 of Bhat et al. (2012). The parameters with
temporal units in Table 3 are: _time-since-trigger_ , _rise time_ , _decay
time_ , and _FWHM_. In all, 34 GRBs comprising 379 pulses are considered here.
We note that rise time, decay time, and FWHM as presented in Table 3 of Bhat
et al. (2012) are tightly correlated and for the argument that follows are
interchangeable without affecting the conclusions. However, we use rise time
to make our argument because, as Bhat et al. (2012) noted, rise times are
observed to be shorter than decay time and FWHM (see Table 3 in Bhat et al.
(2012)). We considered only those light curves from NaI detectors and summed
over the energy acceptance as in Table 3 of Bhat et al. (2012) and in
MacLachlan et al. (2012).

In Fig. 1 we plot the rise time for all 379 pulses (34 GRBs) along the
vertical axis and $\tau_{\beta}$ along the horizontal. Note that for each GRB
for which one $\tau_{\beta}$ is computed, there is a possibility for multiple
pulses and therefore multiple rise times, hence the vertical columns of rise
times for a single value of $\tau_{\beta}$. For a given column of pulse times
the shortest pulse rise times are at the bottom and one finds larger rise
times by moving up the column. An equality line is also shown which is the
locus where $\tau_{\beta}$ equals rise time. Arguing as we do that
$\tau_{\beta}$ represents the minimum variability time scale the space in the
$\tau_{\beta}$-rise time plane below the equality line should be interpreted
as a structureless white noise region. If some method were capable of
discerning light curve structure in the region we define as white noise, then
our assertion of having found a minimum variability time scale will have been
disproven. Indeed, in Fig. 1 there are 27 pulses with rise times below the
equality line. The uncertainties accompanying these 27 rise times are small,
making their intrusion into the white noise region significant.

Table 1: Pulses with rise times smaller than $\tau_{\beta}$ but larger than bin widths. Pulse Number (#), rise (time), $\delta$rise (time), and bin width are taken from Table 3 of Bhat et al. (2012). The column labeled $\Delta$rise (time) is obtained by combining $\delta$rise and bin width in quadrature. The columns diff and % diff refer to the differences between $\tau_{\beta}$ and rise (time). GRB | Pulse # |  $\tau_{\beta}$ [s] |  $\delta\tau^{-}_{\beta}$ [s] |  $\delta\tau^{+}_{\beta}$ [s] | rise [s] |  $\delta$rise [s] |  $\Delta$rise [s] | bin width [s] | diff [s] | % diff
---|---|---|---|---|---|---|---|---|---|---
080825593 | 17 | 0.0775 | 0.0138 | 0.0168 | 0.0660 | 0.0003 | 0.0200 | 0.0200 | 0.0115 | 17.4
080916009 | 9 | 0.2266 | 0.063 | 0.0872 | 0.1670 | 0.0022 | 0.1500 | 0.1500 | 0.0596 | 35.7
080916009 | 15 | 0.2266 | 0.063 | 0.0872 | 0.2190 | 0.0018 | 0.1500 | 0.1500 | 0.0076 | 3.47
080916009 | 20 | 0.2266 | 0.063 | 0.0872 | 0.1930 | 0.0016 | 0.1500 | 0.1500 | 0.0336 | 17.4
080916009 | 22 | 0.2266 | 0.063 | 0.0872 | 0.2260 | 0.0027 | 0.1500 | 0.1500 | 0.0006 | 0.265
080925775 | 10 | 0.1748 | 0.0425 | 0.0562 | 0.1710 | 0.0035 | 0.0501 | 0.0500 | 0.0038 | 2.22
081215784 | 1 | 0.0319 | 0.0043 | 0.005 | 0.0218 | 0.0020 | 0.0054 | 0.0050 | 0.0101 | 46.3

However, a closer inspection of Table 3 of Bhat et al. (2012) reveals that
there are 20 light pulses in Fig. 1 with rise times that are _smaller_ than
the smallest bin widths, in some cases smaller by factors of ten or a hundred.
Moreover, of those 20 pulses there are 16 pulses in Fig. 1 with full widths at
half max that are _smaller_ than the smallest bin widths and indeed those 16
all fall below the equality line. While it seems that inclusion of these
pulses in Table 3 is important for the sake of completeness, we question the
physical reality of these pulses. Note that in MacLachlan et al. (2012) all
light curves are binned at 200 microseconds. Fig. 2 shows the effect of
removing the 20 non-physical pulses.

Figure 1: Minimum variability time scale versus rise time in the Observer
frame. The rise times are taken from Table 3 of Bhat et al. (2012). The line
represents the locus where $\tau_{\beta}$ = rise time. We identify the area
below the line with white noise. The data are expected to press up against the
line from above but not to cross it.  Figure 2: Minimum variability time scale
versus rise time in the Observer frame. We have removed all pulses (20) with
rise times smaller than the light curve bin width.

Note that in Fig. 2 the white noise region has been vacated by all but seven
points and none of the pulse rise times above the equality line have been
disturbed by the bin width cut. For the seven points that remain beneath the
equality line, we show in Table 1 that six are within one sigma of the
equality line.

We make one other point regarding the pulse rise times in Fig. 1 and Fig. 2,
in particular regarding the size of the uncertainties. Of the 379 pulse rise
times reported by Bhat et al. (2012) and used for this meta-analysis, 301 have
uncertainties smaller than the binning of the light curve, in some cases
hundreds or thousands of times smaller. We argue that a conservative estimate
of the uncertainties for the pulse rise times should be no smaller than a bin
width. Thus, we add in quadrature a bin width (as reported in Table 3 of Bhat
et al. (2012)) to the rise time uncertainties (also reported in Table 3 of
Bhat et al. (2012)) and plot the result in Fig. 3.

In Fig. 4 we plot only the smallest rise times for each GRB against
$\tau_{\beta}$. We argue that by rejecting pulse rise times smaller than light
curve bin widths and by folding rise time uncertainties with a bin width we
get strong evidence that Bhat et al. (2012) and MacLachlan et al. (2012) have
tracked the same physical observables over approximately three orders of
magnitude using independent methods.

Figure 3: Minimum variability time scale versus rise time in the Observer
frame as in Fig. 2. We have folded a single bin width into the rise time
uncertainties. Figure 4: Minimum variability time scale versus rise time in
the Observer frame as in Fig. 3 but with only smallest rise times included.
Note that the equality line between $\tau_{\beta}$ and rise time marks a
boundary between scaling processes and white noise and gives substance to the
interpretation of the minimum variability time scale.

##  3 Results and Discussion

For a large sample of short and long Fermi GBM bursts, MacLachlan et al.
(2012) used a technique based on wavelets to determine the minimum variability
time scale, $\tau_{\beta}$. The authors associate this time scale with a
transition from red-noise processes to parts of the power spectrum dominated
by white noise or random noise components. Accordingly, the authors note that
this time scale is the shortest resolvable variability time for physical
processes intrinsic to the GRB. In addition, histograms of the values of
$\tau_{\beta}$ for long and short GRBs were shown to exhibit a clear temporal
offset in the mean $\tau_{\beta}$ values for long and short GRBs.

In a separate analysis, using a particular functional form for pulse shapes,
Bhat et al. (2012) have extracted an extensive set of key pulse parameters
such as rise times, decay times, widths (FWHM), and times since trigger for a
host of bright GRBs detected by Fermi/GBM. Using the FWHM values, these
authors also reported a significant temporal offset between the mean values
for long and short GRBs.

Although neither group offers a concrete explanation for the temporal
difference between the distributions of long and short GRBs, it is noteworthy
that they arrive at a result which is quantitatively in good agreement with
one another, especially having used independent approaches. Both sets of
analyses also suggest scaling trends between characteristic timescales. In the
case of minimum variability timescales the trend is between $\tau_{\beta}$ and
the duration of the burst, typically denoted by $T_{90}$. For the pulse-shape
analysis, the trend is more readily evident and is demonstrated through a
number of positive correlations involving key parameters such as rise times,
decay times and FWHM times.

It is relatively straightforward to interpret the scaling trends in terms of
the internal shock model in which the basic units of emission are assumed to
be pulses that are produced via the collision of relativistic shells emitted
by the central engine. In the case of the pulse-fitting method this is
essentially the default assumption. Indeed, Quilligan et al. (2002) in their
study of the brightest BATSE bursts with $T_{90}>2$ sec were the first to
demonstrate this explicitly by identifying and fitting distinct pulses and
showing a strong positive correlation between the number of pulses and the
duration of the burst. More recent studies, Bhat & Guiriec (2011); Hakkila &
Cumbee (2008); Hakkila & Preece (2011) have provided further evidence for the
pulse paradigm view of the prompt emission in GRBs.

The wavelet analysis does not, however, rely on identifying distinct pulses
but instead uses the multiresolution capacity of the wavelet technique to
resolve the smallest temporal scale present in the prompt emission.
Nonetheless, as MacLachlan et al. (2012) have demonstrated, if the smallest
temporal scale is due to pulse emissions, then we can still get a measure of
the upper bound on the number of pulses in a given burst through the ratio
$T_{90}/\tau_{\beta}$. In the simple model in which a pulse is produced every
time two shells collide, the ratio $T_{90}/\tau_{\beta}$, should show a
correlation with the duration of the burst. Indeed, this correlation was
reported by MacLachlan et al. (2012).

The similar trends of scaling demonstrated by these two methods, not only
suggest the robustness of both methods, but also point, perhaps more
importantly, to an underlying interconnection between key parameters extracted
by the two techniques. In other words, the minimum variability time scale
extracted by the wavelet technique is directly related to key pulse time
parameters such as rise times (as depicted in Fig. 4), under suitably
controlled pulse-fitting methods.

##  4 CONCLUSIONS

Through a meta-analysis of results presented by Bhat et al. (2012) and by
MacLachlan et al. (2012), we have studied the relationship between key
parameters that describe the temporal properties of a sample of prompt-
emission light curves for long and short-duration GRBs detected by the
Fermi/GBM mission. We compare the minimum variability timescale extracted
through a technique based on wavelets, with the pulse-time parameters
extracted through a pulse-fitting procedure. Our main results are summarized
as follows:

a) Both methods indicate a temporal offset between short and long-duration
bursts. The quantitative agreement between the two methods is quite good.

b) Both methods point to scaling trends between characteristic timescales. In
the case of the pulse-fitting method the scaling appears to involve parameters
such as rise times, FWHM, and pulse intervals. For the wavelet technique, the
scaling involves a correlation between the minimum variability time scale and
the duration of the bursts.

c) By demonstrating a strong positive correlation between $\tau_{\beta}$ and
the rise time of the shortest fitted pulses, we provide for the first time, a
direct link between the shortest resolvable temporal structure in a GRB light
curve with that of a key pulse profile parameter.

d) By combining the two techniques, we have shown that one can arrive at a
much tighter demarcation of the boundary between the power spectrum domains
that separate red noise and white noise processes.

##  5 ACKNOWLEDGEMENTS

The NASA grant NNX11AE36G provided partial support for this work and is
gratefully acknowledged. The authors, in particular GAM and KSD, acknowledge
very useful discussions with Jon Hakkila and Narayan Bhat.

## References

  * Bhat & Guiriec (2011) Bhat, P. N., & Guiriec, S. 2011, BASI, 39, 471
  * Bhat et al. (2012) Bhat, P. N. et al. 2012, _ApJ._ , 744, 141 (online-version)
  * Hakkila & Cumbee (2008) Hakkila, J. & Cumbee, R. 2008, arXiv:0901.3171v1 [astro-ph.HE].
  * Hakkila & Nemiroff (2009) Hakkila, J. & Nemiroff, R. J. 2009, _ApJ._ , 705, 372
  * Hakkila & Preece (2011) Hakkila, J. & Preese, R. 2011, _ApJ._ , 740, 104
  * MacLachlan et al. (2012) MacLachlan, G. A., et al. 2012, arXiv:1201.4431v2 [astro-ph.HE]
  * Nemiroff (2000) Nemiroff, R. J. et al. 2000, _ApJ._ , 544, 805
  * Nemiroff (2012) Nemiroff, R. J. 2012, _MNRAS_ , 419, 1650
  * Norris et al. (2005) Norris, J. P. et al. 2005, _ApJ._ , 627, 324
  * Quilligan et al. (2002) Quilligan, F., et al. 2002, _A. & A._, 385, 377


# Observing Jupiter’s radio emissions using multiple LOFAR stations: a first
case study of the Io-decametric emission using the Irish IE613, French FR606
and German DE604 stations

Corentin. K. Louis1, C. M. Jackman1, J.-M. Grießmeier2,3, O. Wucknitz4, D. J.
McKenna1,5, P. C. Murphy5,1, P. T. Gallagher1,5, E. Carley1, D. Ó Fionnagáin6,
A. Golden6,7, J. McCauley5, P. Callanan8, M. Redman9, C. Vocks10
1School of Cosmic Physics, DIAS Dunsink Observatory, Dublin Institute for
Advanced Studies, Dublin 15, Ireland,
2LPC2E - Université d’Orléans /CNRS, France,
3Station de Radioastronomie de Nançay, Observatoire de Paris, PSL Research
University, CNRS, Univ. Orléans, 18330 Nançay, France,
4Max-Planck-Institut für Radioastronomie, Auf dem Hügel 69, 53121 Bonn,
Germany,
5School of Physics, Trinity College Dublin, Dublin 2, Ireland,
6Astrophysics Research Group, School of Mathematics, Statistics and Applied
Mathematics, National University of Ireland Galway, University Road, Galway,
Ireland,
7 Armagh Observatory and Planetarium, College Hill, Armagh, N. Ireland,
8Department of Physics, University College Cork, Cork, Ireland,
9Centre for Astronomy, School of Physics, National University of Ireland
Galway, University Road, Galway, Ireland,
10Leibniz-Institut für Astrophysik Potsdam (AIP), An der Sternwarte 16, 14482
Potsdam, Germany E-mail<EMAIL_ADDRESS>
(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

The Low Frequency Array (LOFAR) is an international radio telescope array,
consisting of 38 stations in the Netherlands and 14 international stations
spread over Europe. Here we present an observation method to study the jovian
decametric radio emissions from several LOFAR stations (here DE604, FR606 and
IE613), at high temporal and spectral resolution. This method is based on
prediction tools, such as radio emission simulations and probability maps, and
data processing. We report an observation of Io-induced decametric emission
from June 2021, and a first case study of the substructures that compose the
macroscopic emissions (called millisecond bursts). The study of these bursts
make it possible to determine the electron populations at the origin of these
emissions. We then present several possible future avenues for study based on
these observations. The methodology and study perspectives described in this
paper can be applied to new observations of jovian radio emissions induced by
Io, but also by Ganymede or Europa, or jovian auroral radio emissions.

###### keywords:

planets and satellites: individual: Jupiter – planets and satellites:
individual: Io – planets and satellites: aurorae

††pubyear: 2021††pagerange: Observing Jupiter’s radio emissions using multiple
LOFAR stations: a first case study of the Io-decametric emission using the
Irish IE613, French FR606 and German DE604 stations–Observing Jupiter’s radio
emissions using multiple LOFAR stations: a first case study of the Io-
decametric emission using the Irish IE613, French FR606 and German DE604
stations

##  1 Introduction

In our solar system, Jupiter is the planet with the most intense radio
emissions, covering the broadest frequency range from a few kHz (where Quasi-
Periodic bursts are seen) up to $40$ MHz (the decametric, or DAM, emissions).
From the ground, only the DAM emissions are observable, as all other lower
frequency radio emissions are blocked by the ionospheric cut-off at $\sim 10$
MHz. These DAM radio emissions were discovered quite early by Burke & Franklin
(1955). A few years later, Bigg (1964) discovered that part of the decametric
emission was controlled by the interaction between Jupiter’s magnetosphere and
the Galilean moon Io. The decametric emissions are known to be generated by
the electron cyclotron maser instability (CMI) in the jovian magnetosphere,
which occurs when a circularly polarised wave resonates with the gyration
movement of electrons with relativistic energies (Twiss, 1958; Hirshfield &
Bekefi, 1963; Wu & Lee, 1979; Zarka, 1998; Treumann, 2006; Louarn et al.,
2017; Louis et al., 2020b). The source regions of the DAM emission are located
above the atmosphere, on magnetic field lines of magnetic apex (distance of
the magnetic field lines at the magnetic equator) between 15 and 50 RJ (1
R${}_{\mathrm{J}}=71492$ km the Jovian radius, Louis et al., 2019a). The CMI
gives rise to emission at the local electron cyclotron frequency
$f_{\mathrm{ce}}$ (which is proportional to the local magnetic field amplitude
B), along a hollow cone (with a thickness of $\sim 1\degr$, Kaiser et al.,
2000) at large angle with respect to the local magnetic field line (from $\sim
75\degr$ up to $90\degr$, Louis et al., 2017b; Pritchett, 1986). From the
observer’s point of view, these emissions have an arc-shape in a time-
frequency map (Marques et al., 2017).

Our knowledge on the CMI emissions allows us to simulate these arcs (Louis et
al., 2019b), which makes it possible to retrieve the energy of the electrons
that produce these emissions (Hess et al., 2008; Louis et al., 2017b), as well
as discover new components, such as decametric emissions induced by the
Galilean moons Europa and Ganymede (Louis et al., 2017c).

However, the physics of the substructures that compose these macroscopic
emissions is less well known. Their study requires observations at high
temporal and spectral resolution. Unfortunately, most instruments do not have
sufficiently high resolution to make the required measurements. For
instruments on board space missions, the telemetry often requires reduced
frequency and time resolution. For ground-based radio telescopes, high spatial
resolution requires tied-array beam forming or interferometric measurements
with a large number of antennas, but also the ability to store very large
amounts of data. The latest generation of radio telescopes, such as LOFAR (van
Haarlem et al., 2013) or NenuFAR (Zarka et al., 2012; Zarka et al., 2015),
provides measurements at very high temporal and spectral resolution.

Only a few authors have looked at these millisecond micro-bursts that composed
the arc-shape radio emission. Zarka et al. (1996) were able to observe these
millisecond bursts of Io-DAM emission using the Nançay Decameter Array (NDA,
Boischot et al., 1980) and the UTR-2 (Ukrainian T-shaped Radio telescope,
Braude et al., 1978) observatory, with a $10$ ms per $13$ kHz resolution, as
well as Hess et al. (2007a) using only the NDA observatory with a $3$ ms per
$50$ kHz resolution. These two studies showed that these millisecond bursts
are due to electron bunches propagating along the magnetic field lines. Abrupt
changes in the drift rate of these millisecond bursts can also reveal electric
potential drops along these field lines. Hess et al. (2007b) were able to
model these millisecond bursts by assuming that electrons are accelerated by
Alfvén waves in the Io flux tube. Zarka (2004) showed the feasibility to do
fast radio imaging of Jupiter’s magnetosphere at radio frequency using a
unique LOFAR station.

In this article, we show that the inclusion of the Irish extension of the
LOFAR telescope (I-LOFAR) and the use of both simulations and improved
probability maps can extend and enhance the observation of these millisecond
structures. In the following, we present the methodology of our observations,
and show the results of a first observation taken in June 2021.

##  2 Methodology

Figure 1: Map showing the location of LOFAR stations across Europe. 38
stations are located in the Netherlands, while 14 (soon 15) stations are
located in 7 (soon 8) other European countries. Here are highlighted the three
stations that were used in the first case study described in Section 3: the
Irish IE613, the French FR606 and the German DE604 stations.

###  2.1 LOFAR stations

The Low Frequency Array (LOFAR) is an international radio telescope array (see
Figure 1, consisting of 38 stations in the Netherlands and 14 international
stations spread over Europe. In this present study, only the DE604, FR606 and
IE613 stations are used (highlighted in red Fig. 1).

At each LOFAR station, a backend is used to record and analyse the raw beam
formed data in real-time. At the DE604 and FR606 stations, raw data packets
were written to disk after lossless compression (using the _Zstandard_
algorithm). At the IE613 station, the REALTA (REAL-time Transient Acquisition,
Murphy et al., 2021) backend was used. Similar successes have already been
achieved with these different backends and used to conduct comparative studies
(see e.g., Grießmeier, J.-M. et al., 2021). After data has been recorded to
disk, _digifil_ (as part of the _dspsr_ package, van Straten & Bailes, 2011)
was used at all stations to generate the the antenna (auto-)correlation
factors X/Y from the underlying voltage data. Finally, a Jovian emission
processing pipeline111https://github.com/CorentinLouis/ILOFAR has been run to
retrieve the Stokes I (intensity) and V (circular polarization) parameters
from the antenna (auto-)correlation factors.

For the study of the jovian decametric radio emissions (extending up to $40$
MHz), only the LOFAR Low Band Antenna (LBA) is used, which allows to make
observations in the $8$-$90$ MHz frequency range, with a temporal resolution
that can go down to $5$ ns.

###  2.2 Optimal observation conditions

To be able to observe Jupiter’s radio emissions in the best possible
conditions, the ionosphere should be in a quiet state to avoid supplementary
interference, thus during night. It is therefore optimal to observe when
Jupiter is near opposition. Jupiter also has to be high enough in the sky
($\gtrsim 10\degr$), to get more signal in the lobe of the antennas. A good
approximation of the international LOFAR antennas effective area can be given
by the following equation (Grießmeier, J.-M. et al., 2021):

$A_{\mathrm{eff}}=A_{\mathrm{eff}_{\mathrm{max}}}~{}\mathrm{cos}^{2}z,$ (1)

where $z$ is the zenith angle of the source and
$A_{\mathrm{eff}_{\mathrm{max}}}$ the maximal frequency-dependent effective
area ($A_{\mathrm{eff}_{\mathrm{max}}}$ = [$3974.0$-$2516.0$] m2 in the range
[$15$-$30$] MHz, see Appendix B of van Haarlem et al., 2013, for more
details).

###  2.3 Emission probability

Figure 2: Example of results given by the online jovian radio probability tool
(https://jupiter-probability-tool.obspm.fr/). In this example, the station
I-LOFAR has been chosen (_Observatory_ entry, upper-left), for an observation
between 7 June 2021 18:00 UTC and 8 June 2021 18:00 UTC (_Start Time_ and _End
Time_ entries, upper-middle). The left-hand panel displays the occurrence
probability map of Io-DAM emission (_Background map_ entry, upper-right), in
function of the Galilean moon position with respect to the observer (phase of
the moon, y-axis) and the position of the observer (Central Meridian
Longitude, or CML, x-axis). The different colors correspond to different Io-
DAM emissions (see text). The position of Jupiter in this map for the chosen
time window is given by the black line (one-hour tick). The miniature image of
Jupiter gives the exact position for the time given in the _Adjust displayed
time_ entry (upper-middle). The right-hand panel displays the ExPRES
simulation for a given moon (_Satellite_ entry, upper-right). Blue corresponds
to emission sources in the northern hemisphere while red corresponds to
emission sources in the southern hemisphere. The thin red dashed line
corresponds to the time given in the _Adjust displayed time_ entry (upper-
middle). The white area gives the time window in which Jupiter is visible
above a certain elevation (_Elevation Threshold_ entry, upper-right).

Observations at very high temporal resolutions are very demanding in terms of
disk space ($\sim 7.6$ Gb per minute of observation in the case of this
study). We must therefore be sure that Jovian radio emission will be
observable during the time window when Jupiter is visible in the sky (at most
8 hours per day, due to the rotation of the Earth).

To predict jovian radio emissions, we use probability maps and simulations,
that have been both gathered in an interactive online tool, called _Jupiter
Radio Probability Tool_ 222https://jupiter-probability-tool.obspm.fr/. An
example is displayed in Figure 2. This tool allows the user to choose (i) an
observatory site from a pre-established list (_Observatory entry_ , upper-
left), or to enter the GPS coordinates (here ‘I-LOFAR’ is chosen as the
Observatory), (ii) a time window (_Start Time_ and _End Time_ entries, upper-
middle), here from 7 June 2021 18:00 UTC to 8 June 2021 18:00 UTC), and
finally (iii) a background map (upper-right entry) for the probability map
(left-panel).

The probability maps (see Fig. 2 left panel) show the position of the visible
emissions as a function of the jovian longitude of the observer (Central
Meridian Longitude, CML) and the phase of the Galilean moon (position of the
satellite in the observer’s frame, counted positively in the direction of
rotation of the satellite, with the origin at the opposition). The example
Figure 2 (left panel) displays the probability map for Io-DAM emission. The
different colors corresponds to different type of Io-DAM emission,
corresponding to radio sources at different positions with respect to the
observer (A: North-East of Jupiter, B: North-West, C: South-East, D: South-
West; see e.g., Fig. 2 of Marques et al., 2017). The position of the observer
in these maps is indicated by the position of the miniature image of Jupiter,
with the black line displaying the location during the chosen time window
(each tick is one hour). Several probability maps have been published to
predict auroral or Galilean moons induced decametric emissions such as Marques
et al. (2017, the one displayed here), Zarka et al. (2018); Leblanc et al.
(1993); Louis et al. (2017c); Zarka et al. (2017).

We also simulate the radio emission (see Fig. 2 right panel), by using the
ExPRES tool (Exoplanetary and Planetary Radio Emission Simulator, Louis et
al., 2019b). This code is here used to produce simulations of the jovian radio
emission linked to the interaction between Jupiter and the Galilean moons (Io,
Europa, Ganymede, see Louis et al., 2020a, for the database), based on the
Cyclotron Maser Instability equations. More precisely, the simulations in this
database were produced using as input parameters the JRM09 magnetic field
model (Connerney et al., 2018, based on in situ measurements from the Juno
mission, ) and the Connerney et al. (1981) current sheet model to reproduce
the magnetic field lines of Jupiter’s magnetosphere, a $3$ keV electron energy
and a loss cone electron distribution function. Using these standard
parameters, the uncertainty for Io-DAM emission is expected to be well within
a $2$ hours window (Louis et al., 2017a) around the predicted arc. The example
of Fig. 2 (right panel) displays two arc-shape emissions related to Io-Jupiter
interaction: the blue one is a northern emission (corresponding to an Io-B
emission) while the red one is a southern emission (Io-D emission). The white
area corresponds to the time when Jupiter is visible above an adjustable
elevation (here above $10\degr$), while the grey-shaded areas correspond to
the time when Jupiter is below this elevation. The observer position
indication, given in the left-hand side panel (with the miniature image of
Jupiter), is here represented by the thin vertical red dashed line (with the
corresponding time in the _Adjust display time_ entry, upper-middle).

Using this online tool, we are thus able to determine when the probability of
observing Io-, Europa-, Ganymede- or auroral decametric emissions is the
highest.

##  3 A first case study: observation of an Io-decametric emission

Figure 3: Observation of an Io-decametric emission, displayed in a time-
frequency map (called _dynamic spectrum_), on different time windows, and for
the stations IE613 (panels a-d,f,h), DE604 (panel e) and FR606 (panel g).
Panels (a) and (b) respectively display Stokes I (corresponding to the
intensity of the emission) and Stokes V (corresponding to the degree of
circular polarization) parameters between 04:10 and 05:30 UTC on 8 June 2021.
Panels (c) and (d) are two zoom-in inside the red boxes, between (c) 05:10:30
and 05:11:30, and (d) 05:10:30 and 05:10:40, showing the data at an higher-
resolution. Panel (f) is a zoom of panel (d) of 1 second, between 05:10:37 and
05:10:38, in the range [$15$-$32$] MHz, at the highest resolution use in this
study, i.e. $81.92$ $\mu\mathrm{s}$ per $12.2$ kHz. Panels (e) and (g) are the
equivalent of panel (f), displaying the data from the DE604 and FR606
stations, respectively. Panel (h) is the equivalent of panel (f), displaying
the Stokes V parameter (degrees of circular polarization). Panels (c) to (g)
share the same intensity colour bar (displayed to the right of panels (d) and
(f)). In panels (e) to (h) the red lines show example of drifting millisecond
bursts. Panel (i) shows a comparison between the highest frequency bursts
observed by the three stations (highlighted in panels (e-h).

As a first case study, and based on the prediction shown in Figure 2, we
observed Jupiter pre-dawn on 8 June 2021, between 04:00 and 07:00 UTC, with
three different LOFAR stations: IE613 in Ireland, FR606 in France and DE604 in
Germany (see Fig. 1 for their respective position). In this study the data
were processed to obtain a resolution of $81.92$ $\mu\mathrm{s}$ per $12.2$
kHz, and only the [$8$-$40$] MHz frequency range is displayed ($40$ MHz being
the upper limit of the Jovian radio emissions).

As predicted by the prediction tool (see Figure 2), we observe (Figs. 3a,b,
IE613 data) an emission produced by the Io-Jupiter interaction. The emission
has a vertex early shape (shape of an ‘opening parentheses’) and displays a
negative circular (Right-Handed) polarization (Fig. 3b), which gives the
information that this emission is an Io-B emission (coming from the North-West
side of Jupiter as seen from the observer point of view, thus from the North-
dawn side of Jupiter as seen from Earth). The main emission (emitted from the
flux tube connected to Io) is observed from $\sim$ 05:00 to 05:30, while
emission observed just before is produced in flux tubes connected to the tail
upstream of Io (called secondary emission).

Figure 3c displays a one minute zoom of Figure 3a between 05:10:30 and
05:11:30. In this dynamic spectrum, we can see that the general arc shape of
this emission has a detailed substructure. If we zoom in even further, in a 10
seconds window (Figure 3d), we then can clearly see that the Io emission is
composed of many millisecond bursts, that present a negative drift (decreasing
frequency with time). The slope of the millisecond bursts gives a direct
indication of the direction of propagation of the electrons responsible for
these emissions. Indeed, the Cyclotron Maser Instability produces emission at
a frequency close to the electron cyclotron frequency $f_{\mathrm{ce}}$, which
is directly proportional to the magnetic field amplitude. Because the magnetic
field strength decreases with altitude, we can therefore determine from the
negative slope of these bursts (the emission frequency decreasing with time)
that the electrons are propagating upward (from the planet to the higher
altitudes).

Figures 3f and 3h show a 1 second zoom in (between 05:10:37 and 05:10:38) in
the [$15$-$32$] MHz frequency range of both Stokes I (intensity, panel f) and
V (circular polarization, panel h) parameters. Superimposed on these two one-
second dynamic spectra are two examples of drifting millisecond bursts (orange
lines), at two different frequency ranges.

The first drifting millisecond burst that is highlighted extends from $30.83$
MHz to $22.32$ MHz, from 05:10:37.274 to 05:10:37.673, thus displaying a drift
of $\sim-21.34$ MHz/s. At these frequencies, using the JRM09 magnetic field
model (based on in situ measurements of the magnetometer onboard Juno,
Connerney et al., 2018, that allow to calculate the Jupiter’s magnetic field
lines), we can determine than a difference of 1 MHz corresponds to a distance
of about $\sim 1000$ km. Using the
$E=1/2\mathrm{m}_{\mathrm{e}}\mathrm{v}^{2}$ formula, we can determine that
this drift corresponds to electrons with an energy of $\sim 1.3$ keV.

The second drifting millisecond burst that is highlighted extends from $20.37$
MHz to $17.04$ MHz, from 05:10:37.289 to 05:10:37.761, thus having a drift of
$-7.06$ MHz/s. This corresponds to electrons with an energy of $\sim 0.14$
keV.

Figures 3e,g display the observation acquired by the DE604 and FR606 stations
during the same $10$ s displayed Fig. 3f. The same bursts are highlighted.

Fig. 3i shows a zoom on the bursts at the highest frequencies, observed by the
three stations (DE604 as a solid line, FR606 as a dashed line and IE613 as a
dotted line). A time shift of $\sim 8$ ms is measured here, with FR606 seeing
the burst before DE604 and then IE613. At that time (8 June 2021 05:10),
Jupiter was located at $157\degr 33^{\prime}02.6"$ in azimuth and $22\degr
27^{\prime}26.9"$ in altitude in the sky. For an European observer, it means
than Jupiter was located on the South section in the sky. Therefore in that
case, it’s more the South/North spreading of the three stations that matters
(see Fig. 1). It is then expected that a radio emission from Jupiter will be
observed first by FR606, then by DE604 and finally by IE613.

##  4 Perspectives

The observing method described in this article for observing Jupiter’s radio
emissions at high resolution could not be more timely. With the Juno mission
we have in situ measurements for up to 2025, followed by the arrival of the
Europa Clipper and JUICE missions in the late 2020s-early 2030s. The medium-
resolution ground support has already existed for many years (Marques et al.,
2017, with daily observation, e.g.,), and the addition of regular high-
resolution measurements will allow comparative in situ and remote
measurements. These results will be compared to simultaneous observation of
the Ultraviolet (UV) emission on Jupiter’s atmosphere (by the Space Telescope
Imaging Spectrograph instrument on-board the Hubble Space Telescope or by the
UV spectrograph on-board the Juno mission, Gladstone et al., 2017), or to
X-ray auroral observation (using e.g., Chandra and XMM). This once-in-a-
generation combination of high fidelity in situ and remote sensing measurement
will allow us to make breakthrough on several key scientific questions for
Jupiter.

We have demonstrated the predictive power of the ExPRES simulations and the
probability maps (gathered on the Jupiter Radio Probability Tool), the
observing capability of the LOFAR network, and the data analysis methods
enabled by software support (e.g., REALTA Murphy et al., 2021) and domain
knowledge (e.g., the search for millisecond bursts in the high resolution data
stream). We now highlight several future avenues for the exploitation of this
rich LOFAR dataset.

The millisecond bursts which we have presented here have the potential to
reveal important information about the nature of electron acceleration regions
at Jupiter. Specifically, examination of the drifting of the bursts can tell
us about the electron population energy that gives rise to this emission, in
particular on the differences in electron energies as a function of frequency,
but also by comparing the energies of the electrons in the main (linked to Io
itself) and secondary (linked to Io’s tail) emissions. Particular attention
will be paid to abrupt changes in the drift rate of the millisecond bursts,
which will reveal electric potential drops along these field lines, and thus
acceleration regions of the electrons. In order to conduct robust statistical
examination of these bursts with future larger LOFAR observations, we plan to
apply automatic detection algorithms (e.g., Hess et al., 2007a) which will
enable examination of the bursts over an extensive frequency range.

Secondly, we plan to unlock the full power of the radio emissions as a tool to
provide information about the nature of the radio source locations themselves.
We plan to do this by combining the latest jovian magnetic field (JRM09,
Connerney et al., 2018) and current sheet (Connerney et al., 2018) models,
with sophisticated inter-station interferometry. This will enable us to
estimate the location of sources in longitude, latitude and altitude. When
this information is combined with in situ Juno measurements, it will represent
an extremely powerful probe of the response of the jovian radio sources to
magnetospheric dynamics.

Thirdly, we plan to use the radio emissions to reveal the electron density
inside the Io torus. In order to do this we will utilise the property that the
CMI perturbation is moving at the Alfven speed $v_{A}=B/\sqrt{\mu_{0}\rho}$
(with $B$ the local magnetic field amplitude and $\rho$ the local electron
density), and that Io is surrounded by a dense plasma torus (produced by its
volcanic activity which releases about one tonne of plasma per second). This
leads to a difference in longitude (called _lead angle_) between the Io
instantaneous flux tube (IFT, connected to Io) and the Io active flux tube
(AFT, where the sources of the main emission are). Once the exact locations of
the sources are known, we will be able to obtain the lead angle in longitude
between the IFT and the AFT. Knowing this will give the possibility to
estimate the electron density inside the Io torus. Knowing the longitude of
the AFT position will make it possible to determine the beaming angle of the
emitting cone, as well as determine the electron energy in a second way (the
first being using the drift rate of the millisecond bursts), by using the
theoretical formula of the cyclotron maser instability

$\theta=\mathrm{arccos}\left(\frac{v/c}{\sqrt{1-\omega_{\mathrm{ce}}/\omega_{\mathrm{ce_{\mathrm{max}}}}}}\right),$
(2)

with $v$ the electron velocity, $\omega_{ce}=2\pi f_{\mathrm{ce}}$ the local
electron cyclotron gyration frequency and $\omega_{ce_{max}}$ the maximal
electron cyclotron gyration frequency at the footprint of the magnetic field
lines.

All these studies are beyond the scope of this paper, and will be the focus of
a future scientific work. The methodology and study perspectives described
here can later be applied to new observations of jovian radio emissions
induced by Io, but also by Ganymede or Europa, or jovian auroral radio
emissions.

## Acknowledgements

This paper is using data obtained with the DE604, FR606 and IE613 stations of
the International LOFAR Telescope, constructed by ASTRON, during station-
owners time. German DE604 station is funded by the Leibniz-Institut für
Astrophysik (Potsdam). These observations were carried out in the stand-alone
GLOW mode (German LOng-Wavelength array), which is technically operated and
supported by the Max-Planck-Institut für Radioastronomie, the
Forschungszentrum Jülich and Bielefeld University. Nançay Radio Observatory
FR606 station is operated by Paris Observatory, associated with the French
Centre National de la Recherche Scientifique and Université d’Orléans. Irish
IE613 station received funding from Science Foundation Ireland (SFI), the
Department of Jobs Enterprise and Innovation (DJEI). The REALTA backend used
at IE613 station is funded by SFI and Breakthrough Listen. The Irish-LOFAR
consortium consists of Trinity College Dublin, University College Dublin,
Athlone Institute for Technology, Armagh Observatory and Planetarium
(supported through funding from the Department for Communities of the N.
Ireland Executive), Dublin City University, Dublin Institute for Advanced
Studies, National University of Ireland Galway and University College Cork.
The authors thank S. Aicardi for developing https://jupiter-probability-
tool.obspm.fr. C. K. Louis’ and C. M. Jackman’s work at DIAS is supported by
the Science Foundation Ireland Grant 18/FRL/6199. P.C.Murphy and D.McKenna are
supported by Government of Ireland Studentships from the Irish Research
Council (IRC). D. Ó Fionnagáin is supported by a Government of Ireland
Postdoctoral Fellowship from the IRC (GOIPD/2020/145). We acknowledge support
and operation of the GLOW network, computing and storage facilities by the FZ-
Jülich, the MPIfR and Bielefeld University and financial support from BMBF
D-LOFAR III (grant 05A14PBA) and D-LOFAR IV (grant 05A17PBA), and by the
states of Nordrhein-Westfalia and Hamburg.

## Data Availability

The data can be access upon request to the authors.

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