# On Docile Spaces, Mackey First Countable Spaces, and Sequentially Mackey
First Countable Spaces

Carlos Bosch Giral and César L. García and Thomas E. Gilsdorf and Claudia
Gómez - Wulschner and Rigoberto Vera
Departamento de Matemáticas, ITAM Río Hondo #1, Col. Progreso Tizapán, México,
DF, 01080, México

###### Abstract.

In this article we discuss the relationship between three types of locally
convex spaces: docile spaces, Mackey first countable spaces, and sequentially
Mackey first countable spaces. More precisely, we show that docile spaces are
sequentially Mackey first countable. We also show the existence of
sequentially Mackey first countable spaces that are not Mackey first
countable, and we characterize Mackey first countable spaces in terms of
normability of certain inductive limits.

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2010 Mathematics Subject Classification. 46A03, 46A13.

Keywords: Mackey first countable, Mackey sequentially first countable, docile
space, bounding cardinal, inductive limit.

##  1\. Introduction.

In this paper, space refers to a Hausdorff locally convex topological vector
space $E=(E,\tau)$ over $\mathbb{K}=\mathbb{C}\,or\,\mathbb{R}$. We assume
that $\,E$ is infinite dimensional unless otherwise indicated. Unspecified
notation follows standard locally convex spaces notation such as in [2]. To
denote $E$ as a linear subspace of another linear space $F$, we use $E\leq F$,
and $E\lneq F$ if $E$ is a proper subspace of $F$.

The concept of a docile space was introduced in [3]:

######  Definition 1.1.

A space $E=(E,\tau)$ is docile if every linear subspace of infinite dimension
contains a $\tau$\- bounded set $B$ such that $span\\{B\\}$ is infinite -
dimensional.

In one of his seminal papers, G. W. Mackey [5] defined the following:

######  Definition 1.2.

A space $E=(E,\tau)$ is first countable, henceforth referred to as Mackey
first countable, if for every sequence of $\tau-$ bounded sets $(B_{n})$,
there exists a sequence of nonzero scalars $(a_{n})$, such that
$\bigcup_{n=1}^{\infty}a_{n}B_{n}$ remains bounded.

In [1], the following was introduced and the authors proved that it is
equivalent to the concept of a docile space:

######  Definition 1.3.

A space $E$ is sequentially Mackey first countable, if given any sequence
$\left(B_{n}\right)$ of bounded sets in $E$, and any sequence
$\left(x_{k}\right)$ from $\bigcup_{n=1}^{\infty}B_{n}$, there exists a
sequence $\left(\alpha_{k}\right)$ of scalars with infinitely many nonzero
terms such that $\left\\{\alpha_{k}x_{k}:k\in\mathbb{N}\right\\}$ is bounded.

Observe that the definition is unchanged if one uses sequences
$\left(x_{k}\right)$ without reference to a sequence $(B_{n})$ of bounded
sets.

Clearly every Mackey first countable space is sequentially Mackey first
countable. In this paper we will prove that there exist spaces that are
sequentially Mackey first countable but not Mackey first countable. We will
also obtain a characterization of Mackey first countability in terms of
increasing sequences of bounded sets.

##  2\. Sequentially Mackey first countable spaces that are not Mackey first
countable.

The existence of sequentially Mackey first countable spaces that are not
Mackey first countable conclusion will be in Corollary 2.2 which is a
consequence of the following main result:

######  Theorem 2.1.

The weak dual of a nonnormable metrizable space of algebraic dimension greater
or equal to $\aleph_{0}$ is not Mackey first countable.

In [4; (5), 393] it is proved that every nonnormable metrizable space is
Mackey first countable and that the strong dual of an infinite dimensional
metrizable space, $(E^{\prime},\beta(E^{\prime},E))$, is not Mackey first
countable. In fact, it is easy to see that any regular inductive (an inductive
limit is regular if every bounded set is contained in and bounded in some step
subspace) cannot be Mackey first countable.

In this paper we will make use of a bounding cardinal, $\mathfrak{b}$, defined
as $\mathfrak{b}=\min\\{|F|\colon F\subset\omega^{\omega}\,$ such that for
each $\,f\in\omega^{\omega}\,$ there exists $\,g\in F\,$ for which $g$ is not
less than $f\\}$, where $\omega$ represents the ordinal of the natural
numbers, $\mathbb{N}$, and $|F|$ denotes the cardinality of $F$.

It is known from ZFC, that $\,\aleph_{1}\leq\mathfrak{b}\leq\mathfrak{c}$,
where $\mathfrak{c}=card(\mathbb{R})$.

From [3; Thm. 7. 4] we have that if the space $E$ is of algebraic dimension
less than $\mathfrak{b}$, then its weak dual,
$(E^{\prime},\sigma(E^{\prime},E))$ is docile. Via [1] we will obtain:

######  Corollary 2.2.

If $E=(E,\tau)$ is metrizable and of dimension $\,\aleph_{o}\,$, then
$(E^{\prime},\sigma(E^{\prime},E))$ is sequentially Mackey first countable,
but not Mackey first countable.

We recall that for an absolutely convex subset $C$ of a space $E$, the linear
subspace generated by $C$ is $E_{C}=span\\{C\\}=\bigcup_{n=1}^{\infty}nC$.
Some simple observations of this construction are: If $a>0$, then $E_{a\cdot
C}=E_{C}$, and if $A\subset C$, then $E_{A}\leq E_{C}$.

The proof of Theorem 2.1 requires two preliminary lemmas:

######  Lemma 2.3.

Suppose $\,C_{1}\subset C_{2}\subset\cdots$ is a sequence of absolutely convex
subsets of a space $E$, and that $\,a_{1},\,a_{2},\cdots$ is a sequence of
positive numbers. If $\,C=\bigcup_{n=1}^{\infty}a_{n}C_{n}\,$, then
$\,E_{C}=\bigcup_{n=1}^{\infty}E_{C_{n}}\,$.

Proof: Because $\,a_{n}C_{n}\subset E_{C_{n}}\,$, we have
$\,C\subset\bigcup_{n=1}^{\infty}E_{C_{n}}\,$, and hence,
$\,E_{C}\subset\bigcup_{n=1}^{\infty}E_{C_{n}}\,$. Next, let
$\,x\in\bigcup_{n=1}^{\infty}E_{C_{n}}$. Then $x\in E_{C_{n}}$, for some
$n\in\mathbb{N}$. It follows that for some $r>0$, $x\in rC_{n}\,$, which
implies that there exists $y\in C_{n}$ with $x=ry$. From this,

$x=ry=\frac{r}{a_{n}}(a_{n}y)\in\frac{r}{a_{n}}a_{n}C_{n}\subset\frac{r}{a_{n}}C\subset
E_{C}.\;\;\;\Box$

If $C$ is bounded, then we can define a norm on $E_{C}$ via the Minkowski
sublinear functional of $C$:

$\rho_{C}(x)=\inf\\{r>0:x\in r\cdot C\\}.$

######  Lemma 2.4.

Under the same hypotheses of the previous lemma, suppose additionally that
each $\,C_{n}$ is bounded in $E$. Consider the increasing sequence of linear
subspaces $E_{1}\subset E_{2}\subset\cdots$, where
$E_{n}=(E_{C_{n}},\rho_{C_{n}}),\,n\in\mathbb{N}$. Suppose that
$F=\bigcup_{n=1}^{\infty}E_{n}$ and that $C=\bigcup_{n=1}^{\infty}a_{n}C_{n}$
is bounded. Then $F=E_{C}$ and the topology induced by the norm $\,\rho_{C}\,$
is weaker than the inductive limit topology formed by the sequence
$(E_{C_{n}},\rho_{C_{n}})$.

Proof: The equality $F=E_{C}$ was proved in the previous lemma. Now, observe
that $a_{n}C_{n}\subset a_{n}C_{n+1}\subset a_{n}C$, and that from

$E_{n}=\bigcup_{i=1}^{\infty}iC_{n}.$

For each $x\in E_{n}=\bigcup_{r>0}rC_{n}$, $x=rz$ for some $r>0$ and $z\in
C_{n}$. From this, we have $\rho_{C_{n}}(x)\leq r$, and

$x=\frac{r}{a_{n}}(a_{n}z)\in\frac{r}{a_{n}}a_{n}C_{n}\subset\frac{r}{a_{n}}C.$

This last line implies that $\rho_{C}(x)\leq\frac{r}{a_{n}}$, that is,
$a_{n}\rho_{C}(x)\leq r$. By taking the infimum, we conclude that
$a_{n}\rho_{C}(x)\leq\rho_{C_{n}}(x)$ implies
$\rho_{C}(x)\leq\frac{1}{a_{n}}\rho_{C_{n}}(x)$, and we deduce that the
topology induced by $\rho_{C}$ on the subspace $E_{n}$ is weaker than the
normed topology of $\rho_{C_{n}}$, that is, $\rho_{C_{n}}$ is weaker than
$\tau_{ind}$, which in turn gives us that $\rho_{C}$ is weaker than
$\tau_{ind}$. $\Box$

Proof of Theorem 2.1. Let $\,V_{1}\supset V_{2}\supset\cdots$ be a decreasing
countable base of zero neighborhoods of the space $E$. We suppose that these
neighborhoods are all absolutely convex and $\tau$ \- closed. For each
$n\in\mathbb{N}$, let $B_{n}=V_{n}^{\circ}$, the polar of the neighborhood
$V_{n}$. By the Alaoglu - Bourbaki theorem, each $B_{n}$ is
$\sigma(E^{\prime},E)$ \- compact, hence, by [4; (5.1), 262], each $B_{n}$ is
both bounded and complete with respect to the strong topology
$\beta(E^{\prime},E)$.

In [4; (8), 394] it is shown that when $E$ is a nonnormable metrizable space,
$(B_{n})$ forms a fundamental sequence of bounded sets in
$\beta(E^{\prime},E)$, that for each $n\in\mathbb{N}$, $B_{n}$ does not absorb
$B_{n+1}$, and that the linear subspace generated by each $\,B_{n}\,$,
($\,E^{\prime}_{B_{n}}\,$) is a proper subspace of that generated by
$B_{n+1}$.

Of course, $\bigcap_{n=1}^{\infty}V_{n}=\\{0\\}$ in $E$ implies that
$\,E^{\prime}=\bigcup_{n=1}^{\infty}B_{n}=\bigcup_{n=1}^{\infty}E^{\prime}_{B_{n}}\,$.

From the previous statements we conclude that the sequence $(B_{n})$ indicates
that $(E^{\prime},\beta(E^{\prime},E))$ is neither Mackey first countable nor
sequentially Mackey first countable. We will prove that:

######  Claim 2.5.

$(E^{\prime},\sigma(E^{\prime},E))$ is not Mackey first countable.

Proof of Claim. Suppose there exists a sequence $(a_{n})$ of positive numbers
such that $\bigcup_{n=1}^{\infty}a_{n}B_{n}\,$ is $\sigma(E^{\prime},E)$ \-
bounded. Consider

$B=\overline{absconv\left\\{\bigcup_{n=1}^{\infty}a_{n}B_{n})\right\\}}^{\,\sigma(E^{\prime},E)},$

the absolutely convex $\sigma(E^{\prime},E)$ \- hull of
$\bigcup_{n=1}^{\infty}a_{n}B_{n}$. We will show that the assumption that $B$
is $\sigma(E^{\prime},E)$ \- bounded leads to the conclusion that $E$ is
finite dimensional.

By Lemma 2.3,
$\,E^{\prime}=\bigcup_{n=1}^{\infty}E^{\prime}_{B_{n}}=E^{\prime}_{B}\,$,
which means that $B$ is absorbing in $E^{\prime}$. Lemma 2.4 implies that the
topology induced by the norm $\,\rho_{B}\,$ in $\,E^{\prime}\,$ is weaker than
$\,\tau_{ind}\,$. Moreover, because $B$ is a $\,\rho_{B}$ \- zero
neighborhood, there exists a $\tau_{ind}$ \- zero neighborhood $U$ such that
$\,U\subset B\,$. Thus, $B$ is a $\tau_{ind}$ \- bornivore, in particular, $B$
is absorbing in $E^{\prime}$

On the other hand, the $\sigma(E^{\prime},E)$ \- completeness of each $B_{n}$
implies that each $\,(E^{\prime}_{B_{n}},\rho_{B_{n}})\,$ is a Banach space,
giving us that the inductive limit of these spaces is barrelled (cf [2; 214]).
Thus, as $\,B\subset E^{\prime}\,$ is an absolutely convex, absorbing,
$\sigma(E^{\prime},E)$ \- closed (hence, $\,\tau_{ind}$\- closed) set, i.e., a
barrel, $B$ is a zero neighborhood with respect to $\,\tau_{ind}$. We conclude
that $\beta(E^{\prime},E)=\tau_{ind}\,$ in $\,E^{\prime}\,$. Meanwhile,
($\,E^{\prime\prime}=(E^{\prime}[\beta^{\prime}])^{\prime}$ is metrizable: [4;
(5), 301] and each $\sigma(E^{\prime\prime},E)$ \- bounded subset is
$\,\beta^{\prime}$-equicontinuous ([2; 3.6.2, 212]). Because $U$ is absorbing
in $E^{\prime}$, its polar in
$\,(E^{\prime},\tau_{ind}^{\prime})^{\prime}=E^{\prime\prime}\,$ is
$\sigma(E^{\prime\prime},E)$ \- compact, which means $U^{\circ}$ is
$\beta(E^{\prime\prime},E)$ \- bounded. From $U\subset B$, $B^{\circ}\subset
U^{\circ}\subset E^{\prime\prime}$, implying that $B^{\circ}$ is
$\beta(E^{\prime\prime},E)$ \- bounded, and $\sigma(E^{\prime\prime},E)$ \-
bounded as well. Because $B$ is $\sigma(E^{\prime},E)$ \- bounded,
$B^{\circ}\subset E^{\prime\prime}$ is absorbing, that is,
$\,E^{\prime\prime}_{B^{\circ}}=E^{\prime\prime}\,$. Furthermore,
$\,B^{\circ}\subset E^{\prime\prime}\,$ is $\,\beta^{\prime}$-equicontinuous
and thus $\,C=E\cap B^{\circ}\subset E\,$ is
$\,\beta^{\prime}$-equicontinuous, hence, $\tau$ \- precompact. The
equicontinuity of $C$ gives us that $C$ is a $\tau$ \- zero neighborhood, and
therefore, $E$ is finite dimensional. This contradiction completes the proof.
$\Box$

Proof of Corollary 2.2. This follows from the Theorem 2.1 above, and from the
proof that such a space is docile ([3; Thm. 7]) given that
$\,\aleph_{o}<\mathfrak{b}$. $\Box$

In [1] the authors proved that a Hausdorff quotient of a Mackey first
countable space is sequentially Mackey first countable. Theorem 2.1 leaves
that conclusion as a partial answer to G. W. Mackey’s original question,
which, in its original form, remains open:

######  Problem 2.6.

(G.W. Mackey, [5]). Is a Hausdorff quotient of a Mackey first countable
locally convex space again Mackey first countable?

##  3\. A characterization of Mackey first countable spaces.

Let $\,B_{1}\subset B_{2}\subset,\cdots$ be an increasing sequence of
absolutely convex closed bounded sets in a space $E$. Put
$\,(F=\bigcup_{n=1}^{\infty}E_{B_{n}},\,\tau_{ind})\,$ as the inductive limit
of the normed linear subspaces generated by this sequence. By [2; 7.3.4, 222],
the space $\,(F,\tau_{ind})\,$ is bornological. In the main result of this
section we will prove that $E$ is Mackey first countable if and only if
$\,(F,\tau_{ind})\,$ is normed.

######  Proposition 3.1.

Suppose $\,\\{B_{n}\\}_{n\in\mathbb{N}}$ is any sequence of closed, convex
subsets of $E$, such that $B_{1}$ is balanced. Define, for each
$\,n\in\mathbb{N}$, $\,B_{n}^{\\#}=\bigcup_{k=1}^{n}B_{k}$. If
$\,\\{a_{n}\\}_{n\in\mathbb{N}}$ is such that $0<a_{n}<1$ for each
$n\in\mathbb{N}$, then $\,\bigcup_{n=1}^{\infty}a_{n}B_{n}\,$ is bounded if
and only if, $\,\bigcup_{n=1}^{\infty}a_{n}B_{n}^{\\#}\,$ is bounded.

Proof. ($\,\Leftarrow\,$): Because $\,B_{n}\subset B_{n}^{\\#}$, this
implication is clear.

($\,\Rightarrow\,$): Without loss of generality, we may suppose that
$a_{1}>a_{2}>\cdots$. Let $\,A\subset E\,$ be $\,\tau$ \- bounded such that
$\,\bigcup_{n=1}^{\infty}a_{n}B_{n}\subset A\,$. Observe that

$\,a_{1}B_{1}=a_{1}B_{1}^{\\#},\;a_{2}B_{2}^{\\#}=a_{2}(B_{1}\cup
B_{2})=a_{2}(B_{1}\cup a_{2}B_{2})\subset a_{1}B_{1}\cup a_{2}B_{2}\subset A.$

Thus,

$\,a_{n}B_{n}^{\\#}=a_{n}(\bigcup_{k=1}^{n}B_{k})=\bigcup_{k=1}^{n}a_{n}B_{k}\subset\bigcup_{k=1}^{n}a_{k}B_{k}\subset
A,$

which shows that $\,\bigcup_{k=1}^{\infty}a_{k}B_{k}^{\\#}\subset A$. $\Box$

######  Corollary 3.2.

Suppose $\,\\{B_{n}\\}_{n\in\mathbb{N}}$ is any sequence of closed, convex
subsets of $E$, with $B_{1}$ balanced. Define $B_{n}^{\\#}$ by:

$\,B_{n}^{\\#}=\overline{absconv\left\\{\bigcup_{k=1}^{n}B_{k}\right\\}}^{\,\tau}.$

If $\,\\{a_{n}\\}_{n\in\mathbb{N}}$ is such that $0<a_{n}<1$ for each
$n\in\mathbb{N}$, then $\,\bigcup_{n=1}^{\infty}a_{n}B_{n}\,$ is bounded if
and only if, $\,\bigcup_{n=1}^{\infty}a_{n}B^{\\#}_{n}\,$ is bounded.

Proof. If $\,\bigcup_{n=1}^{\infty}a_{n}B^{\\#}_{n}\,$ is bounded, then we
obtain again, $\,B_{n}\subset B_{n}^{\\#}$.

Suppose $\,\bigcup_{n=1}^{\infty}a_{n}B_{n}\,$ is bounded. Consider
$\,A\subset E$, an absolutely convex closed, bounded set, such that
$\,\bigcup_{n=1}^{\infty}a_{n}B_{n}\subset A\,$. By the previous proposition,

$\bigcup_{n=1}^{\infty}a_{n}(B_{1}\cup\cdots\cup B_{n})\subset A.$

We observe that

$\overline{absconv\left\\{a_{n}(B_{1}\cup\cdots\cup
B_{n})\right\\}}^{\,\tau}=a_{n}\,B_{n}^{\\#}.$

It follows that $\,a_{n}B_{n}^{\\#}\subset A\,$, for each
$\,n\in\mathbb{N}\,$, hence,

$\bigcup_{n=1}^{\infty}a_{n}B_{n}^{\\#}\subset A.\;\;\Box$

######  Corollary 3.3.

$\sum_{n\in\mathbb{N}}E_{B_{n}}=\bigcup_{n=1}^{\infty}E_{B_{n}^{\\#}}\leq E.$

Proof. For each $\,n\in\mathbb{N}\,$ we have $\,E_{B_{n}}\leq
E_{B_{n}^{\\#}}$, which gives us

$\,\sum_{n\in\mathbb{N}}E_{B_{n}}\leq\bigcup_{n=1}^{\infty}E_{B_{n}^{\\#}}.$

Let $\,x\in\bigcup_{n=1}^{\infty}E_{B_{n}^{\\#}}$. Then $\,x\in
E_{B_{n}^{\\#}}\,$ for some $\,n\in\mathbb{N}\,$. By [4; (1), 173],
$x=\sum_{k=1}^{m}\alpha_{k}x_{k}\,$ with $\,\alpha_{k}\geq 0\,$,
$\,\sum_{k=1}^{m}\alpha_{k}=1\,$ and $\,x_{k}\in\bigcup_{k=1}^{m}E_{B_{k}}\,$
for each $\,k=1,2,\cdots,m\,$. We obtain
$\,x\in\sum_{n\in\mathbb{N}}E_{B_{n}}\,$. $\Box$

######  Theorem 3.4.

The space $(E,\tau)$ is Mackey first countable if and only if, for each
increasing collection $\,\\{B_{n}\\}_{n\in\mathbb{N}}$ of bounded, absolutely
convex closed sets in $E$, the corresponding inductive limit,
$\,ind_{n\in\mathbb{N}}(E_{B_{n}^{\\#}},\rho_{B_{n}^{\\#}})$, is normable.

Proof. Let $\,F=\bigcup_{n=1}^{\infty}E_{B_{n}^{\\#}}\leq E\,$, and let
$\,\tau_{ind}\,$ denote the corresponding inductive limit topology on $F$.

Assume that for each increasing collection $\,\\{B_{n}\\}_{n\in\mathbb{N}}$ of
bounded, absolutely convex closed sets in $E$,
$\,ind_{n\in\mathbb{N}}(E_{B_{n}^{\\#}},\rho_{B_{n}^{\\#}})$, is normable. Let
$\,\\{B_{n}\\}_{n\in\mathbb{N}}$ be a collection of bounded subsets of $E$.
Using $\,\rho\,$ to denote the normed topology on $F$, we have
$\,\tau_{ind}=\rho\,$. If $\,D\subset F\,$ is the $\rho$ \- closed unit ball,
the $(\forall n\in\mathbb{N})(\exists a_{n}>0)$ such that
$\,a_{n}B_{n}^{\\#}\subset D\cap E_{B_{n}^{\\#}}\subset D\,$. From this,
$\,\bigcup_{n=1}^{\infty}a_{n}B_{n}^{\\#}\subset D\,$. Moreover,
$\,B_{n}\subset B_{n}^{\\#}$ implies that
$\,\bigcup_{n=1}^{\infty}a_{n}B_{n}\subset D\,$. Meanwhile, $D$ is $\rho$ \-
bounded, with $\,\tau\leq\tau_{ind}=\rho\,$, and this tells us that $D$ is
$\tau$ \- bounded as well.

Conversely, if $(E,\tau)$ is Mackey first countable, then we will prove that
there exists a norm $\rho$ on $F$ such that $\,\tau_{ind}=\rho\,$. To this
end, let $\,B_{1}\subset B_{2}\subset\cdots$ be an increasing sequence of
closed, absolutely convex bounded subsets of $E$. Then there exists a sequence
$\,\\{a_{n}\\}_{\mathbb{N}}$ in $(0,1]$, with $\,a_{1}>a_{2}>\cdots>0\,$ such
that $\,\bigcup_{n=1}^{\infty}a_{n}B_{n}$ is bounded in $E$. Put

$B=\overline{absconv\left\\{\bigcup_{n=1}^{\infty}a_{n}B_{n}\right\\}}^{\,\tau}.$

######  Claim 3.5.

$\,\rho_{B}=\tau_{ind}\,$.

Proof of claim. We have that $B$ is absolutely convex, closed and bounded. By
Lemma 2.4, $\,F=E_{B}\,$. On the other hand, by the definition of the
inductive limit topology, $\,\rho_{B}\leq\tau_{ind}\,$. Hence,
$\,(F,\rho_{B})\,$ and $\,(F,\tau_{ind})\,$ are both bornological. We conclude
via [2; 3.7.1 (a), 220], that $\,\rho_{B}=\tau_{ind}\,$. $\Box$

######  Proposition 3.6.

The previous result applies to any countable collection of absolutely convex
closed and bounded subsets.

Proof. Consider the family $\wp$ of finite subsets of $\mathbb{N}$. The family
$\wp$ is ordered by inclusion. Moreover, given $\,S\in\wp\,$,
$\,C_{S}=\sum_{k\in S}E_{B_{k}}\,$ satisfies: If $\,S\subset S^{\prime}\,$,
then $\,C_{S}\leq C_{S^{\prime}}\,$. Therefore, we can speak of the inductive
limits of such linear subspaces of the form $\,\\{C_{S}\\}_{S\in\wp}\,$:
$\,G=ind\,C_{S}\,$, with the respective topology $\,t_{ind}\,$.

By Corollary 3.3, $F=G$, and because $\,C_{S}\,$ is a linear subspace of some
$\,E_{B_{n}^{\\#}}\,$, we have: $\,\tau_{ind}\leq t_{ind}\,$. $\Box$

ACKNOWLEDGEMENT: The authors would like to acknowledge research support for
this paper by the Asociación Mexicana de Cultura, A. C.

REFERENCES

[1] Bosch Giral, C., Gilsdorf, T., Gómez-Wulschner, C. Mackey first
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27, (2011), no.4, 737 - 740.

[2] Horváth, J. Topological Vector Spaces and Distributions, Vol. I. Addison -
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[3] Kakol, J., Saxon, S. A., Todd, A. R. Docile locally convex spaces,
Contemp. Math., 341, AMS, (2004), 73 - 77.

[4] Köthe, G. Topological Vector Spaces I. Springer Verlag, NY, (1969).

[5] Mackey, G. W. On infinite- dimensional linear spaces. Trans. AMS, 66, no.
57 (1945), 155 - 207.

[6] Pérez Carreras, P., Bonet, J. Barrelled Locally Convex Spaces. North
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[7] Saxon, S. A., Sánchez Ruiz, L. M. Optimal cardinals for metrizable
barrelled spaces. J. London Math. Soc. (2), 51, (1995), no. 1, 137 - 147.


# The time evolution of $M_{d}/\dot{M}$ in protoplanetary disks as a way to
disentangle between viscosity and MHD winds

Alice Somigliana European Southern Observatory, Karl-Schwarzschild-Strasse 2,
D-85748 Garching bei München, Germany  Fakultat für Physik, Ludwig-
Maximilians-Universität München, Scheinersts. 1, 81679 München, Germany
Leonardo Testi Dipartimento di Fisica e Astronomia, Universita‘ di Bologna,
Via Gobetti 93/2, 40122 Bologna, Italy  INAF-Osservatorio Astrofisico di
Arcetri, Largo E. Fermi 5, I-50125 Firenze, Italy  Giovanni Rosotti
Dipartimento di Fisica, Università degli Studi di Milano, Via Celoria 16,
20133 Milano, Italy  Claudia Toci European Southern Observatory, Karl-
Schwarzschild-Strasse 2, D-85748 Garching bei München, Germany  Giuseppe
Lodato Dipartimento di Fisica, Università degli Studi di Milano, Via Celoria
16, 20133 Milano, Italy  Benoît Tabone Université Paris-Saclay, CNRS, Institut
d’Astrophysique Spatiale, Orsay, France  Carlo F. Manara European Southern
Observatory, Karl-Schwarzschild-Strasse 2, D-85748 Garching bei München,
Germany  Marco Tazzari Institute of Astronomy, University of Cambridge,
Madingley Road, CB3 0HA, Cambridge, UK

###### Abstract

As the classic viscous paradigm for protoplanetary disk accretion is
challenged by the observational evidence of low turbulence, the alternative
scenario of MHD disk winds is being explored as potentially able to reproduce
the same observed features traditionally explained with viscosity. Although
the two models lead to different disk properties, none of them has been ruled
out by observations - mainly due to instrumental limitations. In this work, we
present a viable method to distinguish between the viscous and MHD framework
based on the different evolution of the distribution in the disk mass
($M_{\mathrm{d}}$) - accretion rate ($\dot{M}$) plane of a disk population.
With a synergy of analytical calculations and 1D numerical simulations,
performed with the population synthesis code Diskpop, we find that both
mechanisms predict the spread of the observed ratio $M_{\mathrm{d}}/\dot{M}$
in a disk population to decrease over time; however, this effect is much less
pronounced in MHD-dominated populations as compared to purely viscous
populations. Furthermore, we demonstrate that this difference is detectable
with the current observational facilities: we show that convolving the
intrinsic spread with the observational uncertainties does not affect our
result, as the observed spread in the MHD case remains significantly larger
than in the viscous scenario. While the most recent data available show a
better agreement with the wind model, ongoing and future efforts to obtain
direct gas mass measurements with ALMA and ngVLA will cause a reassessment of
this comparison in the near future.

protoplanetary disks - accretion, accretion disks - planets and satellites:
formation

##  1 Introduction

The gaseous component of protoplanetary disks has traditionally been described
as undergoing viscous accretion (Lynden-Bell & Pringle 1974, Pringle 1981). In
recent years, however, a growing observational evidence is challenging this
picture, as the low levels of turbulence detected in protoplanetary disks
appear incompatible with the observed evolution (Pinte et al. 2016, Flaherty
et al. 2018, Rosotti 2023). The best alternative to the classic viscous
scenario is currently provided by MHD disk winds, originally proposed by
Blandford & Payne (1982). This model has gained increasing popularity in the
recent years, as several studies (see Lesur 2021 for a review) have shown it
to reproduce the key evolutionary features of protoplanetary disks; moreover,
Tabone et al. (2022a) have developed a simple analytical parametrization,
making it a valid alternative to the viscous theory.

A compelling question is which of these mechanism, or which combination of the
two, drives angular momentum transport in protoplanetary disks (Manara et al.,
2023). Answering this question has proven to be a surprisingly difficult task:
even though the two models are in principle well distinguishable through their
characteristic theoretical predictions, the observational counterpart is
lagging behind (e.g., Rosotti et al. 2019b, Ilee et al. 2022). A good example
of this problem is viscous spreading, a fundamental feature of viscous
evolution that causes the gaseous component of disks to expand in radius as
they evolve. As MHD evolution does not show a similar behavior (Zagaria et
al., 2022b), it would in principle be a good candidate for disentangling
between the two predictions: however, the high sensitivity required to detect
it has until now represented a limit. While Class 0 objects are widely
accepted to be born small ($<60$ au: Maury et al. 2019, also supported by the
numerical experiments of, e.g., Lebreuilly et al. 2021) and grow wider in the
first 1-2 Myr of evolution (Najita & Bergin, 2018), whether the radius of
Class II disks increases or decreases with time is widely debated. Dust
continuum radii are observed to be shrinking with time (Hendler et al. 2020,
Zagaria et al. 2022b), as an effect of radial drift, while gas observations
(Ansdell et al. 2018, Sanchis et al. 2021, Toci et al. 2021, Long et al. 2022)
have covered too small of a sample at too low sensitivities to draw firm
conclusions. The advent of ALMA Band 1 (Carpenter et al., 2020) and the next-
generation VLA (ngVLA, Tobin et al. 2018) in the near future will allow to
perform surveys of protoplanetary disks at unprecedentedly long wavelengths,
which will play a crucial role in determining the leading evolutionary
mechanism. At the same time, finding novel approaches to tackle this problem
is crucial to obtain significant results.

In this Letter, we suggest a new method to distinguish between the two models
from the population perspective: through a joint theoretical and population
synthesis approach, we investigate the time evolution of disks in the disk
mass - accretion rate plane, proving it to be a good approach for our goal.
This work is structured as follows: in Section 2 we describe the evolutionary
prescriptions that we adopt and we discuss their numerical implementation. In
Section 3 we present our results and we compare them with the observations.
Finally, in Section 4 we discuss the implications of these results and draw
our conclusions.

##  2 Theoretical model

###  2.1 Secular evolution

The simulations presented in this work have been carried out using the 1D
Python population synthesis code Diskpop. For a detailed description of the
code, as well as its public release, we refer to our upcoming paper
(Somigliana et al. in prep; earlier implementations of the code, its basic
assumptions and features have been described in Rosotti et al. 2019a, Rosotti
et al. 2019b, Toci et al. 2021, Somigliana et al. 2022). The viscous and MHD
evolution are implemented following Lynden-Bell & Pringle (1974) and Tabone et
al. (2022a) respectively. In this section we briefly present both models,
referring to the original papers for a deeper discussion.

In the viscous case, we solve the classic evolution equation

$\frac{\partial\Sigma}{\partial t}=\frac{3}{R}\frac{\partial}{\partial
R}\left(R^{1/2}\frac{\partial}{\partial R}(\alpha_{\mathrm{SS}}c_{s}H\Sigma
R^{1/2})\right);$ (1)

following the prescription by Shakura & Sunyaev (1973), the viscosity $\nu$ is
modeled as $\alpha_{\mathrm{SS}}c_{s}H$, where $\alpha_{\mathrm{SS}}$ is a
dimensionless parameter, $c_{s}$ is the sound speed, and $H$ is the height of
the disk. Furthermore, assuming the viscosity to be a power-law of the disk
radius for ease of solving the equation, $\nu=\nu_{c}(R/R_{c})^{\gamma}$
(where $\nu_{c}=\nu(R=R_{c})$ and $R_{c}$ is a scale radius), the analytical
solution by Lynden-Bell & Pringle (1974) holds.

In the MHD case instead (Tabone et al., 2022a), the evolution equation is
given by

$\begin{split}\frac{\partial\Sigma}{\partial
t}=\frac{3}{R}\frac{\partial}{\partial R}\left(R^{1/2}\frac{\partial}{\partial
R}(\alpha_{\mathrm{SS}}c_{s}H\Sigma R^{1/2})\right)\\\
+\frac{3}{2R}\frac{\partial}{\partial
R}\left(\frac{\alpha_{\mathrm{DW}}\Sigma{c_{s}}^{2}}{\Omega}\right)-\frac{3\alpha_{\mathrm{DW}}\Sigma{c_{s}}^{2}}{4(\lambda-1)R^{2}\Omega},\end{split}$
(2)

where $\Omega$ is the keplerian orbital frequency, $\lambda$ is the magnetic
lever arm parameter, and $\alpha_{\mathrm{DW}}$ is a magnetic equivalent of
$\alpha_{\mathrm{SS}}$. Equation (2) is a generalization of Equation (1) if
the gas surface density evolves not only because of the viscous torque (first
term on the RHS) but also because of the effects of MHD disk winds, which
extract angular momentum and induce a mass loss (second and third term on the
RHS respectively). Assuming that both $\lambda$ and $\alpha_{\mathrm{DW}}$ are
constant across the disk, and that
$\alpha_{\mathrm{DW}}\propto{\Sigma_{c}}^{-\omega}$ (where
$\Sigma_{c}=\Sigma(R=R_{c})$), Equation (2) can be solved analytically (see
Tabone et al. 2022a).

###  2.2 Isochrones

Isochrones are defined as the curves described by a population of objects of
the same age in a given plane. In the case of protoplanetary disks, isochrones
in the $M_{\mathrm{d}}-\dot{M}$ plane have been the focus of recent studies
(Lodato et al. 2017, Somigliana et al. 2020). For viscously evolving disks
(Lodato et al., 2017), the isochrone reads

$\dot{M}=\frac{M_{\mathrm{d}}}{2(2-\gamma)t}\left[1-\left(\frac{M_{\mathrm{d}}}{M_{0}}\right)^{(2-2\gamma)}\right];$
(3)

the only free parameter in Equation (3) is the initial disk mass $M_{0}$,
which only sets the starting point of the isochrone. Nonetheless, at late
stages (when $M_{\mathrm{d}}\ll M_{0}$) all disks in a population are bound to
reach the same locus on the $M_{\mathrm{d}}-\dot{M}$ plane: while this happens
at different ages for each disk, depending on its viscous timescale
$t_{\nu}={R_{c}}^{2}/(3(2-\gamma)^{2}\nu_{c})$, a fully evolved population
($t\to+\infty$) will necessarily sit on the theoretical isochrone of the
corresponding age.

For MHD disks, the isochrone is defined as (Tabone et al., 2022a)

$\dot{M}=\frac{1}{\omega(1+f_{\mathrm{M},0})t}M_{\mathrm{d}}\left[\left(\frac{M_{\mathrm{d}}}{M_{0}}\right)^{-\omega}-1\right];$
(4)

Equation (4) depends not only on $M_{0}$, but also on the equivalent of
$t_{\nu}$ in the MHD winds case, the initial accretion timescale
$t_{\mathrm{acc},0}$ , through $f_{\mathrm{M},0}$ (determined by the disk
radius - see Tabone et al. 2022a for details). The interpretation of the
isochrones in the two models is therefore different: while the viscous curves
for all disks in a population lie on top of each other (except at the early
stages, when $M_{\mathrm{d}}\sim M_{0}$), MHD evolution never loses memory of
the initial conditions. This is because, depending on whether we fix $M_{0}$
or $t_{\mathrm{acc},0}$, we can define two types of isochrones for an MHD
population. As a result, disks with a different $M_{0}$ will occupy an area of
the $M_{\mathrm{d}}-\dot{M}$ plane, rather than sitting on a single curve, and
this will be the case even for evolved populations - which means that it is
not possible to use the isochrones to obtain age estimates for disk
populations. Based on this argument, we investigate whether the evolution of a
population of disks in the $M_{\mathrm{d}}-\dot{M}$ plane could carry tangible
signatures of the evolutionary model.

###  2.3 Population synthesis

In this work we adopt a population synthesis approach, which consists of
generating and evolving a synthetic population of protoplanetary disks via
numerical methods. We employ the Python tool Diskpop, which we expanded from
our previous work (Somigliana et al., 2022) to include MHD disk wind
evolution. In this section, we present a brief outline of the workflow,
referring to the upcoming code release for a detailed description of the
methods and the implementation.

First, we generate $N\sim 100$ stars, whose masses $M_{\star}$ follow the
Kroupa (2001) initial mass function. We then assemble a Young Stellar Object
(YSO) by assigning a disk to each star: to determine the initial mass and
radius of said disk, we assume that the initial disk mass and accretion rate
scale as power-laws of the stellar mass
($M_{\mathrm{d}}\propto{M_{\star}}^{\lambda_{\mathrm{m}}}$ and
$\dot{M}\propto{M_{\star}}^{\lambda_{\mathrm{acc}}}$). In our previous work
(Somigliana et al., 2022) we have demonstrated how
$\lambda_{\mathrm{m},0}\in[0.7,1.5]$ and
$\lambda_{\mathrm{acc},0}\in[1.2,2.1]$ can reproduce the slopes of observed
correlations of disk properties with stellar mass at later ages; we refer to
that paper for a detailed discussion. We determine $M_{\mathrm{d}}$ and
$\dot{M}$ for each disk drawing from a log-normal distribution, centered in
the mean value computed via the power-law correlations and with a width
($\sigma$) of choice; $R_{\mathrm{d}}$ is then derived from considerations on
$\dot{M}$ (see Somigliana et al. 2022 for details). The other relevant
quantities besides $M_{\star}$, $M_{\mathrm{d}}$ and $R_{\mathrm{d}}$ are
fixed in our model: Table 1 shows the parameters that we used in the
simulations presented in this work, based on the disc evolution studies of
Lodato et al. (2017) and Tabone et al. (2022a) for viscosity and MHD winds
respectively. While a detailed study of the parameters space is outside of the
scope of this work, we have tested two more combinations of parameters (shown
by Tabone et al. (2022b) to reproduce the Lupus star-forming region) and we
found that our results are independent on the particular combination chosen.
Once the population of YSOs is generated, it is evolved following the viscous
or MHD prescription via a 1D implementation of the models described in Section
2.1. Although Diskpop allows to numerically solve the evolution equations, in
this work we have used the analytical solutions to Equation (1) and (2); it is
therefore important to note that our results depend on the assumptions needed
to obtain such solutions (e.g., the power-law scaling of viscosity with the
disk radius).

Model | Distributions | IMF |  $\lambda_{\mathrm{m}}$, $\lambda_{\mathrm{acc}}$ |  $\sigma_{\mathrm{M}}$, $\sigma_{\mathrm{R}}$ |  $H/R$ at $R=1$ au | $\alpha_{\mathrm{SS}}$ | $\alpha_{\mathrm{DW}}$ |  $\omega$, $\lambda$ | $<t_{\mathrm{acc},0}>$
---|---|---|---|---|---|---|---|---|---
Viscous | log-normal | Kroupa (2001) | 1.5, 2.1 | 1 dex | 0.03 | $10^{-3}$ | $0$ | $0$ | 0.8 Myr
MHD | 0.65 dex, 0.52 dex | 0111Although the MHD model of Tabone et al. (2022a) allows both $\alpha_{\mathrm{SS}}$ and $\alpha_{\mathrm{DW}}$ to be non-zero, Equation (2) in the $\omega\neq 0$ case can only be solved analytically if $\alpha_{\mathrm{SS}}=0$. | $10^{-3}$ | 0.25, 3
Table 1: Parameters used in the viscous and MHD Diskpop simulations
respectively. $\sigma_{\mathrm{M}}$ and $\sigma_{\mathrm{R}}$ are the width of
the initial disk mass and radius, respectively. These values were chosen
following the works of Lodato et al. (2017) and Tabone et al. (2022a).

It is crucial to point out that disk dispersal is an intrinsic feature of MHD
winds, but not of viscous evolution. Our code includes an observational effect
by considering as dispersed disks with masses lower than $10^{-6}M_{\odot}$;
this simulates a dispersal effect even in the viscous scenario, which would
otherwise generate disks with infinite lifetime, that do not match the
observed disk fraction (see Appendix C). This problem is usually solved in the
literature by adding other physical effects to the purely viscous model, such
as internal photoevaporation (see e.g. Hollenbach et al. 1994, Clarke et al.
2001, Owen et al. 2011, Picogna et al. 2019, Emsenhuber et al. 2023). In order
to account for the statistical effect of reducing our sample throughout the
evolution caused by disk dispersal, we performed 100 simulations for both
setups described in Table 1 and then considered not only the median evolution
of the interesting quantities, but also the interval between the 25th and 75th
percentile (see Section 3).

##  3 Results

In this Section, we show the results of the evolution of viscous and MHD
populations of protoplanetary disks in the $M_{\mathrm{d}}-\dot{M}$ plane: in
particular, we consider the ratio of the two quantities (hereafter
$t_{\mathrm{lt}}$, disk lifetime - see Jones et al. 2012). We first discuss
the expected evolution of the distribution of disk lifetimes from an
analytical point of view (paragraph 3.1), and then we confirm our theoretical
results through Diskpop simulations (paragraph 3.2); finally, we compare our
results with the observations (paragraph 3.3).

Figure 1: Time evolution of a synthetic population of disks, evolved via
viscosity (top) or MHD winds (bottom), in the $M_{\mathrm{d}}-\dot{M}$ plane.
The solid lines show the theoretical isochrones at ages 0.1, 1, and 10 Myr as
per the legend, while the disks in the population at each age are represented
by dots with the same color coding. While viscous disks tend to converge to
the same isochrone at evolved stages, MHD disks show a larger dispersion.

###  3.1 Disk lifetimes distribution

In the traditional viscous picture (Dullemond et al. 2006, Lodato et al.
2017), disks lie on the theoretical isochrone (Equation 3) at a given age $t$
if their initial viscous timescale $t_{\nu,0}$ is much shorter than $t$; as
evolution proceeds, more and more disks reach this stage and therefore the
population converges around the corresponding isochrone. As a consequence, the
spread around the isochrones decreases with time: eventually, once the
population is fully self-similar (i.e., its age is larger than all of the
viscous timescales), the spread will be vanishingly small and the correlation
between $M_{\mathrm{d}}$ and $\dot{M}$ will be perfectly linear. This trend is
illustrated in the top panel of Figure 1: the solid lines show three
theoretical isochrones at different ages, while the dots represent a synthetic
population of $100$ disks obtained with Diskpop evolving in time with the same
color coding. The aforementioned convergence to the theoretical isochrone
starts as early as 1 Myr, while at 10 Myr the population is almost fully
evolved and closely resembles the theoretical curve. From this argument, we
can expect the moments of the distribution of $t_{\mathrm{lt}}$ to evolve in
the viscous case as follows: (i) the mean value of $t_{\mathrm{lt}}$ will
converge towards the actual age of the region, (ii) the spread will decrease
until $t_{\nu}<t$ for every disk in the population, (iii) the skewness will
increase. For a more detailed discussion on the expected and observed
evolution of the skewness, we refer to Appendix A.

The bottom panel of Figure 1 shows a synthetic population of disks evolved via
MHD winds in the $M_{\mathrm{d}}-\dot{M}$ plane. As discussed in paragraph
2.2, the evolved population does not converge to the same isochrone: the large
spread at all ages is such that making a prediction on the time evolution of
the distribution of $t_{\mathrm{lt}}$ is not as straightforward as for a
viscous population. Tabone et al. (2022b) have shown that, assuming an
exponential distribution of $t_{\mathrm{acc},0}$ (which is determined fitting
the observed disk fraction), the distribution of $t_{\mathrm{lt}}$ does not
depend on time; however, this result is specific of the exponential
distribution. If we consider a different distribution of $t_{\mathrm{acc},0}$,
that of $t_{\mathrm{lt}}$ for an evolved population may depend on time: this
is the case for our choice of a log-normal distribution of
$t_{\mathrm{acc},0}$, which can still reproduce both the disk and accretion
fraction (see Appendix B).

###  3.2 Mean and width

Figure 2: Time evolution of the mean (top) and width (bottom) of the
distribution of $t_{\mathrm{lt}}$ for a synthetic population of protoplanetary
disks. The solid lines represent the median values, while the shades cover the
interval between the 25th and 75th percentile out of 100 simulations (to
account for the statistical effect of disk removal). The blue and yellow lines
refer to the viscous and MHD model respectively, with the lighter shades
including the observational uncertainty. While the mean value of the
distributions is not much affected by the presence of such uncertainty or the
choice of the model, the spread shows quite some difference, exhibiting
significantly higher values in the MHD than in the viscous case. The dashed
line in the bottom panel marks the observational uncertainty.

Figure 2 shows the time evolution of the mean (top) and width (bottom) of the
distributions of $t_{\mathrm{lt}}$ for the viscous (blue) and MHD (yellow)
models. The lighter shades of both models include an additional observational
uncertainty, $\sigma_{\mathrm{obs}}$, that we implemented by adding an extra
spread on the disk mass and the accretion rate, of 0.1 dex and 0.45 dex
respectively (as an estimate of the observational uncertainty, see Manara et
al. 2023, Testi et al. 2022). As stated in Section 2, we performed 100 runs
for each simulation: the solid line represents the median, while the shaded
areas around it show the 25th-75th percentile intervals. As the MHD model
removes disks more effectively, the sample size decreases more than in the
viscous case, making the statistical fluctuations between different
simulations larger: this leads the yellow lines to have broader shaded areas.

Considering the mean values of the distributions, adding
$\sigma_{\mathrm{obs}}$ only slightly shifts the curves for both the viscous
and MHD case, resulting in a negligible difference. The two evolutionary
models differ at early stages ($<1$ Myr), but soon reach a common behavior
that makes them indistinguishable within the 25th-75th percentile intervals.
On the other hand, the widths of the distribution (bottom panel) significantly
differ from one case to the other. The viscous case without additional
uncertainty (darker blue) steeply decreases, as expected from viscous theory
(Lodato et al., 2017) and discussed in paragraph 3.1. This is not the case for
the MHD prescription (orange): while the general trend is still decreasing, it
is not as steep as the viscous, and ultimately does not tend to zero but
rather to an evolved value determined by the initial conditions.

The convolution with observational uncertainty in the viscous case (light
blue) significantly shifts the curve up, as well as modifying its shape. The
total width of the distribution is the root sum squared of the intrinsic
spread ($\sigma_{\mathrm{int}}$) and the observational uncertainty
($\sigma_{\mathrm{obs}}$),
$\sigma_{\mathrm{tot}}=\sqrt{{\sigma_{\mathrm{int}}}^{2}+{\sigma_{\mathrm{obs}}^{2}}}$.
The intrinsic spread $\sigma_{\mathrm{int}}$, given by the initial conditions,
tends to zero as discussed above: therefore, we expect the final width to tend
to $\sigma_{\mathrm{obs}}$, which is exactly what we recover. This causes the
evolved population to have a significantly larger spread than that predicted
by theory. On the contrary, despite still being shifted at larger values as an
effect of the additional uncertainty, in the MHD case (yellow) the shape of
the curve is not dramatically modified. This is because
$\sigma_{\mathrm{int}}$ is comparable to $\sigma_{\mathrm{obs}}$ at all times,
which makes this argument strongly dependent on the initial condition: as the
total spread is given by
$\sqrt{{\sigma_{\mathrm{int}}}^{2}+{\sigma_{\mathrm{obs}}}^{2}}$, the behavior
of the MHD case will only be significantly different from the viscous case if
$\sigma_{\mathrm{int}}$ is non negligible with respect to
$\sigma_{\mathrm{obs}}$. In our previous work (Somigliana et al., 2022) we
have shown how initial spreads of 0.65 dex and 0.52 dex for $M_{\mathrm{d}}$
and $R_{\mathrm{d}}$ respectively are able to reproduce the observed spreads
around the correlations with the stellar masses; therefore, we set these
values for the MHD simulation, while we choose a bigger spread of 1 dex for
the viscous case, as it can better reproduce the observed values (see 3.3).

As mentioned in Section 2.3, the purely viscous model does not account for
disk dispersal. Without exploring the whole parameter space, which is beyond
the scope of this Letter, we have run a test model with photoevaporation,
assuming the standard model of Owen et al. (2010), with a mass-loss rate of
$10^{-10}$ M⊙ yr-1 following the latest constraints (Alexander et al., 2023).
The mean and the width of the distribution of $t_{\mathrm{lt}}$ increase with
respect to the purely viscous case, but the difference is minimal and becomes
negligible including the observational uncertainty; therefore, our conclusions
are not affected.

###  3.3 Comparison with the observations

Figure 3: Comparison of the evolution of the mean (top) and width (bottom) for
the viscous (blue) and MHD (yellow) models, including observational
uncertainties, with the observations (gray diamonds). The shaded areas are as
in Figure 2, while the gray bars represent the interval between the 16th and
84th percentiles (top) and the uncertainty on the width (bottom). While both
models overestimate the mean values (see text for details), the evolution of
the width of the distribution suggests a better match with the MHD model.

In paragraph 3.2 we have shown the viscous and MHD predictions for the time
evolution of the mean and width of the distribution of $t_{\mathrm{lt}}$; in
this paragraph, we compare our results with observations of different star-
forming regions. We used the table222The table is available at
http://ppvii.org/chapter/15/. compiled by Manara et al. (2023) for Taurus,
Lupus, Chameleon I and Upper Sco, and the data by Testi et al. (2022) for
L1688 (to limit the contamination from sub-populations with different ages in
the Ophiuchus complex).

Before commenting on the comparison itself, it is important to note that our
simulations do not include dust evolution, making our definition of disk mass
solely based on the gas content of disks; on the other hand, the observed disk
masses rely on sub-mm fluxes, tracing the dust content instead. As the bulk of
disk masses is in the gaseous phase, inferring the total mass from dust
observations requires to i) constrain the dust-to-gas ratio in disks and ii)
assume optically thin emission; however, as the accuracy of these assumptions
is debated, the community is striving towards obtaining more reliable disk
mass estimates (see Bergin et al. 2013, McClure et al. 2016 for HD
observations; Veronesi et al. 2021 for dynamical measurements; Anderson et al.
2022, Trapman et al. 2022 for a combination of gaseous tracers). The results
of the ALMA Large Programs AGE-PRO and DECO will further contribute to this
goal; moreover, the advent of the ALMA Band 1 and ngVLA will allow to move to
longer wavelengths, where dust emission is less optically thick (Tazzari et
al., 2021). In light of these forthcoming developments, our work can be
considered a prediction that will be interpreted to its full potential with
the results of this observational effort. The data comparison presented in the
following is therefore intended as a state-of-the-art, which we anticipate to
revise in the near future.

Figure 3 shows the result of our comparison: the mean and width of the
distribution are shown in the top and bottom panel respectively, and both
include the viscous and MHD (blue and yellow line, as in Figure 2 and 5)
numerical evolution. The gray diamonds represent the observed star-forming
regions. None of the two evolutionary mechanisms reproduces the observed mean
values, which are systematically lower. A potential reason for this mismatch
could be an underestimation of disk masses; a difference of a factor as little
as 3 in the observed masses would be sufficient to explain the discrepancy
with the models - confirming the need to repeat this comparison with more
accurate disk masses estimates. Moreover, Zagaria et al. (2022a) have shown
how taking stellar multiplicity into account can explain the high accretors in
Upper Sco; we expect this effect to shift the theoretical prediction to lower
values of $t_{\mathrm{lt}}$ for evolved populations. Dust growth and evolution
prescriptions, which were not included in this work, are also likely to play a
role as they can better explain the observed disk mass - accretion rate
correlation (Sellek et al., 2020). The width of the distribution, on the other
hand, provides more interesting results. The viscous prediction manages to
marginally recover observed values at the earliest evolutionary stages, but as
such values increase in time, the discrepancy with the viscous expectation
grows larger and larger. This result was already anticipated by Manara et al.
(2020) (see also Manara et al. 2023). It should be kept in mind that our
viscous simulations have a $\sigma_{\mathrm{int}}$ of 1 dex for both the disk
mass and radius (see Table 1); as large as the intrinsic spread can be, the
steeply decreasing viscous trend will always evolve the width of the
distribution to $\sigma_{\mathrm{obs}}$. The MHD simulation instead falls
within the error bars of the earliest observed star-forming region, up until
ages on $\sim 2.5$ Myr. There is an increasing discrepancy for more evolved
populations, up until around 20$\%$ for Upper Sco; however, the oldest
populations also represent the less complete samples, and therefore they carry
a significant bias that should be kept in mind when comparing with
simulations. Moreover, there are caveats to our own simulations, as in the
viscous case we neglect disk dispersal mechanisms (such as internal or
external photoevaporation, e.g. Malanga et al. in prep.) and only consider a
detection threshold in disk masses.

##  4 Discussion and conclusions

In this work, we have investigated how the time evolution of the distribution
of a population of disks in the $M_{\mathrm{d}}-\dot{M}$ plane is impacted by
the evolutionary model, considering the viscous and MHD prescriptions
respectively. We have presented a combination of analytical considerations and
numerical simulations, performed through the 1D population synthesis code
Diskpop, in the case of a log-normal distribution of initial accretion
timescales (which reproduces both the disk and accretion fraction). We find
that, while the mean of the distribution of
$t_{\mathrm{lt}}=M_{\mathrm{d}}/\dot{M}$ is not significantly impacted by the
chosen model, the expected behavior of the width shows considerable
differences depending on the evolutionary prescription; when including the
observational biases in the form of additional uncertainty, this distinctive
behavior is maintained.

Our predictions will be exploited to their full potential through a comparison
with the results of the current observational effort to obtain direct
estimates of disk gas masses; for the time being, we compare our evolutionary
trends with the latest available observational data (based on dust
observations) in different star-forming regions. We find that the purely
viscous case only manages to marginally reproduce the observations at the
earliest ages, while the MHD curve resembles them better. Based on these
results, we suggest the analysis of these distributions as a viable method to
disentangle between the viscous and MHD evolutionary models; our data
comparison hints at a better agreement with the MHD model.

## Acknowledgments

We thank an anonymous referee for their comments that helped us improving the
clarity of the manuscript. This work was partly supported by the Italian
Ministero dell’Istruzione, Università e Ricerca through the grant Progetti
Premiali 2012-iALMA (CUP C52I13000140001), by the Deutsche
Forschungsgemeinschaft (DFG, German Research Foundation) - Ref no. 325594231
FOR 2634/2 TE 1024/2-1, by the DFG Cluster of Excellence Origins (www.origins-
cluster.de). This project has received funding from the European Union’s
Horizon 2020 research and innovation program under the Marie Sklodowska- Curie
grant agreement No 823823 (DUSTBUSTERS) and from the European Research Council
(ERC) via the ERC Synergy Grant ECOGAL (grant 855130), ERC Starting Grant
DiscEvol (grant 101039651) and ERC Starting Grant WANDA (grant 101039452).
Views and opinions expressed are however those of the author(s) only and do
not necessarily reflect those of the European Union or the European Research
Council Executive Agency. Neither the European Union nor the granting
authority can be held responsible for them. B.T. acknowledges support from the
Programme National ‘Physique et Chimie du Milieu Interstellaire’ (PCMI) of
CNRS/INSU with INC/INP and cofunded by CNES.

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##  Appendix A Skewness of the distribution

The skewness of a distribution, defined as the third standardized moment,
measures the asymmetry of the distribution about its mean. As we mentioned in
paragraph 3.1, alongside the mean value and the width, in the viscous case we
expect also the skewness of the distribution of $t_{\mathrm{lt}}$ to evolve in
time; in this Appendix we discuss this theoretical expectation and show the
results of our numerical simulations.

Figure 4: Time evolution of a synthetic population of viscous disks in the
$M_{\mathrm{d}}-\dot{M}$ plane (left panel) and corresponding histograms of
$t_{\mathrm{lt}}$ (right panel). The color coding is as in Figure 1. Full dots
represent disks whose initial viscous timescale is shorter than the age of the
population, and that can therefore be considered evolved.

The left panel of Figure 4 shows a population of viscously evolving disks
(dots) at three subsequent ages, as well as the corresponding theoretical
isochrones (solid lines). Full dots represent disks whose initial viscous
timescale is shorter than the age of the population, which as a whole can
therefore be considered evolved: from viscous theory, such disks are expected
to have reached the self-similar condition and lie on the analytical
isochrone, i.e., to show a linear correlation between the disk mass and the
accretion rate. On the other hand, empty dots represent not-yet-evolved disks,
which lie below the theoretical isochrone. As the population evolves, more
disks satisfy the $t_{\nu}<t$ condition, as can be visualized by the
increasing number of full dots in Figure 4; this implies that more disks lie
on the theoretical isochrone, bringing the population on the
$M_{\mathrm{d}}-\dot{M}$ plane closer to a line. While this causes the width
of the distribution of $t_{\mathrm{tl}}$ to decrease with time, the skewness
on the other hand increases - as we show in the right panel of Figure 4, which
represents the corresponding histograms at all ages. This skewing effect is
due to the fact that younger disks, which do not lie on the isochrone yet,
have a $t_{\mathrm{lt}}$ longer than the actual age of the region, and
therefore contribute to positively skew the distribution - while evolved
disks, which make up the bulk of the population, cluster close to the mean
value. Figure 5 shows the evolution of the skewness of a population of disks
generated and evolved with Diskpop with the same color coding and shaded areas
as Figure 1; the left panel represents the case with no observational
uncertainty, where the viscous distribution (blue) gets more and more skewed
as expected, growing by a factor of 2 between 0.1 and 10 Myr. On the other
hand, the MHD distribution (orange) remains symmetrical within the 25th-75th
percentile for the whole evolution, resulting in a factor 3 difference from
the viscous model for evolved populations. As significant as this theoretical
difference is, including the observational biases (right panel) completely
smooths it out: the two expected observed behaviors are indistinguishable once
convoluted with the additional observational uncertainties.

In conclusion, while the evolution of the skewness makes an interesting
theoretical argument stemming from the different interpretation of isochrones
in the two models, it does not provide a reliable method to compare viscosity
and MHD from the observational point of view.

Figure 5: Time evolution of the skewness of the distribution of
$t_{\mathrm{lt}}$ for a synthetic population of protoplanetary disks, with the
same color coding as Figure 2. Both panels show the comparison between the
viscous and MHD models, without (left) and with (right) the additional
observational uncertainty $\sigma_{\mathrm{obs}}$. Despite the theoretical
predictions of the two models being significantly different (left panel), the
convolution with observational biases completely smooths them out (right
panel).

##  Appendix B Time evolution of the distribution of $t_{\mathrm{lt}}$

As $t_{\mathrm{lt}}$ depends on $t_{\mathrm{acc},0}$ as
$t_{\mathrm{lt}}=(1+f_{M})(2t_{\mathrm{acc},0}-\omega t)$, the evolved
distribution of $t_{\mathrm{lt}}$ is determined by the choice of initial
distribution of $t_{\mathrm{acc},0}$: Tabone et al. (2022b) have shown that,
choosing an exponential distribution for $t_{\mathrm{acc},0}$, the
corresponding distribution of $t_{\mathrm{lt}}$ reads

$\frac{dP}{dt_{\mathrm{lt}}}=\frac{1}{\omega\tau(1+f_{M})}\exp{\left(-\frac{t_{\mathrm{lt}}}{(1+f_{M})\omega\tau}\right)}f_{D}(t),$
(B1)

where $f_{M}$ is defined in Tabone et al. (2022a) and $\tau=2.5$ Myr to fit
the disk fraction, $f_{D}(t)=\exp{(-t/\tau)}$. As $f_{D}$ is only a
normalization factor, (B1) still have an exponential shape; moreover, it does
not depend on time, as well as its mean value. On the other hand, if we pick a
log-normal distribution for $t_{\mathrm{acc},0}$, we can still reproduce both
the disk and the accretion fraction (see Appendix C) but in that case the
evolved distribution of $t_{\mathrm{lt}}$ becomes

$\begin{split}\frac{dP}{dt_{\mathrm{lt}}}=&\frac{1}{\sqrt{2\pi\sigma^{2}}}\frac{1}{t_{\mathrm{lt}}+(1+f_{M})\omega
t}\exp{\Biggl{\\{}-\frac{1}{2\sigma^{2}}\left[\log{\left(\frac{t_{\mathrm{lt}}}{2(1+f_{M})}+\frac{\omega
t}{2}\right)-\mu}\right]^{2}\Biggl{\\}}},\end{split}$ (B2)

where $\mu$ and $\sigma$ are the mean value and width of the initial log-
normal distribution. Notice that Equation (B2) is not a log-normal in
$t_{\mathrm{lt}}$; moreover, it does depend on time, and so does its mean
value and spread.

Figure 6: Disk and accretion fraction (left and right panel respectively) in
our viscous (light blue), MHD (orange) and viscous+photoevaporation (lilac)
simulations, compared with data by Hernández et al. (2007) and Fedele et al.
(2010) (blue squares). The shaded areas are as in Figure 2. The dashed blue
lines show the exponential fits to the data. Following the original paper, we
define the accretion fraction as the fraction of sources with accretion rate
higher than $10^{-11}$ M⊙/yr. Our choice of a log-normal distribution of
initial accretion timescales for the MHD model reproduces both the disk and
accretion fraction, as does the exponential distribution chosen by Tabone et
al. (2022b). The viscous model does not reproduce any of the fractions due to
the lack of a disk dispersal mechanism, while including internal
photoevaporation allows to recovered the observed behavior.

##  Appendix C Impact of internal photoevaporation

As mentioned in the main paper, disk dispersal in an intrinsic feature of MHD
winds. These models manage to reproduce both the disk and accretion fraction,
defined as the fraction of young stars with infrared excess (Hernández et al.,
2007) and accreting (i.e., with $\dot{M}>10^{-11}$ M⊙ yr-1 following Fedele et
al. 2010) objects respectively, as shown by the orange lines in Figure 6. On
the other hand, purely viscous models do not account for disk dispersal. This
leads to a mismatch between the predicted and observed disk and accretion
fraction, represented by the blue lines in Figure 6: the disk fraction is
almost constant to 1, the little decrease being due to the observational
threshold that we introduced in our simulations (considering dispersed disks
with masses lower than $10^{-6}$ M⊙, see Section 2.3), while the accretion
fraction does decrease, but not enough to match the observed values. This
problem is usually overcome in the literature by including internal
photoevaporation, a two-timescale process that introduces a disk dispersal
mechanism, allowing to reproduce the observations as shown by the purple lines
in Figure 6. We ran the test simulation presented in this Appendix using the
standard photoevaporative model of Owen et al. (2012), with a mass-loss rate
of $10^{-1-}$ M⊙ yr-1, consistent with the latest constraints (Alexander et
al., 2023).

Once internal photoevaporation kicks in, it lowers the accretion rates for a
given disc mass, introducing therefore a spread in the
$M_{\mathrm{d}}-\dot{M}$ plane (Somigliana et al., 2020); therefore, it could
in principle affect the conclusions of this work. However, we have tested that
the mean and width of the $t_{\mathrm{lt}}$ distribution in the presence of
photoevaporation do not significantly deviate from the purely viscous
prediction; without observational spread the photoevaporative case lies
between the viscous and MHD models, and becomes indistinguishable from the
viscosity when the observational spread is included.