Then

$\displaystyle\gamma_{*}$
$\displaystyle:=\inf_{\lambda}\mathbb{E}\left[\sup_{h\in\mathcal{H}}\frac{\langle
h-h_{\star},\widehat{\mu}-\mu_{*}\rangle}{\langle
h-h_{\star},\mu_{*}\rangle}\right]^{2}$
$\displaystyle=\inf_{\lambda}\mathbb{E}\left[\sup_{h\in\mathcal{H}}\frac{\langle
h-h_{\star},A(\lambda)^{-1/2}{\zeta}\rangle}{\langle
h-h_{\star},\mu_{*}\rangle}\right]^{2}$
$\displaystyle=\inf_{\lambda}\mathbb{E}\left[\sup_{k>k_{\star}}\frac{\sum_{i=k_{\star}+1}^{k}\frac{{\zeta}_{i}}{\sqrt{\lambda_{i}}}}{(k-k_{\star})h}\vee\sup_{k<k_{\star}}\frac{\sum_{i=k+1}^{k_{\star}}\frac{{\zeta}_{i}}{\sqrt{\lambda_{i}}}}{(k_{\star}-k)h}\right]^{2}$
$\displaystyle\leq\inf_{\lambda}\frac{4}{\epsilon^{2}}\mathbb{E}\left[\max_{k=1,\dots,n}\frac{1}{k}\sum_{i=1}^{k}\frac{{\zeta}_{i}}{\sqrt{\lambda_{i}}}\right]^{2}.$

and

$\displaystyle\max_{k=1,\dots,n}\frac{1}{k}\sum_{i=1}^{k}\frac{{\zeta}_{i}}{\sqrt{\lambda_{i}}}$
$\displaystyle=\max_{j=0,\dots,\log_{2}(n)}\max_{k=[2^{j},2^{j+1})}\frac{1}{k}\sum_{i=1}^{k}\frac{{\zeta}_{i}}{\sqrt{\lambda_{i}}}$
$\displaystyle\leq\max_{j=0,\dots,\log_{2}(n)}\max_{k=[2^{j},2^{j+1})}\frac{1}{2^{j}}\sum_{i=1}^{k}\frac{{\zeta}_{i}}{\sqrt{\lambda_{i}}}$
$\displaystyle=\max_{j=0,\dots,\log_{2}(n)}\left(\frac{1}{2^{j}}\sum_{i=1}^{2^{j}-1}\frac{{\zeta}_{i}}{\sqrt{\lambda_{i}}}+\max_{k=[2^{j},2^{j+1})}\frac{1}{2^{j}}\sum_{i=2^{j}}^{k}\frac{{\zeta}_{i}}{\sqrt{\lambda_{i}}}\right)$
$\displaystyle\leq\max_{j=0,\dots,\log_{2}(n)}\left(\frac{1}{2^{j}}\sum_{i=1}^{2^{j}-1}\frac{{\zeta}_{i}}{\sqrt{\lambda_{i}}}\right)+\max_{j=0,\dots,\log_{2}(n)}\left(\max_{k=[2^{j},2^{j+1})}\frac{1}{2^{j}}\sum_{i=2^{j}}^{k}\frac{{\zeta}_{i}}{\sqrt{\lambda_{i}}}\right).$

We now take $\lambda_{i}=\frac{1}{i\log_{2}(n)}$. Note that
$\sum_{i=1}^{n}\lambda_{i}\geq 1/2$ and for any $k>1$ we have
$\sum_{i=1}^{k-1}\frac{1}{\lambda_{i}}\leq k^{2}\log_{2}(n)/2$.

For any $\nu$ we have

$\displaystyle\mathbb{E}\left[\exp\left(\nu\frac{1}{2^{j}}\sum_{i=1}^{2^{j}-1}\frac{{\zeta}_{i}}{\sqrt{\lambda_{i}}}\right)\right]$
$\displaystyle=\exp\left(\nu^{2}\frac{1}{2^{2j}}\sum_{i=1}^{2^{j}-1}\frac{1}{\lambda_{i}}\right)\leq\exp\left(\nu^{2}\log_{2}(n)/2\right)$

so we apply Proposition 1 with $\sigma_{t}^{2}=\log_{2}(n)$ and
$\mathcal{T}=\\{0,1,\dots,\log_{2}(n)\\}$ to obtain

$\displaystyle\mathbb{E}\left[\max_{j=0,\dots,\log_{2}(n)}\left(\frac{1}{2^{j}}\sum_{i=1}^{2^{j}-1}\frac{{\zeta}_{i}}{\sqrt{\lambda_{i}}}\right)\right]\leq\sqrt{2\log_{2}(n)\log(\log_{2}(2n))}.$

The second term deserves a bit more care. First, note that

$\displaystyle\max_{j=0,\dots,\log_{2}(n)}\left(\max_{k=[2^{j},2^{j+1})}\frac{1}{2^{j}}\sum_{i=2^{j}}^{k}\frac{{\zeta}_{i}}{\sqrt{\lambda_{i}}}\right)$
$\displaystyle\leq\max_{j=0,\dots,\log_{2}(n)}\left(\max_{k=[2^{j},2^{j+1})}\sqrt{\frac{2\log_{2}(n)}{2^{j}}}\sum_{i=2^{j}}^{k}{\zeta}_{i}\right)$
$\displaystyle=\max_{j=0,\dots,\log_{2}(n)}\left(\max_{k=1,\dots,2^{j}}\sqrt{\frac{2\log_{2}(n)}{2^{j}}}\sum_{i=1}^{k}{\zeta}_{i}^{(j)}\right)$

where each $\\{{\zeta}_{i}^{(j)}\\}_{i=1}^{2^{j}}$ are i.i.d. sequences of
$N(0,1)$. Now

$\displaystyle\mathbb{P}\left(\max_{j=0,\dots,\log_{2}(n)}\left(\max_{k=1,\dots,2^{j}}\sqrt{\frac{2\log_{2}(n)}{2^{j}}}\sum_{i=1}^{k}{\zeta}_{i}^{(j)}\right)>t\right)$
$\displaystyle\leq\sum_{j=0,\dots,\log_{2}(n)}\mathbb{P}\left(\max_{k=1,\dots,2^{j}}\sqrt{\frac{2\log_{2}(n)}{2^{j}}}\sum_{i=1}^{k}{\zeta}_{i}^{(j)}>t\right)$
$\displaystyle\leq\sum_{j=0,\dots,\log_{2}(n)}\exp(-t^{2}/4\log_{2}(n))$
$\displaystyle=\log_{2}(2n)\exp(-t^{2}/4\log_{2}(n))$

where the last inequality follows from Doob’s maximal inequality. Thus,

$\displaystyle\mathbb{E}\left[\max_{j=0,\dots,\log_{2}(n)}\left(\max_{k=1,\dots,2^{j}}\sqrt{\frac{2\log_{2}(n)}{2^{j}}}\sum_{i=1}^{k}{\zeta}_{i}^{(j)}\right)\right]$
$\displaystyle\leq\int_{t=0}^{\infty}\mathbb{P}\left(\max_{j=0,\dots,\log_{2}(n)}\left(\max_{k=1,\dots,2^{j}}\sqrt{\frac{2\log_{2}(n)}{2^{j}}}\sum_{i=1}^{k}{\zeta}_{i}^{(j)}\right)>t\right)dt$
$\displaystyle\leq\int_{t=0}^{\infty}\min\\{1,\log_{2}(2n)\exp(-t^{2}/4\log_{2}(n))\\}dt$
$\displaystyle\leq
a+\log_{2}(2n)\sqrt{4\pi\log_{2}(n)}\int_{t=a}^{\infty}\frac{1}{\sqrt{4\pi\log_{2}(n)}}\exp(-t^{2}/4\log_{2}(n))\\}dt$
$\displaystyle\leq
a+\log_{2}(2n)\sqrt{4\pi\log_{2}(n)}\exp(-a^{2}/4\log_{2}(n))\\}$
$\displaystyle\leq\sqrt{4\log_{2}(n)\log(\log_{2}(2n))}+\sqrt{4\pi\log_{2}(n)}$

for $a=\sqrt{4\log_{2}(n)\log(\log_{2}(2n))}$.

Putting it all together we have

$\displaystyle\gamma_{\star}$
$\displaystyle\leq\frac{4}{\epsilon^{2}}\left(\sqrt{2\log_{2}(n)\log(\log_{2}(2n))}+\sqrt{4\log_{2}(n)\log(\log_{2}(2n))}+\sqrt{4\pi\log_{2}(n)}\right)^{2}$
$\displaystyle\leq\frac{194\log_{2}(n)\log(\log_{2}(2n))}{\epsilon^{2}}$

##  Appendix K Implementation Details

In this section we discuss the modifications and implementation details of
Algorithm 2. We consider the slightly modified version present in Algorithm 9.

###  K.1 Optimization

First, we reiterate the following definition from Section 4.2:

$f(\lambda;h;{\zeta}):=\frac{\sum_{i\in[n]}(\tilde{h}_{k}(x_{i})-h(x_{i}))\frac{{\zeta}_{i}}{n\lambda_{i}^{1/2}}}{2^{-k+1}+\widetilde{\operatorname*{\text{err}}}(h,\hat{\eta}_{k-1})-\widetilde{\operatorname*{\text{err}}}(\tilde{h}_{k},\hat{\eta}_{k-1})}.$

####  K.1.1 Mirror Descent

In Algorithm 2, we need to solve the optimization problem in (5), namely,

$\inf_{\lambda\in\Delta_{n}}\mathbb{E}_{\zeta\sim
N(0,I)}[\max_{h\in\mathcal{H}}f(\lambda;h;{\zeta})].$

We use stochastic mirror descent method. Specifically, given a current iterate
$\widehat{\lambda}$ and an i.i.d. sample $\zeta_{1},\cdots,\zeta_{B}$ we
computed an unbiased estimate of the sub-gradient

$g(\widehat{\lambda})=\frac{1}{B}\sum_{i=1}^{B}\nabla_{\lambda}\max_{h\in
H}f(\lambda;h;{\zeta}_{i})\Big{|}_{\lambda=\widehat{\lambda}}\quad,$

and then move to $\widehat{\lambda}_{+}=\Pi(\exp(\log(\widehat{\lambda})-\eta
g(\widehat{\lambda})))$ where $\Pi(x)=x/\|x\|_{1}$ and $\log,\exp$ are applied
element-wise. The step size $\eta$ is chosen by back-tracking line search
where two step sizes are equivalent if the difference in estimated function
values is dominated by the the square root of the empirical variance, with
respect to the finite batch size estimates.

The batch size was grown adaptively. Let
$\bar{f}(\lambda)=\mathbb{E}_{\zeta\sim
N(0,I)}[\max_{h\in\mathcal{H}}f(\lambda;h;{\zeta})]$,
$\lambda_{*}\in\arg\min_{\lambda}\bar{f}(\lambda)$ and suppose the algorithm
is at some current configuration $\widehat{\lambda}$. By convexity of
$\bar{f}$ we have

$\displaystyle\bar{f}(\widehat{\lambda})-\bar{f}(\lambda^{*})$
$\displaystyle\leq\langle\nabla\bar{f}(\widehat{\lambda}),\widehat{\lambda}-\lambda_{*}\rangle$
$\displaystyle=\langle\nabla\bar{f}(\widehat{\lambda})-g(\widehat{\lambda}),\widehat{\lambda}-\lambda_{*}\rangle+\langle
g(\widehat{\lambda}),\widehat{\lambda}-\lambda_{*}\rangle$ $\displaystyle\leq
2\max_{k}\big{|}[\nabla\bar{f}(\widehat{\lambda})-g(\widehat{\lambda})]_{k}\big{|}+\max_{k}\langle
g(\widehat{\lambda}),\widehat{\lambda}-\mathbf{e}_{k}\rangle$
$\displaystyle\approx 2\max_{k}\widehat{\sigma}_{k}+\max_{k}\langle
g(\widehat{\lambda}),\widehat{\lambda}-\mathbf{e}_{k}\rangle$

where $\widehat{\sigma}_{k}^{2}$ is the empirical variance of
$[g(\widehat{\lambda})]_{k}$, namely, the sample variance of
$\\{[\nabla_{\lambda}\max_{h\in
H}f(\lambda;h;{\zeta}_{i})\Big{|}_{\lambda=\widehat{\lambda}}]_{k}\\}_{i=1}^{B}$
divided by $B$. Note that $\widehat{\sigma}_{k}=O(1/\sqrt{B})$. Thus, we
double the batch size $B\mapsto 2B$ whenever
$2\max_{k}\widehat{\sigma}_{k}\geq\max_{k}\langle
g(\widehat{\lambda}),\widehat{\lambda}-\mathbf{e}_{k}\rangle$. This also
motivates our stopping condition: for input $\epsilon$, terminate when
$2\max_{k}\widehat{\sigma}_{k}+\max_{k}\langle
g(\widehat{\lambda}),\widehat{\lambda}-\mathbf{e}_{k}\rangle\leq\epsilon$.
Because $B$ is increasing over time, this stopping condition will always,
eventually, be met.

For a fixed $\zeta_{i}$, to compute the corresponding gradient, we computed
$\bar{h}=\operatorname*{arg\,max}_{h\in\mathcal{H}}f(\lambda;h;{\zeta})$ and
used the fact that
$\nabla_{\lambda}\max_{h\in\mathcal{H}}f(\lambda;h;{\zeta})=\nabla_{\lambda}f(\lambda;\bar{h};{\zeta}_{i})$.
As described in the next section, finding $\tilde{h}$ is difficult due to the
use of surrogate loss functions. We provide a line search method,
$\text{LineSearch}(\lambda,{\zeta})$, that finds an approximate value for it.
Together, the full optimization is as follows:

Goal: Solve for $\max_{\lambda}\mathbb{E}_{\zeta\sim
N(0,I)}[f(\lambda;h;\zeta)]$, where $f$ inherently depends on $\tilde{h}_{k}$
and $\hat{\eta}_{k-1}$.

Input: Tolerance $\epsilon$ for stopping criteria.

Initialize: $\widehat{\lambda}\longleftarrow\frac{1}{d}\mathbf{1}$.

repeat

Sample ${\zeta}_{1},...,{\zeta}_{B}\sim N(0,I)$.

Compute $\bar{h}_{i}=\text{LineSearch}(\widehat{\lambda},\zeta_{i})$ for all
$i\in[B]$ and their corresponding gradient estimates
$\nabla_{\lambda}f(\widehat{\lambda};\bar{h}_{i},{\zeta}_{i})$.

$g(\widehat{\lambda})\longleftarrow\frac{1}{B}\sum_{i=1}^{B}\nabla_{\lambda}f(\widehat{\lambda};\bar{h}_{i},{\zeta}_{i})$.

$\widehat{\lambda}\longleftarrow\Pi(\exp(\log(\widehat{\lambda})-\eta
g(\widehat{\lambda})))$ where $\eta$ is chosen as described above.

if $2\max_{k}\widehat{\sigma}_{k}\geq\max_{k}\langle
g(\widehat{\lambda}),\widehat{\lambda}-\mathbf{e}_{k}\rangle$ then

$B\longleftarrow 2B$.

end if

until $2\max_{k}\widehat{\sigma}_{k}+\max_{k}\langle
g(\widehat{\lambda}),\widehat{\lambda}-\mathbf{e}_{k}\rangle\leq\epsilon$

Algorithm 6 $\text{SMD}(\tilde{h}_{k},\hat{\eta}_{k-1})$.

####  K.1.2 Line Search

Then, computing $\max_{h\in\mathcal{H}}f(\lambda;h;{\zeta})$ is equivalent to

$\displaystyle\min_{r\in\mathbb{R}^{+}}r\quad\text{subject to}\enspace
g(r)\leq 0$

where $a=-2^{-k+1}-\sum_{i\in[n]}(1-2\hat{\eta}_{k,i})\tilde{h}_{k}(x_{i})$,
$b=\sum_{i\in[n]}\frac{{\zeta}}{n\lambda_{i}^{1/2}}\tilde{h}_{k}(x_{i})$,
$c_{i}=1-2\hat{\eta}_{k-1,i}$, $d_{i}=-\frac{{\zeta}}{n\lambda_{i}^{1/2}}$ and

$g(r)=ar+b+\max_{h\in\mathcal{H}}\,\sum_{i\in[n]}(c_{i}r+d_{i})h(x_{i}).$

In particular at the optimal value of $r$, called $r^{\ast}$,
$\operatorname*{arg\,max}_{h\in\mathcal{H}}f(\lambda;h;{\zeta})=\operatorname*{arg\,max}_{h\in\mathcal{H}}g(r^{\ast})$.

As shown in Section 4.2, given access to a weighted classification oracle,
computing $\max_{h\in\mathcal{H}}\,\sum_{i\in[n]}(c_{i}r+d_{i})h(x_{i})$ is
equivalent to a $0/1$-loss minimization problem that is solvable using a
weighted classification oracle. If we had such an oracle then since $g(r)$ is
monotonically decreasing as a function of $r$ we can use a binary search
procedure to solve this optimization problem. Indeed, for any given range
$[r_{\min},r_{\max}]$ where the optimal $r$ lies in, we are checking if
$g(\frac{r_{\min}+r_{\max}}{2})\leq 0$. If that is the case, we just set
$r_{\min}\leftarrow\frac{r_{\min}+r_{\max}}{2}$, otherwise, we set
$r_{\max}\leftarrow\frac{r_{\min}+r_{\max}}{2}$. We then check the sign of
$g(\frac{r_{\min}+r_{\max}}{2})$ again and repeat this procedure until a
sufficient tolerance is met.

However, in practice, we do not have access to a weighted $0/1$ loss oracle
and must employ a convex surrogate loss that may not correctly solve the
weighted classification problem. To be concrete, in all of our experiments we
used Scikit-learn’s LogisticRegression classifier. Given such a surrogate,
which we denote as $\widetilde{\max}_{h\in H}$ to acknowledge that it may not
find the optimal $h$, the resulting function

$\tilde{g}(r)=ar+b+\widetilde{\max_{h\in\mathcal{H}}}\,\sum_{i\in[n]}(c_{i}r+d_{i})h(x_{i})$

may no longer be monotonically decreasing in $r$ hence a binary search
procedure would fail. However intuitively it suffices to look at a large
enough set of $r$’s near a zero $\tilde{g}(r)$. To overcome this issue, we
used the procedure in Algorithm 8.

Algorithm 8 overcomes this issue by considering a large set of potential $r$
values and for each value computing the corresponding
$h_{r}=\operatorname*{arg\,max}\tilde{g}(r)$ and stores these in the array
$S$. It then returns $\operatorname*{arg\,max}_{h\in S}f(\lambda;h;\zeta)$.
The set of $r$ values considered is chosen by a multi-scale procedure that
looks at $r$ values on a finer and finer geometric grid given a budget
$N_{\max}$.

Goal: Solve for
$\widetilde{\max}_{h\in\mathcal{H}}ar+b+\sum_{i\in[n]}(c_{i}r+d_{i})h(x_{i})$.

Compute
$\bar{h}=\widetilde{\max}_{h\in\mathcal{H}}\,\sum_{i\in[n]}(c_{i}r+d_{i})h(x_{i})$
by the relaxed weighted classification oracle.

Set $\widehat{g}\leftarrow ar+b+\sum_{i\in[n]}(c_{i}r+d_{i})\bar{h}(x_{i})$.

$S\leftarrow S\cup\\{\bar{h}\\}$.

Return: $\widehat{g},S$.

Algorithm 7 Oracle(r, S).

Goal: Solve for $\max_{h\in\mathcal{H}}f(\lambda;h;{\zeta})$.

Input: fixed $\lambda$ and ${\zeta}$; maximum number of iterations $N_{\max}$;
tolerance $\epsilon$.

Initialize: $S\leftarrow\\{\\}$, $r\leftarrow 100$, $\gamma\leftarrow 10$,
$\delta\leftarrow\sqrt{2}$, and $t\leftarrow 0$.

$\widehat{g},S\leftarrow Oracle(r,S)$.

while $\widehat{g}<0$ and $t<N_{\max}$ do

$r\leftarrow r/2$. {$r$ is too large}

$\widehat{g},S\leftarrow Oracle(r,S)$.

$t\leftarrow t+1$.

end while

for $j=t,...,N_{\max}-1$ do

$\widehat{g},S\leftarrow Oracle(r,S)$.

if $\widehat{g}>0$ then

$r\leftarrow\gamma\cdot r$. {$r$ is too small, need to increase $r$ so that
$\widehat{g}$ decreases.}

else

$r\leftarrow r/\gamma^{2}$. {Reaches a point where $r$ is too large, scale
back.}

$\gamma\leftarrow\gamma/\delta$. {Start searching with a finer grid scale.}

end if

end for

Return: $\operatorname*{arg\,max}_{h\in S}f(\lambda;h;{\zeta})$.

Algorithm 8 $\text{LineSearch}(\lambda,{\zeta})$.

###  K.2 Sampling

The last portion of our algorithm is the sampling scheme. The algorithm
described in Algorithm 2 does not reuse samples between rounds to compute an
estimate for $\hat{\eta}_{k}$. In practice, this can be very wasteful and we
instead want an estimator based on all samples up to and including those taken
in round $k$. In each round the algorithm computes $\lambda_{k}$ with the goal
of $\lambda_{k}\approx\lambda_{\ast}$ for large enough values of $k$ to ensure
that we are sampling from the optimal distribution in that round. Hence, since
we are recycling samples, we need to ensure that the distribution of _all_
samples taken by the end of round $k$, including those in previous rounds,
match $\lambda_{k}$.

To ensure those, we use a waterfulling technique. We set $p_{1}=\lambda_{1}$
and then for each $k\geq 1$ we set

$\displaystyle p_{k}=\operatorname*{arg\,min}_{q\in\Delta_{n}}\max_{j\leq
n}\max\\{0,k\cdot\lambda_{k,j}-(\sum_{i=1}^{k-1}p_{i,j})-q_{j}\\}.$ (86)

Our algorithm being implemented is shown below:

Input: Budget $T$, tolerance $\epsilon>0$, batch size $N$ (default 250).

$\hat{\eta}_{0}\longleftarrow 0$

for $k=1,2,\ldots,\Big{\lfloor}\log_{2}(\epsilon^{-1})\Big{\rfloor}$ do

$\tilde{h}_{k}\longleftarrow\operatorname*{arg\,min}_{h\in\mathcal{H}}\widetilde{\operatorname*{\text{err}}}(h,\hat{\eta}_{k-1})$.

Optimization

$\lambda_{k}\longleftarrow\text{SMD}(\tilde{h}_{k},\hat{\eta}_{k-1})$.

Sampling

Sample till observing $N$ unique $\\{x_{1}^{(k)},\ldots,x_{N}^{(k)}\\}\sim
p_{k}$ that has not been queried before (from rounds $1,...,k-1$).

Query $x_{1},\ldots,x_{N}$ and observe $y_{1},\ldots,y_{N}$.

Compute an estimate $\hat{\eta}_{k}$ with the naive estimator
$\hat{\eta}_{k}^{(\text{Naive})}$ as defined in Section 5.

end for

Return:
$\operatorname*{arg\,min}_{h\in\mathcal{H}}\widetilde{\operatorname*{\text{err}}}^{(k)}(h)$

Algorithm 9 Fixed Budget ACED with Waterfilling.

###  K.3 Batched IWAL

In this section we explain our implementation of the IWAL
algorithm(Beygelzimer et al., 2010) (and variants such as IWAL1 and oracular
variants) for streaming active algorithms. In round $k$ we assume access to a
(labeled) dataset
$S_{k-1}=\\{(x_{i},y_{i},p_{i})_{i=1}^{n_{k}}\\}\subset\mathcal{X}\times\\{\pm
1\\}\times[0,1]$, with $n_{k}\leq k$. Given a new point from a stream,
$x_{k}$, the decision to label $x_{k}$ is made by computing two hypothesis.
Firstly we compute

$h_{k}=\operatorname*{arg\,min}_{h\in\mathcal{H}}\sum_{i=1}^{n_{k}}\frac{\mathbf{1}\\{h(x_{i})\neq
y_{i}\\}}{p_{i}}$

and second we compute

$h^{\prime}_{k}=\operatorname*{arg\,min}_{\begin{subarray}{c}h\in\mathcal{H}\\\
h_{k}(x_{k})\neq
h^{\prime}_{k}(x_{k})\end{subarray}}\sum_{i=1}^{n_{k}}\frac{\mathbf{1}\\{h(x_{i})\neq
y_{i}\\}}{p_{i}}.$

Thus to compute $h^{\prime}_{k}$ we need access to a weighted classification
oracle for 0/1 loss that can handle a single constraint.

In general including the constraint $h_{k}(x_{k})\neq h_{k}^{\prime}(x_{k})$
is not easy for arbitrary classes and past methods have considered the
optimization

$\operatorname*{arg\,min}_{h\in\mathcal{H}}\sum_{i=1}^{n_{k}}\frac{\mathbf{1}\\{h(x_{i})\neq
y_{i}\\}}{p_{i}}+q\mathbf{1}\\{h(x_{k})\neq h_{k}(x_{k})\\}$

for a sufficiently large weight $q\in\mathbb{R}$ to ensure the
constraint(Karampatziakis & Langford, 2010) .

However in the case of linear classes under the surrogate convex logistic
loss, the main focus of experiments in this paper, we take a different
approach. Assume that $\mathcal{X}\subset\mathbb{R}^{p}$ and that
$\mathcal{H}=\\{h(x)=w^{\top}x+b:w\in\mathbb{R}^{p},b\in\mathbb{R}\\}$.
W.l.o.g. assume that $h_{k}(x_{k})=-1$. Under this convex relaxation, we seek
a classifier where $w^{\top}x_{k}+b=\epsilon$, where $\epsilon\geq 0$ i.e. the
linear predictor flips the predicted sign of $x_{k}$ and has margin
$\epsilon$. Thus we learn (an approximate) $h_{k}^{\prime}$ by solving

$\displaystyle\min_{\begin{subarray}{c}w\in\mathbb{R}^{p},b\in\mathbb{R}\\\
w^{\top}x+b=\epsilon\end{subarray}}\sum_{i=1}^{n_{k}}\frac{1}{p_{i}}\log(1+\exp(-y_{i}(w^{\top}x+b)))$
$\displaystyle=\min_{w\in\mathbb{R}^{p}}\sum_{i=1}^{n_{k}}\frac{1}{p_{i}}\log(1+\exp(-y_{i}(w^{\top}x_{i}-w^{\top}x_{k}+\epsilon)))$
$\displaystyle=\min_{w\in\mathbb{R}^{p}}\sum_{i=1}^{n_{k}}\frac{1}{p_{i}}\log(1+\exp(-y_{i}(w^{\top}(x_{i}-x_{k})+\epsilon)))$

This is a convex optimization problem with no constraints that is easily
solvable using a procedure for logistic regression where we assume that the
intercept is some $\epsilon$ with really small magnitude.

##  Appendix L Full Scale Plots for Performances on the Pool

Figure 5: Full scale of Figure 1 Figure 6: Full scale of Figure 2 Figure 7:
Full scale of Figure 4 Figure 8: Full scale of Figure 3

##  Appendix M Hyperparameters for Baselines

We searched in the following $C_{0}$ that uses the same grid fineness as
(Huang et al., 2015):

Baseline | MNIST | SVHN | FashionMNIST | CIFAR
---|---|---|---|---
IWAL-0&1 | $10^{-7},10^{-6},...,1$ | $10^{-7},10^{-6},...,1$ | $10^{-8},10^{-7},...,1$ | $10^{-7},10^{-6},...,1$
ORA-IWAL-0&1 | N/A | $10^{-4},10^{-3},...,10^{-1}$ | $10^{-8},10^{-7},...,1$ | $10^{-4},10^{-3},...,10^{-1}$
OAC | $.001,.003,.01,.03$ | $.01,.03,.1,.3$ | $.001,.003,...,1$ | $.001,.003,.01,.03$
Table 1: Ranges of $C_{0}$ searched for different experiments. Figure 9:
Sensitivity of OAC to $C_{0}$ on the CIFAR 2 vs 7 dataset (training accuracy).

We used the following amount of passes over dataset for the following
baselines. For OAC, we made sure the number of passes is sufficient enough so
that the algorithm is no longer taking more queries.

Baseline | MNIST | SVHN | FashionMNIST | CIFAR
---|---|---|---|---
IWAL-0&1 and ORA-IWAL-0&1 |  $1$ (N/A for ORA variants) | $2$ | $2$ | $2$
OAC | $5$ | $10$ | $10$ | $10$
Table 2: Ranges of $C_{0}$ searched for different experiments.

For Vowpal Wabbit, we used an initial learning rate of $0.5$ for CIFAR, and
$1$ for every other experiments.

##  Appendix N Generalization Performance on Holdout Set

In the following figures, we show performances of the algorithms on a holdout
test set. We note that there’s no algorithm that is consistently the best
among all four experiments, but ACED is consistently among the top two
algorithms, and is the best in two of the four experiments. We also reiterate
that the OAC curves have stopped taking more queries at the end, so their
final accuracies are inferior than other algorithms in most cases.

Figure 10: MNIST performance on test set Figure 11: SVHN performance on test
set Figure 12: FashionMNIST performance on test set Figure 13: CIFAR
performance on test set


# The fate of icy pebbles undergoing sublimation in
protoplanetary discs

Stefano Spadaccia,1 Holly L. Capelo,1 Antoine Pommerol,1 Philipp Schuetz,2
Yann Alibert,1 Katrin Ros3 and Nicolas Thomas1
1 Space Research and Planetary Sciences Division, Physikalisches Institut,
University of Bern, Sidlerstrasse 5, 3012 Bern, Switzerland
2Lucerne University of Applied Sciences and Arts, Technikumstrasse 21, 6048
Horw, Switzerland
3Lund Observatory, Department of Astronomy and Theoretical Physics, Lund
University, Box 43, 221 00 Lund, Sweden E-mail:
<EMAIL_ADDRESS>
(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

Icy pebbles may play an important role in planet formation close to the water
ice line of protoplanetary discs. There, dust coagulation is more efficient
and re-condensation of vapor on pebbles may enhance their growth outside the
ice line. Previous theoretical studies showed that disruption of icy pebbles
due to sublimation increases the growth rate of pebbles inside and outside the
ice line, by freeing small silicate particles back in the dust reservoir of
the disc. However, since planet accretion is dependent on the Stokes number of
the accreting pebbles, the growth of planetesimals could be enhanced
downstream of the ice line if pebbles are not disrupting upon sublimation. We
developed two experimental models of icy pebbles using different silicate
dusts, and we exposed them to low-temperature and low-pressure conditions in a
vacuum chamber. Increasing the temperature inside the chamber, we studied the
conditions for which pebbles are preserved through sublimation without
disrupting. We find that small silicate particles ($<50$ µm) and a small
quantity of ice (around $15$ % pebble mass) are optimal conditions for
preserving pebbles through sublimation. Furthermore, pebbles with coarse dust
distribution ($100-300$ µm) do not disrupt if a small percentage ($10-20$ %
mass) of dust grains are smaller than $50$ µm. Our findings highlight how
sublimation is not necessarily causing disruption, and that pebbles seem to
survive fast sublimation processes effectively.

###### keywords:

protoplanetary discs – planets and satellites: formation – methods:
miscellaneous – techniques: image processing – techniques: miscellaneous

††pubyear: 2021††pagerange: The fate of icy pebbles undergoing sublimation in
protoplanetary discs–The fate of icy pebbles undergoing sublimation in
protoplanetary discs

##  1 Introduction

Protoplanetary discs are collapsed clouds of gas and dust surrounding young
stars. From micrometer-sized dust particles, planetary embryos form and a
completely new planetary system appears eventually. The direct growth of dust
up to the size of planetesimals ($1-100$ km) is inhibited. Particles bigger
than mm-size tend to disrupt in collisions, and boulder-size objects interact
heavily with the gas in the disc, losing angular momentum and drifting toward
the central star (Weidenschilling, 1997; Blum & Wurm, 2008; Birnstiel et al.,
2012). Streaming instability (Youdin & Goodman, 2005; Johansen & Youdin, 2007)
is a promising mechanism to overcome this inhibition and actually form
planetesimals: mm-cm size aggregates (referred hereafter as ‘pebbles’), when
numerous enough, concentrate in filaments in the disc, these latter eventually
collapsing to form planetesimals.

In this scenario, ice lines in the disc can play a key role in planet
formation. The ice line of a certain volatile species in the disc is the
location where the temperature is low enough to condensate the volatile into
its solid phase (ice). Around the water ice line, the temperature is high
enough (>$170$ K) that the surface energy of water ice is sufficient to allow
icy particles to grow via coagulation (Gundlach & Blum, 2014; Musiolik & Wurm,
2019). Furthermore, ice sublimation at the ice line causes vapor diffusion and
re-deposition outside the ice line, enhancing pebble growth and potentially
triggering streaming instability (Drążkowska & Alibert, 2017; Ros et al.,
2019). Understanding the physics and chemistry of icy pebbles and the phase
changes of volatiles is thus paramount for understanding planet formation
processes close to the ice lines.

What is the composition of an icy pebble? We can retrieve some information
about pebbles from comets, which are thought to be thermally unaltered objects
and preserve pristine materials from the disc in their nuclei. Measurements by
the Rosetta instrument GIADA of the dust ejected from comet
67P/Churyumov–Gerasimenko (67P hereafter) constrain the porosity of the
putative pebbles composing the nucleus of the comet to $\sim 50$ % (Fulle et
al., 2016). Experiments on dust aggregates showed that a highly porous
aggregate ($85$ % porosity) can be compacted down to $60$ % porosity,
simulating compaction by collisions between pebbles in an accretion scenario
(Weidling et al., 2009; Jutzi et al., 2009). Furthermore, Blum et al. (2017)
suggest that pebbles with a size range of $3-6$ mm are present in the nucleus
of 67P and that the interstitial space between pebbles is filled with fluffy
fractal aggregates. The measurements of the CONSERT instrument allowed
deriving a dust-to-ice volume ratio of 67P in the range of $0.4-2.6$ (Kofman
et al., 2015). These data depict a pebble formed outside the ice line as a mm-
size porous aggregate, formed by ice and small dust grains, with a variable
ice-to-dust ratio.

Although pebbles are porous aggregates of refractory and volatile materials,
several studies of dust evolution in protoplanetary discs handle them more
simply as compact spheres made with a specific composition of volatiles and
refractory compounds (Barrière-Fouchet et al., 2005; Gonzalez et al., 2015;
Drążkowska & Alibert, 2017; Schoonenberg & Ormel, 2017). This assumption may
be insufficient to correctly describe coagulation processes and pebble-to-
pebble collisions, but it offers a great simplicity for the purpose of
modeling other dynamical processes.

The fate of icy pebbles undergoing sublimation is important to understand if
the refractory materials embedded in the pebble are released back to the disc
or if the sublimated pebble manages to maintain its integrity without
disrupting. Many theoretical studies assume that pebbles disrupt through ice
sublimation with important consequences on planetesimals growth. Saito &
Sirono (2011) and Ida et al. (2016) showed how disruption of icy pebbles
drifting toward the central star through the ice line enhances the dust local
density and triggers gravitational instability in the vicinity of the ice
line. Schoonenberg & Ormel (2017) use a simple model of pebbles in their
simulations, and find that disruption of pebbles through the ice line and ice
deposition enhance pebble growth, in particular outside the ice line. Their
pebbles are compact spheres made of $50$ % water ice and $50$ % silicates. In
their work, they consider two mixtures of dust and water ice: one type of
pebbles has a compact silicate core and a shell of ice (‘single-seed model’),
while the other one is a homogeneous mixture of µm-sized ice and silicates
particles (‘many-seeds model’). They assume two different behaviors of these
pebbles once they drift through the ice line. The single-seed model sublimates
the ice shell, leaving behind the compact silicate nucleus as a remnant. The
many-seeds model sublimates the ice particles and releases small silicate
particles. In the case of the many-seeds model, Schoonenberg & Ormel (2017)
found an enhancement of dust-to-gas ratio interior to the ice line (but less
than Saito & Sirono (2011) and Ida et al. (2016)) and outside the ice line,
due to the fact that the released small silicate particles couple with the gas
and diffuse outward, sticking to the icy pebbles outside the ice line.
Interestingly, the results from Schoonenberg & Ormel (2017) do not change
considering the porosity of pebbles: increasing porosity would increase the
sublimation rate, but even increasing it by a factor $100$ would not affect
the consequent dust enhancement. Similarly, Hyodo et al. (2021) found
different dust enhancement and planet formation pathways around the ice line,
through pile-up of icy pebbles outside the ice line and of small silicate
grains inside the ice line. The general assumption in these works is that
sublimation of ice is breaking the adhesion forces between dust grains inside
the pebble, causing its disruption. Aumatell & Wurm (2011) created in the
laboratory aggregates of condensed ice crystals and studied their disruption
by sublimation while levitating. They found that those fluffy aggregates are
easily disrupted though a fast sublimation, but they did not include dust
grains in their experimental models. Understanding if icy pebbles are
disrupting throughout sublimation is essential to investigate the processes
that would lead quickly to the formation of planetary embryos in the proximity
of ice lines.

In addition, pebble accretion in general depends strongly on their Stokes
number, and therefore on their size and structure (note that a fluffy
structure could lead to a drag coefficient very different from the usually
assumed drag coefficient of a sphere). Understanding the disruption of pebbles
at the ice line could therefore have important consequences on the formation
timescale of all planets located inside the ice line. For example, Morbidelli
et al. (2015) explained the ‘great dichotomy’ of the Solar System (the fact
that inner planets are dry and small whereas outer planets are wet and large)
by the difference between the accretion rate inside the ice line and outside
it, itself resulting from both the change of mass (reduction of 50 % of the
mass flux due to the evaporation) and of radius. Such a scenario would be put
into question, at least partially, in the case where pebbles would keep their
structure while crossing the ice line.

We present here new experiments on the structure of pebbles composed of water
ice and silicates when they are free to sublimate in vacuum with increasing
temperature. The goal of the present study is to characterize the different
outcomes of sublimated icy pebbles (disruption or preservation), and
understand how the environment and the composition of the pebbles determine
the final outcomes of the sublimation.

The experiments aim at answering these scientific questions:

  1. 1.

Are icy pebbles always disrupting due to sublimation of ice?

  2. 2.

Does the dust-to-ice ratio influence pebble disruption?

  3. 3.

What is the role of the dust size range in the disruption due to sublimation?

The structure of the manuscript is the following. Section 2 presents the
material used, the new pebble production methods and their features, and the
experimental setup used. In section 3, we summarize the experimental results.
Discussion of the results and future experiments are presented in section 4.
Finally, we conclude with our main findings in section 5.

##  2 Material and Methods

The necessity of simulating cold planetary and space environments lead to the
development of the SCITEAS-2 (Simulation Chamber for Imaging the Temporal
Evolution of Analogue Samples version $2.0$) vacuum chamber, which provides a
low-pressure and low-temperature environment for the sublimation of icy
samples, with the possibility to acquire spectral measurements of the samples
over time (Pommerol et al., 2015). Two types of icy particles have been
developed as analogs for icy planetary surfaces: flash-cooling of small water
droplets produces two different spherical particles with size distributions
peaking at $4$ and $70$ µm. These particles can then be mixed with dry dust in
two ways (Pommerol et al., 2019; Poch et al., 2016): through intra-mixture
(i.e. dust particles embedded in icy particles) and inter-mixture (i.e. dust
particles mixed externally with the icy particles). From such ice particles,
it is difficult to create mm-sized aggregates. Therefore, we developed new
methods for creating compact pebble-size mixtures of ice and dust.

The evolution of realistic pebbles at the ice line is a complex problem. The
chemical composition of the dust, how it is mixed with ice, the dust size
distribution, the presence of organics and salts, and the overall amount of
water ice in the pebble are concurring to the outcome of the sublimation
process.

###  2.1 Dust composition

Infrared observations of protoplanetary discs have shown an abundance of
silicates, both in crystalline and amorphous form. In particular, Mg-rich
endmembers of olivine and pyroxene are common (Natta et al., 2006). Alongside
silicates, materials that are more refractory are produced close to the star,
such as corundum, anorthite, hibonite, perovskite, and spinel. These are the
first minerals formed in the protoplanetary disc, and a certain amount can
diffuse in the outer disc through processes such as turbulence, meridional
flows, and X-winds (Watson et al., 2009). The outward diffusion of a certain
amount of highly refractory minerals is demonstrated by their presence in
comets which never left the cold outer disc since their formation (Simon et
al., 2008).

A precise ‘recipe’ for pebble dust mineralogy is difficult to establish, since
we do not know the relative abundances of the different minerals and since
they are not evenly distributed within the disc and undergo different
processes during the disc lifespan. In order to compare our results to
previous theoretical studies of protoplanetary disc evolution, we used two
different silicates as analogs for the mineral dust: olivine and pyroxene. The
list of powders used in the present work is provided in Table 1. Different
grain size ranges have been obtained by grinding the dust in a mortar and dry
sieving it.

The two dust species have been analyzed through scanning electron microscopy
(SEM) to check qualitatively the difference in the grain surfaces and the
amount of fine material of the two silicates (Fig. 1). We do not see evidence
for significant differences in the amount of very fine grains (<$5$ µm) and
the surfaces of the grains appear relatively similar. The only notable
difference is the sharper aspect of the edges of the olivine grains with
respect to pyroxene. Unfortunately, it is not possible to derive the overall
dust size distributions from the SEM images, which could explain some
differences observed between the behaviours of pebbles made of olivine and
pyroxene (see section 4.4). Following the sublimation experiments, we
collected the desiccated dust and measured its near-infrared reflectance
spectrum to look for possible signs of aqueous alteration (Fig. 2). None of
the absorption features indicative of hydration and hydroxilation (around 1.4
and 1.9 µm, see (Noe Dobrea et al., 2003; Pommerol & Schmitt, 2008)) are
visible.

Table 1: List of silicates used in the experiments. Properties | Olivine | Pyroxene
---|---|---
Aspect | Green powder | Gray powder
Initial grain size [ µm] | $0-3000$ | $0-1000$
Density [g cm-3] | $3.3$ | $3-4$
Quarry location | Norway | Brazil
Provider | Microbeads AG |  | UCF & Deep
---
Space Industries
Figure 1: SEM images of olivine and pyroxene grains. A) and B) olivine dust at
different magnifications. C) and D) pyroxene dust at different magnifications.
Figure 2: Visible and near-infrared reflectance spectra of olivine and
pyroxene powders collected from the desiccated pebbles at the end of the
sublimation experiments. The spectra do not show the absorption features
associated with aqueous alteration at 1.4 and 1.9 µm. The shadowed region
indicates the pixel variance of the reflectance in the selected region of
interest defined to derive an average spectrum from hyperspectral cubes.

###  2.2 Sample preparation and analysis

In the rest of the paper, we refer to the two pebble models as pebble type-A
(PA) and pebble type-B (PB). The main difference between the two production
methods is the amount of water ice of the pebble and the relative ice-dust
distribution inside the pebble. Both PA and PB are compact aggregates and
their initial porosity is negligible. Table 2 collects the main features of PA
and PB pebbles.

The two types of pebbles were analysed with X-ray computed tomography (XCT)
technique, to detect the relative distribution of dust and ice inside the
pebbles. The measurements have been carried on a Diondo 2 system manufactured
by Diondo (Hattingen, Germany) at the Lucerne Computed tomography Imaging
facility at Lucerne University of Applied Sciences and Arts. For both
measurements, the microfocus tube XWT$-225$ TCHE+ from X-ray WorX has been set
on an acceleration voltage of $120$ kV and a tube current of $80$ µA and an
aluminum filter with $1$ mm thickness was employed. The detector VAREX
$4343$CT was configured with a capacity of $8$ pF, and integration time of $1$
s. The field of view of the detector was adjusted to match the extension of
the sample with source object distance of $37$ mm, and a source-detector
distance of $851$ mm. For the measurement of the olivine and the pyroxene
sample, $1900$ and $1600$ projections of the object spinning around a fixed
axis in equiangular steps were recorded, respectively. The recorded
projections were reconstructed using the Feldkamp-Davis-Kress algorithm as
integrated in the CERA suite from Siemens (Erlangen, Germany). The virtual
cross-sections had a voxel size of $6.5$ µm isotropically and were analysed
with Volume Graphics Studio Max $3.3$ from Volume Graphics (Heidelberg,
Germany) and imageJ (Schneider et al., 2012).

Table 2: Summary of pebble type-A (PA) and pebble type-B (PB) main features. | PA | PB
---|---|---
Shape | Spheroidal | Irregular spheroid
Ice mass content [ %] | $50\pm 10$ | $15\pm 7$
Typical size [mm] | $1-5$ | $2-7$
Dust distribution | Homogeneous or | Homogeneous
inside the pebble | in one half of the pebble |

####  2.2.1 PA production

The PA production method takes advantage of the behavior of water on
superhydrophobic surfaces. Superhydrophobic surfaces have micrometric ‘spikes’
which, combined with the surface tension of water, allow the liquid to rest on
the top of the asperities. The water flowing on the superhydrophobic surface
captures any dust deposited on the top of the spikes in a process called
‘self-cleaning’ (Marmur, 2004). An inclined plane coated with superhydrophobic
paint (Cytonix WX$2100$ aerosol) is placed above a bowl full of liquid
nitrogen (Fig. 3). On the surface, some dust is deposited with a spoon,
distributing it on the surface as homogeneously as possible. A droplet of
distilled water, created with a micropipette, rolls and captures the dust
deposited on the plane, and then sinks in liquid nitrogen where the water
freezes.

The droplets created with the micropipette are typically about $2-3$ mm in
diameter, but sometimes a droplet impinging the plane can split and create
secondary droplets of about $1$ mm in diameter, which capture the dust in the
same way as the bigger droplets.

The dust collected by the droplets on the plane would tend to remain on the
surface of the droplet because of the centrifugal force and surface tension of
water. On the other hand, if more dust is collected, the grains that are
already embedded in water are gradually pushed toward the center of the
droplet. In the limit case, if the droplet is saturated with dust, the water
does not interact with the superhydrophobic plane anymore and the drop stops
on the inclined plane. When a droplet sinks into liquid nitrogen, the inverse
Leidenfrost effect (Thimbleby, 1989; Adda-Bedia et al., 2016) prevents the
droplet to freeze instantly, and the dust captured by the droplet can move
under gravity inside the liquid core of the pebble. In this way, it may happen
that the frozen pebble has a heterogeneous distribution of the dust inside it,
with more dust on one half due to the gravity acting on the dust grains during
the freezing process. This heterogeneity may not happen when the droplet is
small (like secondary droplets): in this case, the freezing process is fast
and the grains are ‘trapped’ homogeneously inside the pebble (Fig. 4). In this
work, we refer by "heterogeneity" to the overall misplacement of grains with
respect to the ice inside the pebble, while we make no assumption on the
homogeneity of the spatial distribution of the grains with respect to each
other.

The ice-dust ratio was estimated by weighing the pebble before and after the
evaporation of the water. The ice mass was found to be in the range of
$\left(50\pm 10\right)$ % of the whole pebble mass. It can happen that a
droplet does not catch much dust and has a higher ice content, but since the
experiments involve several pebbles in the same sample holder, an average
value is more indicative.

Figure 3: The procedure for PA production. A droplet created with a
micropipette rolls over a superhydrophobic (SH) inclined plane, and it catches
the dust deposited on it. Then, the droplet sinks into liquid nitrogen (N2
liquid), where water freezes in a few seconds. Figure 4: X-ray tomography of
PA icy pebbles analogues made with olivine (dust size smaller than 50 µm). The
circle in the center is the cross section of an icy pebble. a) This PA clearly
shows a heterogeneous spatial distribution of ice (dark gray material on the
top) and dust (light gray material on the bottom), well distinguishable from
each other. b) This PA contains an internal homogeneous distribution of dust
grains. The scale bar for both images is 1 mm.

####  2.2.2 PB production

The second type of pebbles uses the capability of liquid water of linking
small dust particles through capillary bridges, also called ‘wet granulation’
(Iveson et al., 2001). A $1$ cm thick layer of dust is settled in a small
vessel. A droplet of water is produced with a micropipette above the dust. The
droplet impinges the dust layer with enough momentum to get a fast
interpenetration between water and dust particles. The final wet aggregate is
stable because of the capillary and viscous forces of water bridges acting on
adjacent dust particles. Finally, the aggregate is carefully taken with a
spoon and sunk in liquid nitrogen, where the water bridges freeze. Any dust
particle that is not connected to the others through capillary bridges is lost
in liquid nitrogen due to its turbulent motion at room temperature.

Weighing PBs made of different types of dust before and after the evaporation
of the water resulted in an ice mass equivalent to $\left(15\pm 7\right)$ % of
the total mass. The final shape of a PB is generally an irregular spheroid
with a homogeneous distribution of dust inside the pebble (Fig. 5). Sometimes
a wet aggregate disrupts into two or three pieces before being put into liquid
nitrogen, resulting in a wider distribution of sizes of the final icy
aggregates, ranging from $2$ to $7$ mm in diameter.

Granulation processes are intensively studied for industrial processes. The
above procedure creates a compact pebble, where water is creating films on the
surface of the particles and is filling the voids between particles (Iveson et
al., 2001).

Figure 5: X-ray tomography of an icy PB made with pyroxene (dust size 100-300
µm). The icy bonds keep the dust grains (bright gray) together and the dust
distribution inside the pebble is homogeneous. The scale bar is 3 mm.

###  2.3 Experimental setup

We performed all the sublimation experiments with the SCITEAS-$2$ (Simulation
Chamber for Imaging the Temporal Evolution of Analogue Samples version $2.0$)
vacuum chamber (Fig. 6). We evacuated the chamber through a primary dry vacuum
pump and a turbomolecular pump, which allowed to reach a high vacuum ($\sim
10^{-7}$ mbar). At the center of the chamber, we cooled a copper plate with a
Helium cryocooler, reaching temperatures down to $40$ K. The cylindrical
sample holder was placed on the copper plate and a thin sheet of graphene
between the two components guaranteed good thermal contact. The icy pebbles
were in contact with the sample holder, that had a higher temperature than the
cooled copper plate (around $100$ K at the beginning of the decompression). A
PT$100$ thermal sensor measured the temperature at the bottom of sample
holder, recording the temperature of the surface below the icy pebbles.

During the sample production, we pre-cooled down the copper base inside
SCITEAS-$2$ by pouring liquid nitrogen into the sample holder. Once the
samples were ready, we kept them in liquid nitrogen to avoid frost formation.
When the copper plate of the He cryocooler was almost at the same temperature
as the liquid nitrogen, we placed the samples in the cold sample holder, which
was inserted in the SCITEAS-$2$ chamber. Then we plugged the PT100 sensor
inside the chamber and closed the upper cover. Immediately afterwards, we
turned on the primary pump to quickly evacuate the chamber, and the
temperature-pressure reading started. At a pressure of about $10^{-2}$ mbar,
we turned on the turbomolecular pump, allowing to reach about $10^{-6}$ mbar
in a few minutes. The samples rested a few hours in these conditions until we
achieved an equilibrium pressure and temperature inside the chamber. We then
turned off the cryocooler and the samples slowly heated up by the thermal
radiation coming from the window above and the walls of the chamber. The
temperature increased in about $25$ hours to room temperature. At this point,
we turned off the pumps to ventilate slowly the chamber. For different types
of pebbles, more experiments have been carried out.

Figure 6: Scheme of the SCITEAS-$2$ setup. A camera is placed in front of a
window above the vacuum chamber. The chamber is evacuated through a primary
pump and a turbomolecular pump, and the sample holder is in contact with a
copper head cooled down with a He-cryocooler.

The temperature profile of the sample holder during the experiment evolves
logarithmically over time, from a temperature of around $100$ K to a final
temperature of about $280$ K. The total pressure inside the chamber settles
down to $10^{-7}$ mbar before turning off the He-cryocooler. When the
temperature starts to increase, the total pressure inside the chamber
increases slowly up to $\sim 10^{-3}$ mbar due to the sublimation of ice
condensed on the cold copper plate at the closure of the chamber. At around
$10$ hours after the beginning of the experiments, it slowly decreases down to
$\sim 10^{-6}$ mbar.

####  2.3.1 Disruption measurement

After the sublimation process inside the chamber, pebbles can either keep
their spheroid shape or disrupt completely or partially in piles of dust. Both
gravity and sublimation participate to this disruption process (see section
4.3). In Fig. 7 we show how pebbles look like once removed from the chamber
when the experiment is finished. The preserved pebbles show a clear spheroid
shape conserving overhangs and protrusions, while the disrupted pebbles are
just piles of dust and lost their initial spheroidal shape.

Figure 7: Different outcomes of the experiment: pebbles can preserve their 3D
shape or disrupt in piles of dust. a) Preserved PA made of olivine with grain
size range < 50 µm. b) Preserved PB made with pyroxene. c) preserved PB made
with olivine. d) Disrupted PB made with coarse dust of pyroxene. Note a
partially disrupted pebble on the top of the image, surrounded by a halo of
settled dust whit a bigger projected area than the original pebble. All scale
bars are adjusted to be about 1 cm.

Analysing the pebbles outside the chamber is difficult.The vibrations caused
while removing the sample holder from the vacuum chamber can disrupt weak
preserved pebbles, and the re-compression of the chamber after the experiment
could as well participate in the disruption of weakly preserved dust
aggregates. To address this problem, we developed a method for evaluating the
disruption of the aggregates directly inside the chamber, without handling the
samples. To determine quantitatively the level of preservation or disruption
of the pebbles, we placed a camera above the window, pointing at the sample.
Two images were taken at the beginning and at the end of the experiment,
allowing the detection of differences in the projected area of the pebbles
before ($A_{\mathrm{b}}$) and after ($A_{\mathrm{a}}$) sublimation. For
geometrical reasons, when a spheroidal aggregate disrupts into a pile of dust,
its projected area increases. We define then the disruption parameter $D$ as
follows:

$D=\frac{A_{\mathrm{b}}}{A_{\mathrm{a}}}.$ (1)

If the ratio of the areas is close to $1$, then the pebble shape did not
change and the pebble survived the sublimation of the ice. If the ratio is
$<1$ then the projected area increased and the pebble disrupted in a pile of
dust (Fig. 8). Disruption is measured at the end of the sublimation, before
the re-compression of the chamber and its ventilation. This method allows to
recognize small changes in the projected areas of disrupted pebbles, leading
to a detection of disruption. For each type of pebbles and dust combination,
we have run sufficient sublimation experiments to obtain a sample with good
statistic relevance. The pebbles that do not disrupt during the experiment,
can resist months at room temperature and ambient pressure, until some
external forces stronger than gravity break the aggregates (handling,
vibrations).

Figure 8: a) Setup for observation of pebble disruption. The ratio between
$A_{\mathrm{b}}$ and $A_{\mathrm{a}}$ provides information on the level of
disruption of the pebble. The images before and after sublimation are used to
calculate the ratio between projected areas. b) PAs of olivine before
sublimation. c) The same pebbles after sublimation. The disruption created
piles of dust with bigger cross-sectional area. The scale bar is 1 cm.

##  3 Results

The sublimation outcome of icy pebbles depends on the grain size ranges of the
dust, the ice content (i.e. PA or PB), the silicate type, and the dust-ice
distribution inside the pebbles. The experiments aim at distinguishing the
effects of each of these contributions to the final preservation or disruption
of the pebble after sublimation.

The disruption parameter $D$ is averaged between pebbles of the same type (PA
or PB, dust type, and grain size range). The maximum error related to the
average $\bar{D}$ is:

$\Delta\bar{D}=\sqrt{{\sigma_{P}}^{2}+4\,\%\bar{D}}$ (2)

where $\sigma_{P}$ is the standard deviation of $D$ between pebbles of the
same type, and, assuming $2\,\%$ relative error on the experimental evaluation
of the pebble area on the image, the term $4\,\%\bar{D}$ is the absolute error
of the average disruption parameter measured for pebbles of the same type.
When the measurement of $D$ with its maximum error is consistent with $D=1$,
then the pebbles are not disrupting.

###  3.1 Pebbles of olivine and pyroxene with different grain size ranges

We created the two types of pebble (PA or PB) with different grain sizes for
the two dust types (olivine and pyroxene). The grain size ranges are obtained
by grinding the powder in a mortar and dry sieving the dust with nylon sieves.
Three ranges are used in the present work: grains smaller than $50$ µm
(referred to as ‘fine component’), $50-100$ µm, and $100-300$ µm (referred to
as ‘coarse component’). Olivine pebbles made with dust grains smaller than
$50$ µm are not affected by sublimation (Fig. 9). At $50-100$ µm, we already
see that PAs do not maintain their integrity, while PBs are still preserved.
For coarser dust, $100-300$ µm, both PAs and PBs are not able to counter ice
sublimation and gravity and they disrupt. Pyroxene pebbles (Fig. 10), on the
other hand, show complete disruption for PAs, for all the size ranges. PBs
however are following the same pattern as olivine pebbles: the pebbles survive
up to $100$ µm, and the coarse component is disrupting under sublimation
pressure and gravity.

Big error bars are usually associated to the presence, in the same sample, of
both disrupted and preserved pebble (e.g. PA with olivine). This is leading to
a bigger standard deviation of $D$ within the sample.

Figure 9: Disruption parameter depending on grain size range of olivine, for
both PA and PB. When the measurement of $D$ is consistent with $1$, then the
pebbles survived the sublimation process. Figure 10: Disruption parameter
depending on grain size range of pyroxene, for both PA and PB. When the
measurement of $D$ is consistent with $1$, then the pebbles survived the
sublimation process.

###  3.2 Pebbles of olivine and pyroxene with two-component dust size ranges

Disruption of pebbles with $100-300$ µm dust grains of olivine and pyroxene is
observed for both PA and PB. Mixing a certain amount of the fine component
($<50$ µm) with the coarse component ($100-300$ µm) can change this outcome
drastically (Fig. 11 \- 12):

  1. 1.

PBs made with olivine survive if $20$ % of the dust mass is the fine
component;

  2. 2.

PAs made with pyroxene are always disrupting, even if they are made with $100$
% of fine dust (see also Fig. 10);

  3. 3.

PBs made with pyroxene do not disrupt through sublimation when at least $5-10$
% mass of dust is $<50$ µm;

  4. 4.

PAs made with olivine disrupt unless the fine dust component is up to $90-100$
% of the total dust mass.

This is an indication of the capability of fine dust to cement bigger
particles together after sublimation. If the icy pebbles contain a higher
amount of ice (PAs), then the amount of fine dust needed to cement bigger
grains increases or in extreme case sublimation disrupts the pebbles even if
they are made of small grains only.

Figure 11: Disruption parameter depending on the amount of fine component
($<50$ µm) mixed with coarse component ($100-300$ µm) of olivine for both PA
and PB. When the measurement of $D$ is consistent with $1$, the pebbles
survived the sublimation process. Figure 12: Disruption parameter depending on
the amount of fine component ($<50$ µm) mixed with coarse component ($100-300$
µm) of pyroxene for both PA and PB. When the measurement of $D$ is consistent
with $1$, the pebbles survived the sublimation process.

##  4 Discussion

In this work, we present sublimation experiments with two different icy pebble
models to investigate how the initial amount of ice in the pebble, the dust
type, and the dust size ranges are affecting the disruption of the icy pebble
through the sublimation process. In the case of an intimate mixture of ice and
dust, icy aggregates are thought to be unstable upon sublimation of the ice
and undergo disruption releasing small silicate particles (Saito & Sirono,
2011; Schoonenberg & Ormel, 2017; Hyodo et al., 2021). Our experiments show
that it is possible to maintain the integrity of pebbles after the complete
sublimation of the ice intermixed between the dust grains.

###  4.1 Sublimation rate

Inside the vacuum chamber SCITEAS-$2$, the temperature below the pebbles rises
from $\sim 120$ K to $280$ K in about $20$ hours. The sublimation of the ice
occurs during this time, although the experimental setup does not allow
detecting exactly when it ends.

The mean size of PAs is about $1.5$ mm in radius, with an ice volume ratio of
$77$ %. The equivalent ice would be contained into an equivalent sphere with a
radius of around $1.4$ mm. The linear decrease rate of ice in the direction of
the surface normal $R_{\mathrm{ice}}$, can be calculated with the Hertz-
Knudsen formula (Knudsen, 1909),

$R_{\mathrm{ice}}=\frac{V_{\mathrm{m}}(p-p_{\mathrm{sat}})}{\sqrt{2\pi\mu
R_{\mathrm{g}}T}}$ (3)

where $V_{\mathrm{m}}$ is the molar volume, $p$ is the vapor pressure above
the surface, $p_{\mathrm{sat}}$ the saturation pressure, $\mu$ is the molar
mass of ice, $R_{\mathrm{g}}$ is the gas universal constant, and $T$ is the
temperature of the ice. The saturation pressure $p_{\mathrm{sat}}$ is related
exponentially to the temperature $T$ (Haynes et al., 1992)

$p_{\mathrm{sat}}=6.034\cdot
10^{12}\mathrm{g\,cm^{-1}s^{-2}}\cdot\mathrm{e}^{-\frac{5938\mathrm{\,K}}{T}}.$
(4)

The total pressure inside the vacuum chamber varies over time (from $10^{-7}$
mbar to $10^{-3}$ mbar), depending on the sublimation processes of the
condensed gasses trapped inside. Since at the beginning of the experiment some
vapor condenses inside the chamber on the cold copper plate, when the
temperature increases during the experiment, the pressure increases as well,
until all the condensed ice is sublimated. Assuming that the pressure inside
the chamber is due to the sublimation of such ice, the vapor partial pressure
$p$ is exactly the pressure measured during the experiment inside the vacuum
chamber. Once the partial vapor pressure is known, the rate of linear decrease
of ice can be calculated as a function of the temperature only. Assuming that
the ice of the pebbles is at the same temperature of the sample holder on
which they lay and since the temperature of the sample holder is known, it is
possible to calculate at what temperature the sublimation happens and how much
time it requires. For our experiments, the sublimation time of a compact ice
sphere with a radius of around 1.4 mm is between 4 and 6 hours and it happens
on average when the temperature spans from $\sim 190$ to $240$ K. This gives a
sublimation rate of about $2\cdot 10^{-1}$ mm/hour.

At the water ice line temperature, the sublimation rate is considerably slower
than in the laboratory. If we assume a protoplanetary disc with gas surface
density $\Sigma(r)$ and a temperature profile $T(r)$ described by

$\Sigma(r)=1000\,\mathrm{g\,cm^{-2}}\left(\frac{r}{\mathrm{au}}\right)^{-1}$
(5) $T(r)=280\mathrm{\,K}\left(\frac{r}{\mathrm{au}}\right)^{-\frac{1}{2}}$
(6)

where $r$ is the radial distance from the central star, and the amount of
water (in both ice and vapor phases) with respect to the hydrogen and helium
content of the disc is equal to $Z_{\mathrm{H_{2}O}}=0.005$, it is possible to
calculate the partial vapor pressure $P_{v}$ from:

$P_{v}=\frac{k_{\mathrm{b}}Z_{\mathrm{H_{2}O}}T(r)\Sigma(r)}{\mu_{\mathrm{H_{2}O}}\sqrt{2\pi}H_{\mathrm{gas}}(r)}$
(7)

where $k_{\mathrm{b}}$ is the Boltzmann constant, $\mu_{\mathrm{H_{2}O}}$ is
the weight of a molecule of water and $H_{\mathrm{gas}}(r)$ is the disc scale
height given by

$H_{\mathrm{gas}}(r)\approx
0.033\,\mathrm{au}\left(\frac{r}{\mathrm{au}}\right)^{\frac{5}{4}}$ (8)

Given these parameters, the ice line is around $2.7$ au, and the sublimation
rate at $2.5$ au (just inside the ice line), is about $10^{-4}$mm/hour.

Considering an amount of water equal to $Z_{\mathrm{H_{2}O}}=0.005$ in the
disk, the water vapor pressure at the ice line is within the range of the
vacuum chamber pressure (around $10^{-6}$ mbar), but the temperature of the
sample holder increases quickly above the ice line temperature ($\sim 170$ K)
up to $280$ K in 20 hours. This means that pebbles undergoing sublimation in
the laboratory are at least three orders of magnitude faster in sublimating
their ice than pebbles just inside the ice line, highlighting the fact that
survivability of icy pebbles is possible at a much higher sublimation rate
than the one found at the ice line. It is probable that some of the pebbles
that are disrupting in our experiment could survive a sublimation process in
conditions of slower sublimation rates and in the absence of gravity.
Nonetheless, temperature fluctuations in protoplanetary discs could lead to
faster sublimation rates than the one at the ice line (see e.g. McNally et al.
(2014) and Bodénan et al. (2020)).

Previously, theoretical works have assumed that pebbles disrupt after
sublimation freeing micrometer-size dust particles inside the ice line,
leading to enhancement of planet formation just inside and outside the ice
line. What would happen if a pebble similar to a PA (in size and porosity)
passes the ice line without disrupting? If pebbles are not disrupting through
sublimation, then planet formation due to pebble accretion could be enhanced
inside the ice line. This is a direct effect of the change of the Stokes
number of pebbles or dimensionless friction time of the pebble in the Epstein
regime,

$St=\frac{a\rho_{\mathrm{p}}}{H_{\mathrm{gas}}\rho_{\mathrm{gas}}}$ (9)

with $a$ particle radius, and $\rho_{\mathrm{p}}$ and $\rho_{\mathrm{gas}}$
being the pebble and gas density. When the ice sublimates, the pebble changes
its apparent density, while maintaining the same size. The Stokes number of a
sublimated olivine pebble with $1.5$ mm of radius and 77 % porosity (see
section 4.2) is about $5\cdot 10^{2}$ times the Stokes number of a µm-size
olivine grain. Since the effective radius for accretion onto planetesimal
depends on the Stokes number of the accreting pebbles according to
$r_{\mathrm{eff}}\propto St^{\frac{1}{3}}$, their accretion efficiency would
be increased inside the ice line (Lambrechts & Johansen, 2012; Morbidelli et
al., 2015). On the other hand, a higher Stokes number means a higher drift
velocity toward the star, decreasing the efficiency of planet formation from
pile-up of material close to the ice line. It should be noted, however, that
higher velocities of pebbles could result in disruptive pebble-pebble
collisions, so pile-up of fine dust material could still be possible.

###  4.2 Role of porosity

Many studies show that porosity is important for pebble evolution and for hit-
and-stick growth (Dominik & Tielens, 1997; Ormel et al., 2007; Garcia &
Gonzalez, 2020). For our experiments, we use two pebble models that have
negligible porosity at the beginning of the sublimation experiment. At the end
of the experiment however, the porosity of the preserved pebbles increases
since all the ice sublimated, but they maintain their shape with insignificant
changes. We can therefore calculate the final porosity of pebbles
$\Phi_{\mathrm{f}}$:

$\Phi_{\mathrm{f}}=1-\frac{V_{\mathrm{dust}}}{V_{\mathrm{pebble}}}=\frac{i}{i+(1-i)\frac{\rho_{\mathrm{ice}}}{\rho_{\mathrm{dust}}}}$
(10)

Where $V_{\mathrm{dust}}$ is the volume occupied by the dust,
$V_{\mathrm{pebble}}$ is the total volume of the pebble, $\rho_{\mathrm{ice}}$
and $\rho_{\mathrm{dust}}$ are the ice and dust densities respectively, and
$i$ is the ice mass percentage inside the initial pebble. The final porosity
of PAs is $\left(77\pm 7\right)$ % and the final porosity of PBs is
$\left(37\pm 12\right)$ %. PBs porosity is close to the expected porosity of
pebbles in the nucleus of 67P (Fulle et al., 2016), and their initial dust-to-
ice volume ratio is $1.7\pm 0.4$ which is consistent with the limits found by
Kofman et al. (2015).

After sublimation, preserved pebbles are porous, highlighting that such kind
of dry porous aggregates would survive sublimation with an amount of ice
filling the space between dust grains up to the initial mass ratio (50 and 15
% mass of ice for PAs and PBs, respectively). In the limit case, the ice is
filling all the space inside the dry porous pebble, but sublimation is not
strong enough to push apart the dust particles.

###  4.3 Role of gravity

In the laboratory, sublimation and gravity are both concurring to the
disruption of pebbles. Ice sublimation pushes the grains apart and tends to
disrupt the pebble, while gravity disrupts any grain-to-grain adhesion that is
weaker than the gravity force of the dust column above it inside the pebble.

Hereafter, we refer to gravity as a "disrupting" force. There are many
configurations of granular materials where the dust is kept in a certain
position because of gravity which acts as a "constructive force". For example,
a dust pile maintains its shape because of the static friction between grains,
which counters actively the tangent force of gravity and prevents the grains
to slide further. On one hand, if we consider a spheroidal aggregate of dust,
the top hemisphere of the pebble is similar to a pile of dust, where friction
and adhesion between particles are preventing them to slide and roll down. On
the other hand, the bottom hemisphere can be seen as an upside-down pile of
dust. Gravity and friction forces would tend to disrupt such configuration of
dust in a pile of dust, and the weight of the top-half hemisphere of dust
contributes to the disruption of the bottom-half hemisphere. This is not
observed for preserved pebbles (see Fig. 7 a-b-c). The bottom-half of the
pebble is well preserved and overhangs and protrusions are conserved in the
final shape. Therefore, gravity has a disruptive role in transforming pebbles
in piles of dust and preserved pebbles survived both gravity and sublimation
forces during the experiment.

While it is clear that preserved pebbles survived both forces, disruption of
pebbles in our experiments could be caused by both sublimation and gravity and
there is a regime for which sublimation is more important than gravity. If we
assume a spherical dust grain resting on an icy surface, we can compare the
gravity force to the drag force of the sublimated ice acting on the small
grain. The drag force $F_{\mathrm{d}}$ in Epstein regime is given by

$F_{\mathrm{d}}=\frac{\pi}{3}\rho_{\mathrm{vap}}{R_{\mathrm{p}}}^{2}\sigma_{\mathrm{th}}\nu$
(11)

with $\rho_{\mathrm{vap}}$ vapor density just above the surface of the pebble,
$R_{\mathrm{p}}$ dust particle radius, $\sigma_{\mathrm{th}}$ thermal
velocity, and $\nu$ flux velocity of the sublimating gas. The flux velocity
can be retrieved applying the conservation of mass at the surface of the
pebble passing from ice to vapor

$\mu\rho_{\mathrm{ice}}R_{\mathrm{ice}}=\rho_{\mathrm{vap}}\nu$ (12)

where $\rho_{\mathrm{ice}}$ is the ice density (in mol $\mathrm{m}^{-3}$),
$R_{\mathrm{ice}}$ is the sublimation velocity (see equation 3) and
$\rho_{\mathrm{vap}}$ is given by the ideal gas law

$\rho_{\mathrm{vap}}=\frac{p_{\mathrm{sat}}}{RT_{\mathrm{ice}}}$ (13)

with $p_{\mathrm{sat}}$ saturation pressure (see equation 4), and
$T_{\mathrm{ice}}$ temperature of the ice. The thermal velocity
$\sigma_{\mathrm{th}}$ is

$\sigma_{\mathrm{th}}=\sqrt{\frac{8RT_{\mathrm{ice}}}{\pi\mu_{\mathrm{H_{2}O}}}}$
(14)

If we assume that the temperature of the ice is the same of the sample holder,
it is possible to calculate the drag force during the experiment and compare
it to the gravity force acting on different grain sizes. Fig. 13 shows how the
ratio between drag force and gravity force is varying depending on the
temperature of the ice, for different grain diameters. In the temperature
range where sublimation is important for the pebbles, gravity is the
disrupting force in the case of coarse grains, while the drag force due to
sublimation dominates the gravity force for small grains (up to $100$ µm)
above $\sim 225$ K. This means that all the preserved pebbles with grains
smaller than $100$ µm overcame sublimation-driven disruption. Gravity force
can be still important for preserved pebbles if inside the pebble there is a
heterogeneous dust-ice distribution. In this case, if the pebble survives
sublimation, it can still collapse under gravity because of a structural
deficiency, like a hole left by the ice inside the core of the pebble. This is
why the error on the disruption parameter for PAs is large, although the
general trend for different dust size ranges is not affected because of the
presence of pebbles that are homogeneous and do not have any structural
weaknesses when the ice is gone.

A detailed description of the sublimation-driven disruption process should
take into consideration what happens inside the pebble, and the possible
formation of over-pressure in pores of the pebbles, but this kind of model is
beyond the purpose of the present work, and will the be topic of future
investigations.

Figure 13: Ratio between sublimation drag force $F_{\mathrm{d}}$ and gravity
force $F_{\mathrm{g}}$ acting on particles of different diameters vs.
temperature of the ice. The sublimation of the pebble happens between $190$
and $240$ K, and in this range of temperatures, gravity is the main disruption
force for coarse grains, while sublimation is important in the higher range of
temperatures for grains smaller than $100$ µm. The black continuous line is a
reference for the case in which sublimation drag force and gravity are equal.

###  4.4 Role of grain size ranges and mineralogy

For our experiments, we used two types of silicates: olivine and pyroxene. The
dust grain size ranges considered in this work span from $0$ to $300$ µm,
divided in three categories: ‘fine component’ (less than $50$ µm), $50-100$
µm, and ‘coarse component’ ($100-300$ µm). In protoplanetary discs before
growth to pebbles size, dust is mostly present as sub- µm or µm-size covering
a narrower size range than our ‘fine component’ (Bouwman et al., 2001; Boekel
et al., 2003). Nonetheless, our experiments show that pebbles can overcome
sublimation with a dust size up to $50$ µm in the case of olivine PAs and
$100$ µm in the case of PBs.

A pebble made with the coarse component always disrupts, but when mixed with a
certain amount of fine component it can be preserved through sublimation. The
minimum amount of dust fine component needed for preservation varies with the
mineral species, being higher for PB olivine (about $20$ % of the total dust
mass) and lower for PB pyroxene ($10$ % of the total dust mass). The pyroxene
PAs are always disrupting with any percentage of fine dust, while olivine PAs
increase their stability through sublimation when the fine component is
between $90$ and $100$ % of the total dust mass. Some olivine PAs do not
disrupt, increasing the variability of the disruption parameter within the
sample, and therefore the total error associated with the measurement.
Therefore, the measurement of olivine PAs shows an increasing disruption
parameter with increasing amount of fine component, but the exact minimum
amount of fine component needed to avoid disruption is difficult to retrieve.
In general, the introduction of fine dust mixed with coarse dust increases the
pebble stability through sublimation. The small particles could act as bridges
between bigger particles, increasing their adhesion capability in a way
similar to the one described by Seiphoori et al. (2020). Future laboratory
work should address this point, by observing the microscopic morphology of the
icy pebbles and their sublimated remnants in loco.

Olivine and pyroxene pebbles have slightly different behaviors. PAs with dust
size smaller than $50$ µm are preserved for olivine and disrupted for pyroxene
(Fig. 9 \- 10). Furthermore, when fine and coarse dusts are mixed, olivine PBs
seem to necessitate a higher percentage of fine component than pyroxene PBs to
be preserved (Fig. 11 \- 12). Explaining these differences is challenging,
since several factors are governing the adhesion strength of particles. The
ability of grains to aggregate and to form bonds between each other is given
by properties of the material (such as surface energy, Poisson ratio and
Young’s modulus), grain shape characteristics (surface roughness, crystal
faces) and physical and chemical state of grain surfaces (adsorption layers,
chemical reactions, electric charges). Our SEM measurement do not show
significant alterations of the grain surfaces before and after the experiment,
and the spectra of the grains after the experiment exclude the presence of
large quantity of hydroxylated material (Fig. 2), which could also weaken
grains bonding (Quadery et al., 2017). Although the SEM images do not show
clear difference in the amount of very fine material (<$5$ µm), a difference
in particles size distribution could explain the different behaviour of
olivine and pyroxene, leading to a more adhesive behaviour of pyroxene over
olivine (Fig. 11 \- 12). It should be noted, however, that the adhesion
capability of the two species are determined also by other parameters cited
above (e.g. grain roughness).

Future experiments will address the different behavior of pyroxene and
olivine, and the role of ice interaction with grains of different sizes.

##  5 Conclusions

Different studies show that it is possible to enhance the growth of dust
inside and outside the ice line, if icy pebbles are assumed to completely
disrupt by sublimation inside the ice line (Saito & Sirono, 2011; Schoonenberg
& Ormel, 2017; Hyodo et al., 2021). We researched experimentally the outcome
of two icy pebble models undergoing sublimation in low-temperature and low-
pressure conditions varying the dust mineralogy, the water ice content, and
the dust size range.

Our findings can be summarized as follows:

  1. 1.

Icy pebbles can survive sublimation. We demonstrate that a range of
combinations of dust type (olivine or pyroxene), dust size ranges and ice
content lead to preserved pebbles upon sublimation of the ice.

  2. 2.

Pebbles with low ice content survive sublimation better than the ones with
higher ice content. Our results show that pebbles with 15 % ice mass are more
resistant to disruption than pebbles with 50 % ice mass for different dust
size ranges.

  3. 3.

Icy pebbles survive better sublimation if the dust particles are smaller than
$50$ µm. If the pebble is made with coarse particles ($100-300$ µm), a minimum
amount of fine dust allows it to avoid disruption. The minimum amount of fine
dust increases with the amount of ice and is dependent on the silicate type.

These results are relevant for planet formation processes close to the water
ice line and downstream of it, providing useful information for modeling the
behavior of ice-dust aggregates in protoplanetary discs when sublimation is
occurring. In particular, they provide elements for modelling the sublimation
outcome of icy pebbles depending on their ice content, dust type, and dust
size ranges, which play a major role in the disruption process. Verifying
these results for other dust types, narrower dust size distributions, and in
presence of organics is an important future extension of this work.

## Acknowledgements

This work has been carried out within the framework of the NCCR PlanetS
supported by the Swiss National Science Foundation. We thank the two anonymous
reviewers for the useful insights which helped improving the manuscript.

## Data Availability

All the camera images of the sublimation process are available in figshare
from
https://figshare.com/articles/figure/The_fate_of_icy_pebbles_undergoing_sublimation_in_protoplanetary_disks/16912399.

## References

  * Adda-Bedia et al. (2016) Adda-Bedia M., Kumar S., Lechenault F., Moulinet S., Schillaci M., Vella D., 2016, Langmuir, 32, 4179
  * Aumatell & Wurm (2011) Aumatell G., Wurm G., 2011, MNRAS, 418, L1
  * Barrière-Fouchet et al. (2005) Barrière-Fouchet L., Gonzalez J.-F., Murray J. R., Humble R. J., Maddison S. T., 2005, A&A, 443, 185
  * Birnstiel et al. (2012) Birnstiel T., Klahr H., Ercolano B., 2012, A&A, 539, A148
  * Blum & Wurm (2008) Blum J., Wurm G., 2008, ARA&A, 46, 21
  * Blum et al. (2017) Blum J., et al., 2017, MNRAS, 469, S755
  * Bodénan et al. (2020) Bodénan J.-D., Surville C., Szulágyi J., Mayer L., Schönbächler M., 2020, ApJ, 901, 60
  * Boekel et al. (2003) Boekel R. v., Waters L. B. F. M., Dominik C., Bouwman J., Koter A. d., Dullemond C. P., Paresce F., 2003, A&A, 400, L21
  * Bouwman et al. (2001) Bouwman J., Meeus G., Koter A. d., Hony S., Dominik C., Waters L. B. F. M., 2001, A&A, 375, 950
  * Dominik & Tielens (1997) Dominik C., Tielens A. G. G. M., 1997, ApJ, 480, 647
  * Drążkowska & Alibert (2017) Drążkowska J., Alibert Y., 2017, A&A, 608, A92
  * Fulle et al. (2016) Fulle M., et al., 2016, MNRAS, 462, S132
  * Garcia & Gonzalez (2020) Garcia A. J. L., Gonzalez J.-F., 2020, MNRAS, 493, 1788
  * Gonzalez et al. (2015) Gonzalez J. F., Laibe G., Maddison S. T., Pinte C., Ménard F., 2015, Planet. Space Sci., 116, 48
  * Gundlach & Blum (2014) Gundlach B., Blum J., 2014, ApJ, 798, 34
  * Haynes et al. (1992) Haynes D. R., Tro N. J., George S. M., 1992, J. phys. chem. (1952), 96, 8502
  * Hyodo et al. (2021) Hyodo R., Guillot T., Ida S., Okuzumi S., Youdin A. N., 2021, A&A, 646, A14
  * Ida et al. (2016) Ida S., Guillot T., Morbidelli A., 2016, A&A, 591, A72
  * Iveson et al. (2001) Iveson S. M., Litster J. D., Hapgood K., Ennis B. J., 2001, Powder Technol., 117, 3
  * Johansen & Youdin (2007) Johansen A., Youdin A., 2007, ApJ, 662, 627
  * Jutzi et al. (2009) Jutzi M., Michel P., Hiraoka K., Nakamura A. M., Benz W., 2009, Icarus, 201, 802
  * Knudsen (1909) Knudsen M., 1909, Annalen der Physik, 333, 75
  * Kofman et al. (2015) Kofman W., et al., 2015, Science, 349, aab0639
  * Lambrechts & Johansen (2012) Lambrechts M., Johansen A., 2012, A&A, 544, A32
  * Marmur (2004) Marmur A., 2004, Langmuir, 20, 3517
  * McNally et al. (2014) McNally C. P., Hubbard A., Yang C.-C., Mac Low M.-M., 2014, ApJ, 791, 62
  * Morbidelli et al. (2015) Morbidelli A., Lambrechts M., Jacobson S., Bitsch B., 2015, Icarus, 258, 418
  * Musiolik & Wurm (2019) Musiolik G., Wurm G., 2019, ApJ, 873, 58
  * Natta et al. (2006) Natta A., Testi L., Calvet N., Henning T., Waters R., Wilner D., 2006, arXiv:astro-ph/0602041
  * Noe Dobrea et al. (2003) Noe Dobrea E., Bell J., Wolff M., Gordon K., 2003, Icarus, 166, 1
  * Ormel et al. (2007) Ormel C. W., Spaans M., Tielens A. G. G. M., 2007, A&A, 461, 215
  * Poch et al. (2016) Poch O., Pommerol A., Jost B., Carrasco N., Szopa C., Thomas N., 2016, Icarus, 266, 288
  * Pommerol & Schmitt (2008) Pommerol A., Schmitt B., 2008, Journal of Geophysical Research: Planets, 113
  * Pommerol et al. (2015) Pommerol A., Jost B., Poch O., El-Maarry M. R., Vuitel B., Thomas N., 2015, Planet. Space Sci., 109-110, 106
  * Pommerol et al. (2019) Pommerol A., et al., 2019, Space Sci. Rev., 215, 37
  * Quadery et al. (2017) Quadery A. H., Doan B. D., Tucker W. C., Dove A. R., Schelling P. K., 2017, ApJ, 844, 105
  * Ros et al. (2019) Ros K., Johansen A., Riipinen I., Schlesinger D., 2019, A&A, 629, A65
  * Saito & Sirono (2011) Saito E., Sirono S.-i., 2011, ApJ, 728, 20
  * Schneider et al. (2012) Schneider C. A., Rasband W. S., Eliceiri K. W., 2012, Nature Methods, 9, 671
  * Schoonenberg & Ormel (2017) Schoonenberg D., Ormel C. W., 2017, A&A, 602, A21
  * Seiphoori et al. (2020) Seiphoori A., Ma X.-g., Arratia P. E., Jerolmack D. J., 2020, PNAS, 117, 3375
  * Simon et al. (2008) Simon S. B., et al., 2008, Meteoritics & Planet. Sci., 43, 1861
  * Thimbleby (1989) Thimbleby H., 1989, Phys. Educ., 24, 300
  * Watson et al. (2009) Watson D. M., et al., 2009, ApJS, 180, 84
  * Weidenschilling (1997) Weidenschilling S. J., 1997, Icarus, 127, 290
  * Weidling et al. (2009) Weidling R., Güttler C., Blum J., Brauer F., 2009, ApJ, 696, 2036
  * Youdin & Goodman (2005) Youdin A. N., Goodman J., 2005, ApJ, 620, 459