\section{Introduction}
The upper limit for the window of a planet's habitability is set by the lifetime of its host star. If the increasing luminosity during the red-giant phase isn't enough to compromise the biosphere of the planet, the death of the host star will surely spell the end for any living beings in that planetary system. A sufficiently intelligent civilization has several pathways for survival. Shorter term solutions which cope with the changing luminosity of their host star include terraforming their planet, migrating to another planet in their system, or something like planetary migration, e.g., asteroid deflection which increases the orbital radius and keeps the flux on the planet constant \citep{Korycansky_2001}. A longer term solution could involve star-lifting (SL), the process of removing mass from the host star in order to slow down the rate of nuclear fusion and prevent a dramatic increase in luminosity. This idea was first popularized in 1985 by David Criswell \citep{Criswell_1985} and a simple SL analysis was explored for a Sun-like star in \citet{Matloff_2017}. Matloff assumes the mechanism for star-lifting will involve a laser which increases the star's mass ejection via stellar winds and explores several scenarios, with the most conservative approach resulting in a ${\sim} 3\%$ mass loss over a 600 myr period.

In this paper, we set aside the engineering complications of this method and derive the required mass {\it vs.} age and $\dot{M}$ {\it vs.} age relations which maintain a constant bolometric irradiance on a habitable planet. If this removed mass stays interior to the orbit of the habitable planet, the orbital radius of the planet should remain approximately unchanged so keeping the star's luminosity constant ({\it isoluminosity}) is equivalent to keeping the flux on the planet constant. If the mass is ejected out of the system, the planet will drift and mass should be removed at a slower rate ({\it isoirradiance}) in order to keep the flux on the planet constant.

This paper is organised as follows. In \S~\ref{sec:methods} we describe the difference between {\it isoluminosity} and {\it isoirradiance}, introduce the MIST dataset, and share our numerical techniques. In \S~\ref{sec:results} we present our results for the mass {\it vs.} age and $\dot{M}$ {\it vs.} age relations, give a worked example for a Sun-like star, and compare extended lifetimes to the natural lifetimes for each technique. In \S~\ref{sec:discussion} we discuss these results, compare SL to other habitability-extending pathways, and consider detection methods.



\section{Methods}
\label{sec:methods}



\subsection{Accounting for changes in orbital radius during SL}
SL could be implemented in several ways. We will divide these into two classes. If SL is implemented in a way that relocates mass from a star onto a compact object in a close orbit with that star, keeping the mass interior to the planet and spherically symmetric, the orbit of the planet will not change via Newton's Shell Theorem. In this case, keeping flux on the planet constant is equivalent to keeping the luminosity constant, and we call this implementation {\it isoluminosity}.

Alternatively, SL could be implemented in a way which dramatically increases the rate the star loses mass through stellar winds \citep{Matloff_2017}.  A mechanism like this would result in roughly continuous mass loss which is gradual enough to keep the habitable planet in a stable circular orbit, though this would cause the orbital radius of the planet to increase as the mass leaves the volume enclosed by the planet's orbital radius. Assuming this is done so that there is net torque on the habitable planet, the planet's orbital angular momentum will be conserved. 

For planet mass $m_{\rm p}$, orbital radius $r$, stellar mass $M_*$ and gravitational constant $G$, angular momentum of the planet follows $J_{\rm p} = m_{\rm p}rv_{\rm p} = m_{\rm p} \sqrt{GM_*r}$ meaning the quantity $M_*r$ is kept constant. With $M_* \propto \frac{1}{r}$, the incident flux on the planet $F_{\rm inc}$ follows
\begin{align}
    F_{\rm inc} \propto \frac{L_*}{r^2} \propto L_*M_*^2
\end{align} 
where $L_*$ is the luminosity of the host star. So, for a given age, we solve for the mass such that $\mathcal{F} \equiv LM_*^2$ is held constant. Though both implementations hold bolometric irradiance constant for a planet at any orbital radius, we call this implementation where the lifted mass is moved exterior to the orbital radius {\it isoirradiance}. 

Both methods have their merits (see \S~\ref{sec:discussion} for a  comparison of methods for extending habitability) so it will be difficult to predict which mechanism an advanced civilization would prefer. Our work explores both mechanisms for mass loss, where {\it isoluminosity} sets an upper bound for the required mass loss rate $\dot{M}$ and {\it isoirradiance} will set an approximate lower bound.





\subsection{MIST evolutionary tracks}
We calculate mass {\it vs.} age relations using evolutionary tracks provided by MESA Isochrones \& Stellar Tracks (MIST), generated through the Modules for Experiments in Stellar Astrophysics code (MESA) \citep{Dotter_2016, Choi_2016, Paxton_2010, Paxton_2013, Paxton_2015}. Our dataset contains 906 evolutionary tracks for initial mass $M_0$ with $0.1 \leq M_0/{\rm M}_\odot \leq 1.6$ with increments of $dM_0{=}0.01$ and metallicity [Fe/H] ${\epsilon}  \ \{0.5, 0, -1,-2,-3,-4\}$.  We also use two different sets of isochrones provided by Synthetic Clusters Isochrones \& Stellar Tracks (SYCLIST): a set of isochrones ranging from $0.2-1.5{\rm M}_{\odot}$  \citep{Mowlavi_2012} and another set ranging from $0.5-3.5 {\rm M}_{\odot}$  \citep{Amard_2019}. Both sets run from $10^7-10^{11}$ yrs, in steps sizes of $d {\rm log}_{10}(\text{age}) = 0.05$, generating 81 isochrones each with $Z=Z_{\odot}$. The results of these data sets are similar in the overlapping ranges for mass and age, though all data presented in this paper uses MIST as it covers a wider range of interest.

Our analysis focuses on main sequence (MS) stars (corresponding to phase $0$ in the MIST data) and we filter out PMS (phase $-1$) and everything beyond the MS phase (phase $>=2$). Using only the main-sequence phase gives well-defined non-overlapping tracks. In contrast, the red-giant branch of the tracks overlap with the MS tracks of other stars and cause interpolation issues or derived mass {\it vs.} age relations which are non-physical. While trying to avoid this by interpolating in two phases, one over the MS tracks until the engineered star leaves the MS then another over the RGB tracks, we find that the RGB phase quickly leads to a natural mass loss which exceeds the required SL mass loss, and only provides a marginal benefit to stellar lifetime. Given this, we simplify our analysis and derive similar results by focusing on the MS branch.



\begin{figure*}
 \includegraphics[width=\columnwidth]{worked_example_astart=4603000000.0isoirr=True.pdf}
\includegraphics[width=\columnwidth]{worked_example_astart=4603000000.0isoirr=False.pdf}
 \caption{An example of the mass {\it vs.} age relation that a star like our Sun would follow if star-lifting began today, $A_{\rm start}=4.6$ Gyr, for {\it isoirradiance} (left) and {\it isoluminosity} (right). The natural evolutionary tracks are plot behind this, with a nearly constant mass (nearly horizontal evolution) but increasing $\mathcal{F}$ (colored) as the stars age. For a given track, we interpolate both mass and age for a fixed $\mathcal{F}$. Our Sun's main-sequence lifetime is roughly 10 Gyr, though a Lazarus Sun would stay on the MS for $16$ Gyr with {\it isoirradiance} and $20$ Gyr with {\it isoluminosity}. The required $\dot{M}_\odot$ is shown in the bottom panels, in units of $M_{\rm ceres}$yr$^{-1}$. Terminal Sun marks the point where our Lazarus Sun leaves the main-sequence and star-lifting is no longer viable.}
 \label{fig:worked_example}
\end{figure*}





\subsection{Interpolation and Fitting}
Towards the goal of constructing mass {\it vs.} age relations for a fixed incident flux, we consider 2-dimensional and 1-dimensional interpolation. 2D interpolation is done with \texttt{scipy}'s griddata function which allows 2D interpolation on non-uniform data \citep{2020SciPy-NMeth}. We also consider bilinear interpolation, two 1D interpolations along both mass and age. For a fixed $F$ (which is equal to $\mathcal{F}$ for {\it isoirradiance} and $L_*$ for {\it isoluminosity}), and a track $i$ which contains $F$, we interpolate mass $M_i$ and age $t_i$ in the $F$ {\it vs.} $M_i$ and $F$ {\it vs.} $t_i$ relations. After applying both techniques on every track which contains $F$, we fit a 3rd degree polynomial to $\{t_i\}$ {\it vs.} $\{M_i\}$ which results in our mass {\it vs.} age relation. We repeat this across several $F$, where $F$ is calculated for a desired initial mass and age when SL starts, $A_{\rm start}$. We find that the 2D interpolation produces results which are physically unrealistic, (e.g. a star of $0.2 {\rm M}_{\odot}$ must increase mass at some ages in order to fix $F$), but the bilinear interpolation produces smooth and physically realistic results.

Calculating a smooth $\dot{M}$ requires fitting the interpolated mass {\it vs.} age relations, and we find that the 3rd degree polynomial fits the data extremely well, with the exception of stars with an initial mass $>1.4 {\rm M}_\odot$, where interpolation leads to an artificially jagged evolution of mass {\it vs.} age, so we restrict ourselves to initial masses $<1.4 {\rm M}_\odot$ in this study. For stars with mass $M_*$ in the range $1 \leq M_*/{\rm M}_{\odot} \leq 1.4$, the fit is reliable for most of the star's life, though for some stars with mass $M \gtrsim 1.2 {\rm M}_{\odot}$, this results in a physically unrealistic $\dot{M} < 0$ at early times and we set these artifacts to $\dot{M} = 0$. 

We note that by interpolating across the MIST evolutionary tracks, we are only approximating the required mass {\it vs.} age relations and ignoring the issues associated with metallicity and the delineation between the core and the shell. For low mass convection driven stars, an advanced civilization capable of star-lifting might be able to extract the heavier elements from the well-mixed star and keep the metallicity on track with the derived relations. For more massive stars, though removing mass from the outer shell should slow fusion, the core of a star which has been star-lifted to a mass $M$ will have more metals than a star which naturally has the same mass at a similar age. We can imagine an advanced civilization finding a solution to this too, but if not, these calculations are an approximation and a more careful analysis should focus on the effects of this increased core metallicity.



\section{Results}
\label{sec:results}
\subsection{Mass {\it vs.} age relations}

Fig~\ref{fig:worked_example} shows a worked example for the Sun, applying both {\it isoirradiance} and {\it isoluminosity}, with SL beginning today, $A_{\rm start}=4.6$ Gyr. The derived mass {\it vs.} age relation is plot against the MIST evolutionary tracks which are colored according to $\mathcal{F}$ ($L_*$) which is held constant during interpolation for {\it isoirradiance} ({\it isoluminosity}). The MIST evolutionary tracks are nearly horizontal, as the total mass loss during the main-sequence is much smaller than the total mass. Every interpolated point is fit with a 3rd order polynomial, which is then used to calculate $\dot{M}$ in the bottom panels. {\it Isoirradiance} requires a slightly more modest star-lifting rate of $\dot{M} \approx 0.02 \ M_{\rm Ceres} \text{yr}^{-1}$, and extends the Sun's MS lifetime to ${\sim} 16$ Gyr, increasing the total MS lifetime by 6 Gyr. {\it Isoluminosity} initially requires a larger SL rate, resulting in a MS lifetime of ${\sim} 20$ Gyr, increasing the total MS lifetime by ${\sim} 10$ Gyr. We mark the point when our modified Lazarus Sun leaves the main-sequence, which we dub Terminal Sun, and star lifting is no longer viable.



Derived mass {\it vs.} age and $\dot{M}$ {\it vs.} age relations for {\it isoirradiance} and {\it isoluminosity} are shown in Fig~\ref{fig:m_vs_a} with [Fe/H] $\epsilon \ \{0,-2,-4\}$ and $A_{\rm start}=2$ Gyr. We focus on the first 20 Gyr since most stars with initial mass $>0.8 {\rm M}_\odot$ do not live beyond this window. For most stars, we find that the initial $\dot{M}$ required increases with initial mass, and for a fixed track $\dot{M}$ typically monotonically decreases with age, with the exception of high mass high metallicity stars, where the required $\dot{M}$ sometimes increases with age. We choose a small $A_{\rm start}$ in order to capture the behavior of these higher mass stars, who would see very little benefit to a later start time (as most would be nearing the end of their MS life) and low mass stars live long enough that this early $A_{\rm start}$ has a negligible effect on their evolution.

\subsection{Extended lifetimes}
In order to show the benefit of SL for all stars, we compare MS stellar lifetimes in Fig~\ref{fig:lifetime_ratios} for $A_{\rm start}=2$ Gyr, plotting the difference between the star-lifted MS lifetime $\tau_{\rm SL}$ and the expected MS lifetime $\tau_{\rm expected}$ against the initial mass of the star. We also subplot $\tau_{\rm SL}$ and $\tau_{\rm expected}$ directly. 

We note many interesting phenomenon in this figure. Both methods see similar patterns, though they differ in magnitude. For higher mass stars with $M_* \gtrsim 0.8 {\rm M}_\odot$, we see a modest lifetime increase of $0.1 {-} 10$ Gyr. As the initial mass grows, the increased lifespan decrease due to the fact that $A_{\rm start}$ is approach the end of their main-sequence lifetimes.


Compared to higher initial mass stars, both methods see a jump in $\tau_{\rm SL}-\tau_{\rm expected}$ near $0.6 \ {\rm M}_\odot$ (with the mass where this occurs varying by metallicity), which is likely due to stars where heat transfer is radiation-driven being star-lifted into a convection-driven mass range. This gives these stars access to fuel which they wouldn't see otherwise, greatly increasing their lifetime. In the case of {\it isoirradiance}, this transition happens around $0.65{\rm M}_\odot$ for [Fe/H]${=}-4$ and $0.5{\rm M}_{\odot}$ for [Fe/H]${=}0.5$, where their slightly more massive counterparts see a lifetime increase nearly an order of magnitude smaller. Lower metallicity stars likely see this transition at a higher mass due to the required SL rate roughly increasing with decreasing metallicity, making it easier for these stars to achieve the convection-driven state.

We see the greatest benefit of SL for the least massive stars, who can extend their lifetimes up to ${\sim}1$ Tyr. Though the increased lifespan typically grows with decreasing initial mass,  we see extremely low mass stars with $M_*<0.2 {\rm M}_{\odot}$ benefit less than than stars with mass $M_* {\sim} 0.3 {\rm M}_{\odot}$. This is due to all of these stars being engineered towards the lower mass limit of the MIST evolutionary tracks where their mass approaches $0.1 {\rm M}_{\odot}$, meaning they all achieve a similar $\tau_{\rm SL}$. As the initial mass approaches $0.1 {\rm M}_{\odot}$, $\tau_{\rm SL} - \tau_{\rm expected}$ approaches 0. See \S~\ref{sec:discussion} for a discussion on these stars and the implications for an advanced civilization who want to maximize the lifetime of their star, independent of the flux incident on their planet.


Fig.~\ref{fig:terminal_age} shows the lifetimes of stars with [Fe/H]${=}0$ as a function of initial mass for both methods, compared to the lifetime without SL. We see convergence at the higher end of the mass range as $A_{\rm start}$ apporaches the main-sequence lifetime of these stars. We also see convergence at the lower end of the mass range, where SL causes these stars to approach the hydrogen-burning limit, in this case the minimum mass for the MIST tracks ($0.1 {\rm M}_{\odot}$) and the extended lifetime approaches the lifetime of this $0.1 {\rm M}_{\odot}$ star. We also show the movement of a few of these stars in mass-luminosity space in Fig.~\ref{fig:funny_HR}.

\begin{figure*}
 \includegraphics[width=2\columnwidth]{m_vs_a_a_start=2000000000.0_iso=True.pdf}
 \includegraphics[width=2\columnwidth]{m_vs_a_a_start=2000000000.0_iso=False.pdf}

 \caption{Mass {\it vs.} age and $\dot{M}$ {\it vs.} age for {\it isoirradiance} (top) and {\it isoluminosity} (bottom) with [Fe/H] $\epsilon \{-4,-2,0\}$. Colors indicate ${\rm log}_{10}(\mathcal{F}/\mathcal{F}_\odot)$ ({\it isoirradiance}) and ${\rm log}_{10}(L/{\rm L}_\odot)$ ({\it isoluminosity}) when star-lifting begins, $A_{\rm start} = 2$ Gyr, with these values held constant across the engineered evolution. Mass {\it vs.} age lines are calculated by interpolating over the MIST evolutionary tracks for a fixed $\mathcal{F}$ or $L_*$. $\dot{M}$ is calculated using the derivative of a 3rd degree polynomial fit to mass  {\it vs.} age and shown in units of $M_{\rm Ceres}$ yr$^{-1}$, where $M_{\rm Ceres} = 9.1\times 10^{20} kg$. Some stars with initial mass $>{\rm M}_{\odot}$ show a chaotic $\dot{M}$ evolution, due to the turbulent mass {\it vs.} age evolution. We manually inspect these and find that the polynomials provide an excellent fit to interpolated mass {\it vs.} age, though some ages require a non-physical $\dot{M} <0$, so we set $\dot{M}=0$ where these artifacts occur.}
 \label{fig:m_vs_a}
\end{figure*}





\begin{figure*}
 \includegraphics[width=1.95\columnwidth]{lifetime_gains/astart_2.0gyr_lmsq.pdf}
 \includegraphics[width=1.95\columnwidth]{lifetime_gains/astart_2.0gyr_l.pdf}
 \caption{Initial mass $M_0$ {\it vs.} $\tau_{\rm SL} -\tau_{\rm expected}$ for the main-sequence lifetime with star-lifting ($\tau_{\rm SF}$) and without star-lifting ($\tau_{\rm expected}$). Color indicates metallicity. Top: {\it Isoirradiance}, holding $\mathcal{F} = L_*M_*^2$ constant. Bottom: {\it Isoluminosity}, holding $L_*$ constant.}
 \label{fig:lifetime_ratios}
\end{figure*}




\section{Discussion}
\label{sec:discussion}










\subsection{Star-lifting {\it vs.} Alternative Methods}

Before considering detection of SL, it is important to compare the benefit and energy requirements of SL to other methods, with the intention of gauging the likelihood that SL is chosen over these other methods. While it is difficult to predict what an advanced civilization would be capable of, and what an implementation of alternative life-extension strategies might look like, we can make rough approximations for their energy requirements.

For {\it isoluminosity}, assuming the material is kept interior to the orbit of the planet but moved to a radius $a$ where $a \gg R_*$, the energy required each year follows
\begin{align}
    \dot{E} &\approx U_f - U_i = GM_* \frac{dM_*}{dt} [-\frac{1}{a}+ \frac{1}{R_*}]\approx GM_* \frac{dM_*}{dt} \frac{1}{R_*}
\end{align} For our Sun, with a required SL rate of $\frac{dM_*}{dt}  {\sim} 2{\times}10^{19}$kg $\text{yr}^{-1}$, this would require $\dot{E} \sim 4\times 10^{37}$ erg $\text{yr}^{-1}$, or ${\sim} 3.3\times10^{-4} {\rm L}_\odot$. For an implementation which uses {\it isoirradiance}, \citealt{Matloff_2017} approximates the energy required to lift mass from a star's photosphere to be $1.9 \times 10^{20}$ erg $\text{kg}^{-1}$. For our Sun, this results in $\dot{E} \sim 4\times 10^{39}$ erg ${\rm yr}^{-1}$, or ${\sim} 0.03 {\rm L}_\odot$.

Taking place over billions of years, the total energy required would be on the order of $10^{46}$ erg for {\it isoluminosity} and $10^{48}$ erg for {\it isoirradiance}. This is several orders of magnitude larger than the total energy required for migrating Earth via asteroid deflection in order to keep the incident flux constant, ${\sim} 10^{40}$ erg \citep{Korycansky_2001}. Though SL is much more expensive, the energy required for this is readily available from the star and offers a long-term solution by extending the lifetime of the host star, whereas the upper limit for habitability via orbital migration is still set by the natural MS lifetime.  Further, assuming more than one habitable planet in the system through luck or terraforming, SL extends the lifetime of the whole planetary system by keeping the flux on every planet in that system constant (assuming the mass is kept interior for {\it isoluminosity}). For a multi-planetary civilization, SL may be preferred, as a technique like orbital migration requires a transfer of orbital energy from one planet to another, which makes it difficult to preserve multiple habitable planets in the same system. Orbital migration could also lead to a destabilization of asteroids or orbits of other planets \citep{Korycansky_2001}. Spin up or spin down torque considerations also have to be made with the orbital migration technique. Alternatively, \citealt{McInnes_2002} show that migration could be achieved with a large reflective sail which is much less massive than the required asteroid, though SL would be advantageous for similar reasons.


Rather than SL, the civilization might choose to relocate. Relocating to a planetary system in the local Milky Way (e.g $\sim 100$ ly away) with peak speed $0.1c$ would require $> 10^4$ yrs. A journey like this for a human-like civilization with ${\sim} 10^{10}$ inhabitants, assuming relocation of each inhabitant, and a ship mass of $5\times 10^4$ kg per inhabitant (which tracks the ISS mass-capacity ratio, though this is likely very conservative for an interstellar journey) would require a ship of mass ${\sim} 10^{15}$ kg, if not much larger. Ignoring the Solar System escape velocity (as $ v_{\rm esc} \ll 0.1c$), the total energy required for migration of the entire civilization is roughly $KE \approx \frac{1}{2}10^{15} (3\times 10^7)^2 = 4.5 \times 10^{36}$ erg, though this is likely extremely conservative. While this is cheaper than SL, the logistics of achieving near-light-speed travel, finding a fuel source for accelerating this much mass, and survival of the inhabitants during the journey may make this infeasible for a large population. Migrating a small fraction of the population is much more likely, where the majority of the population left behind would benefit from SL.

Terraforming is another option. Among other requirements, generating a magnetic field would be an extremely energy demanding necessity. \citet{Bamford_2022} find the minimum energy stored in a magnetic field to create a habitable Mars-like planet to be $10^{24}$ erg, though this doesn't include the energy required to ramp up the magnetic field. They find that kick-starting Mars' iron core into an active magnetic dynamo would require $10^{33}$ erg. While technically possible, this is ultimately another short term solution where the upper limit for habitability is set by the MS lifetime of the host star.

Another life extension strategy involves dumping metal-rich objects onto a star, increasing metallicity and slowing fusion, assuming the metals could be mixed with the core. This could be achieved by kicking objects near the host star out of their orbit and should be relatively energy efficient, though the total amount of material in the planetary system may be a limiting factor. For our own Solar System, we could dump at most a fraction of a percent of a Solar mass onto the Sun. Though this would increase the host star lifetime by an amount roughly equal to the natural lifetime discrepancy between the original and enriched star, the luminosity of the star would still increase with time and a relocation method like asteroid deflection or a reflective sail would be required.


\subsection{Maximizing stellar lifetimes}
Though there are several methods for extending the habitability of a planetary system, we can combine them to get the most out of the host star. We find that both star-lifting types have stars which approach the hydrogen burning limit, though it is possible to SL at much higher rates in order to approach this limit for more massive stars. Assuming metallicity concerns and the problem of mixing could be overcome, the limit for the most massive star that could be pushed into the hydrogen burning limit is set by the maximum star lifting rate, which is determined by the energy available. For the initial mass range explored in this work, the stars radiate much more energy than required for {\it isoirradiance} or {\it isoluminosity}, so even a Sun-like star could be engineered into a red dwarf.

Implementing this would set the lifetime of their host star to the lifetime of these ${\sim} 0.1 {\rm M}_{\odot}$ stars - trillions of years. This lifetime could be extended even further through a method like star-dumping, increasing the metallicity of the star. Further, the harvested material could be used to feed the star while the heavier elements produced in the star are extracted. This means that a star which is aggressively star-lifted into a red dwarf would be able to live approximately $\tau \sim N \times \tau_{\rm RD}$ where $\tau_{\rm RD}$  is the lifetime of a metal rich $0.1{\rm M}_{\odot}$ late M-dwarf, ${\sim} 5$ Tyr, and $N$ is the ratio of initial mass and final mass. Combining these techniques for a Sun-like star could result in a habitable lifetime on the order of 50 Tyr, ${\gtrsim}3500$ times the age of the universe.

As an extremely long-term solution, this may be an appealing option for an advanced civilization. For a Sun-like star, an initial SL rate 10 times the derived $\frac{dM_*}{dt}  {\sim} 2{\times}10^{19}$kg $\text{yr}^{-1}$ would be more than enough to avoid an early MS termination and would at most require ${\sim} 33\%$ of the stars yearly output. This would decrease the output of the star, but the required SL rate and energy demand would also decrease. SL at this aggressive rate would also solve the problem of metal mixing, where metals at the core are efficiently mixed once the star dips below the convection-dominated mass threshold. Of course, the problems of energy capture and storing this lifted mass in a stable orbit are massive hurdles which require a leap of the imagination.


\subsection{Detection of star-lifting}

Though difficult, there are several ways which we could observe SL. Implementing {\it isoluminosity} would likely cause dips in the apparent brightness as the star-lifted material orbits between the distant observer and the host star. This was considered in other works (e.g. \citealt{Boyajian_2016, Wright_2015}). An {\it isoirradiance} implementation could also cause noticeable effects. If this is implemented through an increased stellar wind, it may be possible to detect the unusually high wind directly. Some of the most massive stars explored in this work require a mass loss rate which matches or exceeds previous lower limits for stellar wind detection, ${\sim} 10^{-10} {\rm M}_{\odot}\text{yr}^{-1}$ \citep{Lamers_1999}. Further, extremely long period observations could detect these unusual changes in mass, where a star's mass and luminosity evolution might follow one of the unusual tracks shown in Fig~\ref{fig:funny_HR}. 

SL could also take the form of an unusually large population of low metallicity stars in extremely old clusters -- stars which should have expired given the age of the cluster may still exist if they have been engineered. Though these stars would be hard to distinguish from a star which naturally has a similar mass, age, and metallicity, a population distribution which is skewed towards the lower mass range could hint at SL, assuming the civilization has engineered on the galactic scale.


Contradictory ages derived from gyrochronology and asteroseismology could also be a sign of SL. For example, if we correctly calculate the age via asteroseismology for an engineered star, we won't necessarily find a similar age via gyrochronology. The spin of the star for a fixed mass and age could vary wildly depending on the implementation of SL and where that removed mass is placed. Inferring different ages in a binary pair may also hint at SL.






\begin{figure*}
 \includegraphics[width=1.6\columnwidth]{terminal_age_astart=2000000000.0.pdf}
  \caption{Initial mass {\it vs.} lifetime for both methods (dashed for {\it isoluminosity} and solid for {\it isoirradiance}) and natural evolution without SL (dotted), where [Fe/H]$=0.0$ and $A_{\rm start} = 2$ Gyr.}
 \label{fig:terminal_age}
\end{figure*}


\begin{figure*}
 \includegraphics[width=1.8\columnwidth]{funny_HR.pdf}
  \caption{Evolution in Luminosity-Mass space for both methods of SL (dashed for {\it isoluminosity} and solid for {\it isoirradiance}) and the natural evolution without SL (dotted). {\it Isoluminosity} holds luminosity constant by definition, so movement is perfectly horizontal and we see this method result in the lowest final MS mass. {\it isoirradiance} allows for a slight increase in luminosity in order to offset the increased orbital radius of the host planet after mass is ejected out of the planetary system. Natural evolution appears vertical, though there is a slight mass loss during the MS due to natural stellar winds.}
 \label{fig:funny_HR}
\end{figure*}




\section*{Acknowledgements}

We thank Ben Cassese, M.-M. Mac Low, Jamie Tayar, and Doug Scoggins for helpful comments and discussion.

MS thanks Ben Cassese for helpful comments on the draft of this manuscript.

We thank the authors of the freely available code Matplotlib \citep{Hunter:2007}, SciPy \citep{2020SciPy-NMeth}, and NumPy \citep{harris2020array}.

Special thanks to donors to the Cool Worlds Lab: Mark Sloan,
Douglas Daughaday,
Andrew Jones,
Elena West,
Tristan Zajonc,
Chuck Wolfred,
Lasse Skov,
Graeme Benson,
Alex de Vaal,
Mark Elliott,
Stephen Lee,
Zachary Danielson,
Chad Souter,
Marcus Gillette,
Tina Jeffcoat,
Jason Rockett,
Scott Hannum,
Tom Donkin,
Andrew Schoen,
Jacob Black,
Ms. Reza Ramezankhani,
Steven Marks,
Gary Canterbury,
Nicholas Gebben,
Mike Hedlund,
Dhruv Bansal,
Jonathan Sturm,
Rand Corporation,
Leigh Deacon,
Ryan Provost,
Brynjolfur Sigurjonsson,
Benjamin Paul Walford,
Nicholas De Haan,
\& Joseph Gillmer.

\section*{Data Availability}
The data used in this paper can be downloaded from the \href{http://waps.cfa.harvard.edu/MIST/interp_tracks.html}{MIST website}. The code and data produced during the preparation of this manuscript is available at this \href{https://github.com/mscoggs/star_lifting}{github repository}.


\bibliographystyle{mnras}

\section{Notes}









