APE Code: Analytical Protostellar Simulator

• APE Code is a semi-analytical tool that models the physical, thermal, and chemical evolution of protostellar systems from collapse to disk and outflow formation.
• It integrates analytical collapse dynamics, stellar evolution, disk formation, particle tracking, and radiative transfer to generate accurate synthetic observations and chemical maps.
• The modular design bridges core physics with environmental factors, facilitating simulations across diverse star-forming regions and aiding observational calibration.

The Analytical Protostellar Environment (APE) Code is a semi-analytical modeling tool developed to rapidly simulate the physical, thermal, dynamical, and chemical evolution of protostellar systems, encompassing their formation from Bonnor–Ebert spheres through to the development of stars, disks, and outflows. By integrating analytical prescriptions with chemical and radiative transfer modeling interfaces, APE supplies detailed physical conditions for chemical models and synthetic observations, facilitating direct interpretation of interferometric data and assessment of the environment’s impact on chemical complexity and star/planet formation.

1. Structural and Evolutionary Modeling

APE employs as its starting point a critical Bonnor–Ebert sphere—an isothermal, pressure-confined and self-gravitating sphere. The code analytically follows its free-fall collapse, deriving radial profiles for density and velocity via closed-form or semi-analytical expressions. For density normalization, APE applies: $D = \frac{\rho}{c}$ with $c$ as the central density, and non-dimensionalizes physical radii using characteristic scales combining the sound speed and gravity: $x_{\mathrm{be}} = \frac{r}{s/\sqrt{4\pi G c}}$ where $s$ is the isothermal sound speed and $G$ the gravitational constant.

Following the collapse phase (after a free-fall time, once densities exceed $\sim10^{-8}$ g cm$^{-3}$), APE “creates” a central object. The protostar’s mass, radius, luminosity, and effective temperature are tracked in time using interpolated stellar evolution models that encode accretion physics and energy transport, incorporating, for instance, the impact of deuterium shell burning and accretion shocks on radius inflation.

APE represents circumstellar disks using an $\alpha$-disk prescription. The disk mass is a specified fraction of the accreted mass, while disk radii are estimated by analytic scaling relations informed by non-ideal MHD simulations, with the magnetic mass-to-flux ratio ($\lambda$) as a key parameter: $$\mathrm{disk}(t) = 28.0 \,\mathrm{au} \left(\frac{\eta_A}{10^{18}\,\mathrm{cm}^2\,\mathrm{s}^{-1}}\right)^{2/9} \left(\frac{M_{\mathrm{acc}}(t)}{0.1\,M_\odot}\right)^{1/3} \lambda^{4/9}$$ Here, $\eta_A$ is the ambipolar diffusion coefficient. The mass surface density follows a power-law with exponential taper: $$\Sigma(c,t) = \Sigma_0(t) \left(\frac{c}{\mathrm{disk}(t)}\right)^{-1} \exp\left(-\frac{c}{\mathrm{disk}(t)}\right)$$ APE incorporates an outflow component by carving a conical low-density cavity whose opening angle and velocity (set by the escape velocity near the star) are analytically prescribed.

2. Temperature and Radiative Transfer

APE determines the dust (and, by extension, gas) temperature structure through radiative equilibrium estimates. For the envelope: $$T_{\mathrm{grain}}(r) = \left[\left(\frac{R_*}{r}\right)^2 T_*^4 + T_{\mathrm{mc}}^4\right]^{1/4}$$ where $R_{*}$ and $T_{*}$ are the protostellar radius and effective temperature, and $T_{\mathrm{mc}}$ is the background molecular cloud temperature.

This analytical temperature estimate, although simplified (neglecting shadowing and complex radiative transfer effects), is found to agree well with more detailed dust RT codes (e.g., RADMC–3D) in outer regions ($r \gtrsim 50$ au). APE can also export physical grids to dedicated RT codes for more precise synthetic observations or comparisons with observed SEDs and molecular line emission.

3. Particle-Tracking Dynamics and Interface to Chemical Models

APE’s “particle mode” enables detailed tracking of virtual fluid elements (“particles”) through the evolving protostellar environment. Particle positions are integrated via

$\frac{dr}{dt} = v_r(r)$

with

$v_r(r) = -\sqrt{\frac{GM_{\mathrm{in}}(r)}{2r}}$

modifications allow inclusion of support from rotation, magnetic fields, and thermal pressure, making $v_r$ a generalized infall velocity. Adaptive time-stepping ensures dynamical fidelity and captures rapid transitions in temperature and density: $$dt = \min \left( \frac{1}{f}\frac{r}{v_r},\, dt_0 \right)$$ where $dt_0$ is a maximum allowed step and $f$ a safety factor.

Particle histories (density, temperature as functions of time) serve as input for gas-grain chemical models such as Nautilus, enabling full three-phase chemical evolution (gas, surface, mantle) to be simulated for each trajectory.

4. Dust Evolution and Opacity Calculation

APE incorporates an evolving dust grain model, generalizing from the canonical Mathis–Rumpl–Nordsieck (MRN) ISM distribution. The dust size distribution evolves as a function of local density and temperature, either by direct adoption of MHD simulation results or through a growth parameter formalism. The net distribution at each location is then used to compute wavelength-dependent opacities.

Rosseland mean opacity at a location is calculated as: $$\kappa_R = \frac{\int_{a_{\min}}^{a_{\max}} n(a)\, m(a)\, \kappa_R(a)\, da}{\int_{a_{\min}}^{a_{\max}} n(a)\, m(a)\, da}$$ with $n(a)$ the size distribution and $\kappa_R(a)$ the grain-size dependent opacity.

The resulting opacity maps feed back into temperature calculation (for radiative transfer and grain-surface chemistry), ensuring self-consistency across thermal and chemical modeling domains.

5. Chemical and Synthetic Observation Capabilities

APE is purpose-built to interface with chemical and radiative transfer codes, making it a central tool for producing predictions that are directly comparable to observations. In “snapshot mode,” APE provides 2D density, temperature, velocity, extinction, and dust property maps at a chosen evolutionary stage. These are used for both chemistry and synthetic imaging.

With Nautilus, the code computes gas-phase and grain-surface abundances for hundreds of species, tracking the chemical evolution along each particle’s time-dependent trajectory. For synthetic observation, APE outputs are ingested by radiative transfer codes (e.g., RADMC–3D) to generate spectral cubes, which are then processed by simulated interferometry tools matching instrument parameters (e.g., ALMA configuration). This capability allows the reproduction of observed signatures such as molecular emission morphologies (e.g., X-shaped structures in CO, CS, H$_2$CO) and SEDs, enabling rigorous model–data comparison.

A representative example is modeling a Class I object with a 2 $M_\odot$ progenitor, evolved for 150 kyr with specified $\lambda$ and dust properties, producing CO, CN, CS, H$_2$CO, and CH$_3$OH abundance maps, and matching observed emission morphologies from IRAS 04302+2247.

6. Integration of Environmental Factors and Turbulence

APE incorporates environmental effects such as external pressure (via clump surface density $\Sigma$) in setting the initial core structure: $$R_c = 3.3 \times 10^{-2}\,\mathrm{pc} \left(\frac{M_c}{2\,M_\odot}\right)^{1/2}\left(\frac{\Sigma}{0.1\,\mathrm{g\,cm}^{-2}}\right)^{-1/2}$$ with pressure estimated as $P \simeq 0.88 G\Sigma^2$. The star formation timescale is strongly pressure-dependent, scaling as $t_{*f} \propto M_c^{1/4}\Sigma^{-3/4}$. By modulating $\Sigma$, APE simulates low-pressure (e.g., Taurus) and high-pressure (e.g., Orion, IRDCs, Galactic Center) star-forming environments, capturing the resulting effects on accretion rates, luminosities, SED morphology, and outflow feedback.

APE can be modified to account for turbulence and non-axisymmetric accretion, as revealed by recent simulations that highlight the importance of sheetlike overdensities and anisotropic infall (rather than pure solid-body rotation). Incorporating turbulent velocity distributions, gravitational collimation, and asymmetric source terms into the analytical framework is crucial for more accurately modeling angular momentum transport and mass delivery onto disks.

7. Role in Astrochemical Evolution and Observational Calibration

APE enables systematic investigation of how physical evolution governs the formation, destruction, and inheritance of complex organic molecules (COMs). Coupled APE–Nautilus studies show that light COMs (e.g., CH$_3$OH, C$_2$H$_5$OH) are predominantly inherited from the prestellar phase, while heavier O- and N-bearing COMs form during collapse or in the inner disk. Variations in initial cloud temperature (from 10 K to 15 K) substantially enhance COM formation in the prestellar phase, notably for c–C$_2$H$_4$O and N-bearing species; by contrast, increasing the initial cloud mass (e.g., from 2 to 5 $M_\odot$) has less impact on early disk chemical abundances.

APE simulation outputs—abundance maps, line ratios, and synthetic images—can be matched to multi-molecular observational tracers (e.g., DCO$^+$, H$_2$CO, c–C$_3$H$_2$, C$_2$H) to calibrate the physical modeling. This approach allows constraints on temperature structures, mass reservoirs, and fragmentation propensities, as well as on specific chemical pathways. The bulk envelope temperature is not a strong discriminator for fragmentation, with mass and density exerting principal control, highlighting the need for accurate structural modeling in synthetic prediction and observational analysis.

Summary Table: Major Functional Modules in APE

Module	Purpose	Output
Analytical Collapse & Disk	Initial conditions, time evolution of core/disk	Density/velocity maps
Particle Tracking	Trajectories for parcels (history of T, $\rho$)	Histories for chemistry
Dust Grain Modeling	Evolving size distribution, compute opacities	Opacity maps
Chemistry Interface	Time-dependent 3-phase chemistry (e.g., Nautilus)	Abundance maps
Radiative Transfer & Imaging	RT & synthetic interferometry for molecular lines	Spectral cubes/images

APE is a modular, fast, and extensible analytical framework bridging core collapse theory, disk/outflow evolution, dust processing, chemistry, and radiative transfer. Its structure supports detailed, self-consistent predictions of observables and chemical evolution, making it an effective platform for interpreting current and future observations of protostellar systems and their emerging planetary disks.