A Deep Learning Framework for TTV Characterization: A Case Study of the Kepler-88 System

Chen Chen; Jiapeng Gao; Gongjie Li; Lingkai Kong; Molei Tao

astro-ph.EPSep 6, 2024

A Deep Learning Framework for TTV Characterization: A Case Study of the Kepler-88 System

Authors: Chen Chen, Jiapeng Gao, Gongjie Li, Lingkai Kong, Molei Tao

Organizations: School of Earth and Atmospheric Sciences, Georgia Institute of Technology, Atlanta, GA, 30332, USA · Center for Relativistic Astrophysics, School of Physics, Georgia Institute of Technology, Atlanta, GA 30332, USA · School of Mathematics, Georgia Institute of Technology, Atlanta, GA, 30332, USA · Machine Learning Center, Georgia Institute of Technology, Atlanta, GA, 30332, USA

Abstract

Transit timing variation (TTV) provides rich information about the mass and orbital properties of exoplanets, which are often obtained by solving an inverse problem via Markov Chain Monte Carlo (MCMC). In this paper, we design a new data-driven approach, which potentially can be applied to problems that are hard to traditional MCMC methods, such as the case with only one planet transiting. Specifically, we use a deep learning approach to predict the parameters of non-transit companion for the single transit system with transit information (i.e., TTV, and Transit Duration Variation (TDV)) as input. This approach is constructed using the Transformer-based architecture that can extract long-range interactions from TTV sequential data. We focus on two-planet systems based on Kepler-88 as a starting point, with the inner planet fixed to the properties of Kepler-88 and the mass and orbital elements of the outer planet varied. Applying to Kepler-88, we obtained an overall fractional error of $\lesssim$ 10% on semi-major axis and eccentricity, and $\sim 10-30\%$ on planet mass (assuming that the stellar and transiting planet parameters are known), based on a training set of a million simulations. When applied to the test set, our model achieves low fractional errors of less than a few percent for systems with TTV semi-amplitudes ranging from $\sim 0.3$ s to $\sim 5$ min. Extending our approach to a broader range of planetary architectures, including systems with substantially weaker or stronger TTV signals and those containing additional planets or stellar companions, remains an important direction for future work.

Figures & tables

Figure 1: How TTV changes as the semi-major axis varies. Peaks demonstrate the lack of regularity (i.e. smoothness) in the response. Results obtained from high accuracy N-body simulations [ 10 ] . The system parameters are based on [ 38 ] , with the semi-major axis varying within $10\%$ of the given value of 0.15292 AU (1000 different values simulated), and other parameters fixed. x-axis is the difference between the tested and given [ 38 ] semi-major axis values. y-axis is $\chi^{2}=\sum\frac{(V_{fixed}-V_{sim})^{2}}{abs(V_{sim})}$ , where $V_{fixed}$ is the simulated TTV with fixed parameters based on [ 38 ] while $V_{sim}$ represents the simulated TTV with varied semi-major axis.

	Planet b	Planet c
Mass	$8.7M_{E}$	$10M_{E}-10M_{J}$
Semi-major axis a	0.095 AU	0.117 - 5 AU
Eccentricity e	0.05593	0.01 - 0.1
Inclination	$0.945^{\circ}$	lower bound - $7.7^{\circ}$
Argument of Periastron	$-179.41^{\circ}$	$0^{\circ}-360^{\circ}$
Longitude of ascending node	$270^{\circ}$	$0^{\circ}-360^{\circ}$

Table 1: Summary of parameters of Kepler-88 System for N-body simulation

Figure 2: Fractional error distributions for predictions on the validation set using the pure Transformer, Transformer+LSTM, and Transformer+GRU architectures, shown in orange, green, and blue, respectively. The top-left panel shows the distribution for the semi-major axis ( $a$ ), the top-right panel for the mass, the bottom-left panel for the eccentricity ( $e$ ), and the bottom-right panel for the inclination (i). The MSE calculated on the validation set are $6.130\times 10^{-5}$ , $6.241\times 10^{-5}$ , and $2.475\times 10^{-3}$ , respectively for Transformer+GRU and Transformer+LSTM and pure Transformer.

Figure 3: Deep learning model architecture. $N_{G}$ and $N_{F}$ denotes the number of layers of GRU and FF, respectively. $N_{T}$ is the number of layers of transformer encoder. $N_{FD}$ is the number of FF layers in decoders. Token is an extra input of the transformer encoder, which is initialized as a tensor with zeros. There are three model variants in this work: the model with TDV input and without noise, the model without TDV input and without noise, and the model without TDV input and with noise. The numbers of layers, hidden dimensions, and other hyperparameters differ among these variants, as described in Sections II.3 and III.1 and implemented in the released code. As a representative example, for the model with TDV input and without noise, Input 1 has dimension $[64,135,3]$ , corresponding to batch size, number of transit epochs, and the three input features: transit time, TTV, and TDV. Input 2 has dimension $[64,1,2]$ . The GRU output has dimension $[64,135,256]$ . After concatenating the GRU output, the encoded Input 2 token, and an additional zero token, the Transformer output has dimension $[64,137,256]$ . The final output dimension is $[64,4]$ .

Input	Model	a	Mass	e	i
GRIT	with TDV	0.1525 AU (0.3%)	176.4 $M_{E}$ (11.2%)	0.0628 (11.6%)	$4.2^{\circ}$ (9.9%)
Observation	with TDV	0.1468 AU	68.9 $M_{E}$	0.0720	$2.7^{\circ}$
GRIT	without TDV	0.1419 AU (7.22%)	257.4 $M_{E}$ (29.5%)	0.0547 (2.81%)
Observation	without TDV	0.1539 AU	369.8 $M_{E}$	0.0625

Table 2: Prediction on Kepler-88 without noise. The observation input is the transit information from [ 25 ] . For the input from GRIT, the parameters of a, mass, e and i based on [ 38 ] are 0.15292 AU, 198.8

M_{E}

, 0.05628, and

3.8^{\circ}

, respectively.

Input	a	Mass	e
GRIT	0.1454 AU (4.92%)	123.985 $M_{E}$ (37.6%)	0.0565 (0.403%)
Observation	0.1458 AU	194.9 $M_{E}$	0.0410

Table 3: Prediction on Kepler-88 with noise. The observation input is the transit information from [ 25 ] . For the input from GRIT, the parameters of a, mass, e and i based on [ 38 ] are 0.15292 AU, 198.8

M_{E}

, 0.05628, and

3.8^{\circ}

, respectively.

Figure 4: Fractional error distributions on the testing set prediction for the model with TDV in the input. Top left shows the distribution for semi-major axis (a). Top right shows the distribution for the mass. Bottom left shows the distribution for the eccentricity (e). Bottom right shows the the distribution for the inclination (i).

Figure 5: Fractional error distributions on the testing set for the model without TDV in the input. From left to right, it shows the distribution for semi-major axis (a), mass, and eccentricity (e), respectively.

Figure 6: The relationship between the true value and the fractional error by the model with TDV. Each dot shows the fractional error for a specific system.

Figure 7: The relationship between the true value and the fractional error by the model without TDV. Each dot shows the fractional error for a specific system.

Figure 8: The fractional error versus the TTV semi-amplitude. Each dot shows the fractional error for a specific system. The orange dash-dotted line shows the running $90^{th}$ percentile, and the red line shows the running median. The fractional error are typically low $\lesssim$ a few percent for a wide range of TTV semi-amplitude of $\sim 10^{-4}-0.1$ hr.

		$\sim$ 3AU, $\sim$ 1 $M_{J}$	$\sim$ 3AU, $\sim$ 5 $M_{J}$	$\sim$ 0.2 AU, $\sim$ 20 $M_{E}$
a	T	3.0255 AU	2.9932 AU	0.1782 AU
	P	3.0155 AU	2.9770 AU	0.1971 AU
	FE	0.3305%	0.5441%	10.5684%
Mass	T	0.9924 $M_{J}$	4.9825 $M_{J}$	17.7183 $M_{E}$
	P	0.9901 $M_{J}$	4.9770 $M_{J}$	24.0963 $M_{E}$
	FE	0.2348%	0.1111%	35.9967%

Table 4: Prediction on selected example systems from testing set. T denotes true value, P represents prediction, and FE is the abbreviation of fractional error.

		$\sim$ 3AU, $\sim$ 1 $M_{J}$	$\sim$ 3AU, $\sim$ 5 $M_{J}$	$\sim$ 0.2 AU, $\sim$ 20 $M_{E}$
a	T	3.0255 AU	2.9932 AU	0.1782 AU
	P	3.0263 AU	3.0044 AU	0.2111 AU
	FE	0.0277%	0.3742%	18.4295%
Mass	T	0.9924 $M_{J}$	4.9825 $M_{J}$	17.7183 $M_{E}$
	P	1.0081 $M_{J}$	4.9337 $M_{J}$	30.3714 $M_{E}$
	FE	1.5827%	0.9792%	71.4125%

Table 5: Prediction on selected example systems from testing set. T denotes true value, P represents prediction, and FE is the abbreviation of fractional error.

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