## 1 Introduction  
Glaciers and ice in polar regions have essential information about the past as well as recent geological history and climate change [1]. An unexplored subglacial lake, which is a lake formed under a glacier, attracts researchers from various fields, such as biology, astrobiology, and biochemistry [1, 2, 3]. After discovering unknown life and information in the Vostok glacier in Antarctica, a great deal of research has been conducted to reach the glacier and obtain microbial samples. Therefore, it is essential to develop polar exploration technology (i.e., drilling technology), facilitating quick penetration toward subglacial lakes without contamination in an ecosystem of the polar regions. There are various conventional drilling techniques, such as wire-line drilling, hot-water drilling, and coiled-tube drilling that can penetrate the ice and reach a subglacial lake [4]. However, it is reported that such a conventional drilling method can cause contamination in both the polar regions and in retrieved samples [5, 6]. An electro-thermal drilling probe (ETDP) is a promising way to facilitate the exploration of the subglacial lake without contamination and has been widely used to penetrate the glacier [7, 8, 9]. A heated probe tip is attached in the head of the ETDP, as shown in Figure 1. The ice which is in contact with the heated probe tip starts to melt and forms a thin liquid film [10, 11, 12]. Subsequently, the ETDP pushes the thin liquid film due to its weight and can slowly move downward through the close-contact melting process. The rate of penetration (ROP), which indicates the speed at which the heated probe tip deepens the borehole, has been one of the key parameters in designing the shape and operating conditions of an ETDP [12; 13; 14; 15]. Many researchers have analyzed the effect of factors, such as operating conditions and melting-tip shape, on the ROP. [PERSON] et al. [12] found that the ROP increased with heating power and the cross-sectional area of heated probe tip owing to a faster phase change with the increase in heating power. [PERSON] et al. [13] showed that a copper heated probe tip could penetrate ice faster because of its higher thermal conductivity. In addition, they conducted a parametric study to investigate the effect of the heated probe tip shape on the ROP. From the results, the heated probe tip with a 60\({}^{\circ}\) cone shape shows the highest ROP. They also found that the ROP increased with the weight of ETDP because heavier a ETDP can squeeze the molten liquid layer more strongly and move faster. [PERSON] et al. [15] provided several lab-scale experimental results showing that an ETDP with a spherical shape penetrates an ice-chamber faster with a higher heating power. The simple model was suggested to predict ROP based on the required heating power for melting ice. They over-estimated the ROP by about 30% compared to the experimental results. [PERSON] et al. [14] studied the effects of the shape and material of the drill head and heater temperature on the rate of penetration via lab-scale experiments. They also estimated the total required thermal energy considering the thermal loss with various conditions through a one-dimension heat conduction problem. From the experimental results, a conical shape showed a higher ROP than a flat shape due to the large contact area with ice. The studies mentioned above broadly analyzed the effects of design factors (shape and operating conditions) of ETDPs on the ROP. Unfortunately, there is a lack of information to understand the underlying physics of the drilling process considering the close-contact melting characteristics of ETDP. The heated probe tip melts the ice and forms a thin liquid layer [12]. The ETDP pushes the molten liquid layer and can move in a downward direction. The thin liquid layer plays a role in thermal resistance and influences heat transfer characteristics during the melting process. However, such close-contact melting characteristics of ETDP are not easily directly measured.