Neutron stars (NSs) provide a unique laboratory to study matter under extreme densities. Recent observations from gravitational and electromagnetic waves have enabled constraints on NS properties, such as tidal deformability (related to the tidal Love number) and stellar compactness. Although each of these two NS observables depends strongly on the stellar internal structure, the relation between them (called the Love–C relation) is known to be equation-of-state insensitive. In this study, we investigate the effects of a possible crystalline phase in the core of hybrid stars (HSs) on the mass–radius and Love–C relations, where HSs are a subclass of NS models with a quark matter core and a nuclear matter envelope with a sharp phase transition in between. We find that both the maximum mass and the corresponding radius increase as one increases the stiffness of the quark matter core controlled by the speed of sound, while the density discontinuity at the nuclear-quark matter transition effectively softens the equations of state. Deviations of the Love–C relation for elastic HSs from that of fluid NSs become more pronounced with a larger shear modulus, lower transition pressure, and larger density gap and can be as large as 60%. These findings suggest a potential method for testing the existence of distinct phases within HSs, though deviations are not large enough to be detected with current measurements of the tidal deformability and compactness.

中子星（NS）提供了一个独特的实验室来研究极端密度下的物质。最近对引力波和电磁波的观测使得对 NS 特性的限制成为可能，例如潮汐可变形性（与潮汐洛夫数相关）和恒星致密性。尽管这两个 NS 可观测量中的每一个都强烈依赖于恒星的内部结构，但它们之间的关系（称为 Love- C关系）已知对状态方程不敏感。在这项研究中，我们研究了混合星（HS）核心中可能的晶相对质量半径和Love -C关系的影响，其中HS是具有夸克物质核心和核物质的NS模型的子类包络线之间有急剧的相变。我们发现，最大质量和相应的半径随着声速控制的夸克物质核心的刚度的增加而增加，而核-夸克物质转变处的密度不连续性有效地软化了状态方程。弹性 HS 的 Love- C关系与流体 NS 的 Love-C 关系的偏差随着较大的剪切模量、较低的转变压力和较大的密度差距而变得更加明显，并且可能高达 60%。这些发现提出了一种测试 HS 内不同阶段是否存在的潜在方法，尽管偏差还不够大，无法用当前的潮汐变形性和致密性测量来检测。