The fundamental plane of blazars based on the black hole spin-mass energy

We examine the fundamental plane of 91 Blazars which include FSRQs and BL Lacs with known X-ray luminosity ($L_{R}$), radio luminosity ($L_X$), and black hole mass measurements ($M$) to reflect the relationship between jet and accretion for blazars. The fundamental plane of Blazars are log$L_{R}$=${0.273}_{+0.059}^{-0.059}$log$L_X$+${0.695}_{+0.191}^{-0.191}$log$M$+${25.457}_{+2.728}^{-2.728}$ and log$L_{R}$=${0.190}_{+0.049}^{-0.049}$log$L_X$+${0.475}_{+0.157}^{-0.157}$log$M$+${28.568}_{+2.245}^{-2.245}$ after considering the effect of beam factor. Our results suggest that the jet of blazars has connection with accretion. We set the black hole spin energy as a new variable to correct the black hole mass and explore the effect of black hole spin on the fundamental relationship. We find that the fundamental plane of Blazars is effected by the black hole spin, which is similar to the previous work for AGNs. We additionally examine a new fundamental plane which is based on the black hole spin-mass energy ($M_{spin}$). The new fundamental plane (log$L_{R}$=${0.332}_{+0.081}^{-0.081}$log$L_X$+${0.502}_{+0.091}^{-0.091}$log$M_{spin}$+${22.606}_{+3.346}^{-3.346}$ with R-Square=0.575) shows that $M_{spin}$ has a better correlation coefficient comparing to the $M$ for fundamental plane of Blazars. These results support that the black hole spin should be considered as a important factor for the study of fundamental plane for Blazars. And these may further our understanding of the Blandford-Znajek process in blazars.