The Abundance of SiC$_{2}$ in Carbon Star Envelopes$^\star$: Evidence that SiC$_{2}$ is a gas-phase precursor of SiC dust

Silicon carbide dust is ubiquitous in circumstellar envelopes around C-rich AGB stars. However, the main gas-phase precursors leading to the formation of SiC dust have not yet been identified. The most obvious candidates among the molecules containing an Si--C bond detected in C-rich AGB stars are SiC$_2$, SiC, and Si$_2$C. We aim to study how widespread and abundant SiC$_2$, SiC, and Si$_2$C are in envelopes around C-rich AGB stars and whether or not these species play an active role as gas-phase precursors of silicon carbide dust in the ejecta of carbon stars. We carried out sensitive observations with the IRAM 30m telescope of a sample of 25 C-rich AGB stars to search for emission lines of SiC$_2$, SiC, and Si$_2$C in the 2 mm band. We performed non-LTE excitation and radiative transfer calculations based on the LVG method to model the observed lines of SiC$_2$ and to derive SiC$_2$ fractional abundances in the observed envelopes. We detect SiC$_{2}$ in most of the sources, SiC in about half of them, and do not detect Si$_2$C in any source, at the exception of IRC +10216. Most of these detections are reported for the first time in this work. We find a positive correlation between the SiC and SiC$_2$ line emission, which suggests that both species are chemically linked, the SiC radical probably being the photodissociation product of SiC$_2$ in the external layer of the envelope. We find a clear trend in which the denser the envelope, the less abundant SiC$_2$ is. The observed trend is interpreted as an evidence of efficient incorporation of SiC$_2$ onto dust grains, a process which is favored at high densities owing to the higher rate at which collisions between particles take place. The observed behavior of a decline in the SiC$_2$ abundance with increasing density strongly suggests that SiC$_{2}$ is an important gas-phase precursor of SiC dust in envelopes around carbon stars.