Final answer:
The critical temperature to initiate the proton-proton cycle in the cores of stars like the Sun is approximately 10 million degrees Kelvin (10³ K). This critical temperature allows protons to overcome the electric potential barrier and begin the fusion process, which generates the star's energy.
Step-by-step explanation:
The critical temperature required to initiate the proton-proton cycle at the cores of stars is highly dependent on the mass and composition of the star. In stars like our Sun, which has a mass smaller than 1.2 times the mass of the Sun, the proton-proton chain is the dominant energy production mechanism. The temperature inside such stars needs to be high enough so that protons can overcome the electrical potential barrier and allow nuclear fusion to occur. This fusion process involves protons colliding directly to form helium nuclei. The critical temperature for the proton-proton cycle to initiate is around 10 million degrees Kelvin (10³ K).
It is important to note that in the sun's core, for example, a proton will typically undergo numerous collisions over approximately 14 billion years before fusing with another proton due to the electrical potential barrier between positively charged protons. However, the rate of fusion reactions increases significantly with temperature, rising roughly with the temperature to the fourth power, which explains the sensitivity of the fusion rate to temperature changes. Eventually, as the core temperature of a pre-star entity reaches the critical point, the commencement of nuclear fusion marks its transition into a star.