What values should I use for s.d. in power.t.test() for one sample t-test? In the power analysis of a one-sample t-test (the t-test is used to test a hypothesis about the population mean $\mu$ of a r.v. $\xi\sim N(\mu,\sigma^2)$ where we do not know $\sigma$), we need to assume a distribution for $\xi_i$ where $\xi_1,\xi_2,\cdots,\xi_n$ is a simple random sample of $\xi$. Often it is assumed that $\xi_i\sim N(\mu_*,\sigma_*),\forall i=1,2,\cdots,n$ where $\mu_*,\sigma_*$ are some specified constant real numbers. It follows that the sample mean $\overline{\xi}\sim N(\mu_*,\frac{\sigma_*^2}{n})$. Then we could use the critical value for sample mean $\overline{\xi}_c$ derived from the t-distribution to calculate $\beta$ and the power of the test.
However, problems arises when we are trying to work out the needed sample size $n$ for a specified power. I am told to use the power.t.test() function in R to calculate it. However, I am not sure what should I put in as the values for parameter sd.
Question 1:
Can we assume that $\sigma=\sigma_*$, or use the sample variance of a random sample as $\sigma$, and let sd equal to $\sqrt{\dfrac{\sigma^2+\sigma_*^2}{2}}$?
Question 2:
How does the power.t.test() function work for the when it is used in the power analysis of a one-sample t-test?

The usual use of power.t.test() is to determine the sample size when planning an experiment, and in that role it is sensible to input the standard deviation that you expect to find in the data.
Given that power is a function of effect size and that you do not know the effect size in an experiment that you have not yet run, it is very helpful to examine power as a graphical function. See the graphs in section 2.5 Power and Expected P-values of this chapter: https://link.springer.com/chapter/10.1007/164_2019_286
