Question

Audrey, an astronomer is searching for extra-solar planets using the technique of relativistic lensing. Though there are believed to be a very large number of planets that can be found this way, actually finding one takes time and luck; and finding one planet does not help at all with finding planets of other stars in the same part of the sky. Audrey is good at it, and finds one planet at a time, on average once every three months. a.) Find the expected value and

Answer 1

Astronomers like Audrey use indirect methods such as relativistic lensing to detect extra-solar planets due to the difficulty of direct observation. The Kepler mission enabled the extrapolation of data to the entire galaxy regarding exoplanet frequency. The Doppler effect was a key technique in early discoveries of such planets, exemplified by the detection of a planet orbiting 51 Pegasi.

Step-by-step explanation:

Audrey, an astronomer, is searching for extra-solar planets using the technique of relativistic lensing. This is one of the methods employed by astronomers to detect planets around other stars indirectly, as direct observation can be extremely challenging due to the vast distances and the small size of the planets relative to their host stars. By observing the bending of light from a distant star caused by a planet's gravity, astronomers can infer the presence of the planet, much like seeing the effect of a mosquito flying around a spotlight without directly seeing the mosquito itself.

The Kepler mission was a significant project launched by NASA that revolutionized our knowledge of exoplanets, with the primary goal of estimating the frequency of Earth-like planets around other stars. Although it focused on a small fraction of stars, the data collected was sufficient to draw conclusions about the entire galaxy due to the representativeness of the observed stars.

The discovery of planets via the Doppler effect, as was the case with the planet around 51 Pegasi in 1995, is another example of astronomers using indirect methods to detect these distant worlds. These discoveries, along with transiting exoplanets observed by the Kepler space telescope, have contributed to our growing catalog of planets outside our solar system.

Answer 2

- the expected value is 8

- the standard deviation is 2.8284

Step-by-step explanation:

Given the data in the question;

The model N(t), the number of planets found up to time t, as a poisson process,

∴ N(t) has distribution of poisson distribution with parameter (λt)

so

the mean is;

λ = 1 every month = 1/3 per month

E[N(t)] = λt

E[N(t)] = (1/3)(24)

E[N(t)] = 8

Therefore, the expected value is 8

For poisson process, Variance and mean are the same,

Var[N(t)] = Var[N(24)]

Var[N(t)] = E[N(24)]

Var[N(t)] = 8

so the standard deviation will be;

σ[N(24)] = √(Var[N(t)] )

σ[N(24)] = √(8 )

σ[N(24)] = 2.8284

Therefore, the standard deviation is 2.8284