Question

Time standards are now based on atomic clocks. A promising second standard is based on pulsars, which are rotating neutron stars (highly compact stars consisting only of neutrons). Some rotate at a rate that is highly stable, sending out a radio beacon that sweeps briefly across Earth once with each rotation, like a lighthouse beacon. Pulsar PSR 1937 + 21 is an example; it rotates once every 1.557 806 448 872 75$\pm$3 ms, where the trailing$\pm$3 indicates the uncertainty in the last decimal place (it does not mean$\pm$3 ms). (a) How many rotations does PSR 1937 + 21 make in 7.00 days? (b) How much time does the pulsar take to rotate exactly one million times and (c) what is the associated uncertainty?

Short answer

(a) Number of rotations in 7.00 days is$3.88\times {10}^8$

(b) Time taken by the Pulsar for 1 million rotations is role="math" localid="1651833489654" $1557.80644887275s$

(c) Associated uncertainty is$\pm 3\times {10}^{-11}\text{s}$

Step-by-step solution

Given data

The time period of rotation of the Pulsar,${\text{P}}_t=5578064487275ms$

The uncertainty per rotation is$\pm 3\times {10}^{-17}s$

Understanding the atomic clock

Pulsars are rapidly rotating stars that emit radio pulses with a very regular frequency.Atomic clocks measure time by measuring the frequency of radiation emitted by atoms.An atom's natural oscillations work like the pendulum in a grandfather clock.

(a) Determination of the number of rotations in 7 days

The frequency is reciprocal of time. Therefore,

$\text{f=}\frac{1rotation}{1.55780644887275\times {10}^{-3}s}$… (i)

Multiply this frequency by the given time 7days in seconds to get the number of rotations.

Thus, the Pulsar makes$3.88\times {10}^8$rotations in 7 days.

Step 3:(b) Determination of the time taken by Pulsar for 1 million rotations

Using equation (i), time is calculated as:

$n=f\times t$

Here number of rotations is $1\times {10}^6$. So

Thus, the time taken by the Pulsar to rotate one million times is$\text{1557.80644887275s}$

Step 3:(c) Determination of the uncertainty

The time uncertainty per revolution is$\pm 3\times {10}^{-17}s$

For one million revolutions, associated uncertainty is,

role="math" localid="1651834524162" $$\begin{gathered} \text{Associated Uncertainty}=\pm 3\times {10}^{-17}\times 1\times {10}^6 \\ =\pm 3\times {10}^{-11}\text{s} \end{gathered}$$

Thus, the associated uncertainty is.${\text{±3×10}}^{-11}s$