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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 \(1.55780644887275 \pm 3 \mathrm{~ms}\), where the trailing \(\pm 3\) indicates the uncertainty in the last decimal place (it does \(n o t\) mean \(\pm 3 \mathrm{~ms}\) ). (a) How many times does PSR \(1937+21\) rotate in \(7.00\) days? (b) How much time does the pulsar take to rotate \(1.0 \times\) \(10^{6}\) times, and (c) what is the associated uncertainty?

Step by step solution

01

Convert the Period to Seconds

Given that the period of the pulsar PSR 1937+21 is \(1.55780644887275 \text{ ms}\). Convert milliseconds to seconds by dividing by 1000. Thus, \(1.55780644887275 \text{ ms} = 1.55780644887275 \times 10^{-3} \text{ s}\).

02

Calculate Total Rotations in 7 Days

First, convert 7 days to seconds. There are 24 hours in a day, 60 minutes in an hour, and 60 seconds in a minute: \(7 \text{ days} = 7 \times 24 \times 60 \times 60 \text{ s} = 604800 \text{ s}\). Now, divide the total time in seconds by the period to find the number of rotations: \(N = \frac{604800 \text{ s}}{1.55780644887275 \times 10^{-3} \text{ s}} \approx 3.88475 \times 10^{8}\).

03

Time to Rotate 1 Million Times

Given the pulsar needs to rotate \(1.0 \times 10^6\) times, calculate the total time for these rotations by multiplying the period by the number of rotations: \(T = 1.55780644887275 \times 10^{-3} \text{ s} \times 1.0 \times 10^6 = 1557.80644887275 \text{ s}\).

04

Calculate the Uncertainty

The uncertainty in the period is \( \pm 3 \times 10^{-14} \text{ s}\). For \(1.0 \times 10^6\) rotations, the total uncertainty is calculated by multiplying the uncertainty per single rotation by the number of rotations: \( \Delta T = 3 \times 10^{-14} \text{ s} \times 1.0 \times 10^6 = 3 \times 10^{-8} \text{ s}\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Atomic Clocks

Atomic clocks are the most accurate timekeeping devices we have today. They use the vibrations of atoms, typically cesium or rubidium, to measure time very precisely. Atomic clocks are crucial for many modern technologies, especially GPS and telecommunications. The precision of an atomic clock is so high that it can measure time to within a few billionths of a second. This stability and accuracy make atomic clocks indispensable for scientific research and global navigation systems. By using the regular oscillation of atoms, atomic clocks can achieve unparalleled accuracy, vastly improving our ability to measure time on Earth and in space.

Pulsars

Pulsars are highly magnetized, rotating neutron stars that emit beams of electromagnetic radiation out of their magnetic poles. These beams of radiation are detected on Earth as pulses of radiation, which occur at very regular intervals, making pulsars extremely reliable natural clocks. The first pulsar was discovered in 1967, and since then, many have been found across the universe. One of the most interesting aspects of pulsars is their stability; some pulsars have extremely stable rotation periods, comparable to the accuracy of atomic clocks. This makes them potential candidates for timekeeping standards in space.

Rotating Neutron Stars

Neutron stars are the remnants of massive stars that have exploded in supernovae. They are incredibly dense, with masses often greater than that of the Sun but packed into a sphere only about 20 kilometers in diameter. When these stars rotate, they do so at extremely high speeds, sometimes hundreds of times per second. The rapid rotation, combined with their immense density, means that neutron stars have significant gravitational and magnetic fields. This intense rotation and gravitational pull make neutron stars fascinating objects for studying the fundamental laws of physics.

Time Calculation

Time calculation in astronomy often involves converting between different units or applying precise measurements to deduce information. For example, in the given problem involving pulsar PSR 1937+21, we needed to convert the rotation period from milliseconds to seconds and then calculate the total number of rotations over a span of days. Accurate time calculation allows astronomers to predict events, track objects in space, and synchronize clocks for scientific experiments. The process often includes steps like unit conversion, multiplication, and division to achieve precise results.

Uncertainty in Measurement

Uncertainty in measurement is a fundamental concept in science. It acknowledges that all measurements have limitations and thus includes an estimate of possible error. For example, the period of pulsar PSR 1937+21 is given with an uncertainty of 3 milliseconds in the last decimal place. When calculating large numbers of rotations, this small uncertainty can become significant. Scientists must account for this uncertainty to ensure accurate and reliable results. This is often done by including an error margin in their final calculations and interpretations.