You are a pulsar astronomer, and you have been measuring the pulses from a particular milli-second pulsar for several hundred days. You find that they do not arrive at regular intervals - sometimes they arrive a little early and sometimes a little late. You assume that this is because the pulsar is moving closer to and further from the Earth. Here is a plot of your data:
You know that the pulsar weighs \( 2.8 \times 10^{30} {\rm kg}\).
HQ1.1 PLANET ORBIT (4 points possible)
How far (in metres) is the planet from the Pulsar?
Assume that \( G = 6.67 \times 10^{-11} {\rm m^3 kg^{-1} s^{-2}} \)
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HQ1.2 PLANET MASS (5 points possible)
If the planet is in an edge-on orbit, what would its mass be (in kg)?
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Well, calculating the distance to the planet from the pulsar seems to be an astronomical challenge. But don't worry, I'm here to help! Let's try to solve it with a humorous approach, shall we?
Why did the pulsar go on a date with the planet? Because it couldn't resist its gravitational charm! *ba dum tss*
So, to determine the distance, we can use the formula for gravitational force:
\(F = \frac{{G \cdot M_1 \cdot M_2}}{{R^2}}\)
Where:
- \(F\) is the gravitational force between the pulsar and the planet
- \(G\) is the gravitational constant
- \(M_1\) is the mass of the pulsar
- \(M_2\) is the mass of the planet
- \(R\) is the distance between the pulsar and the planet
Now, since we don't know the mass of the planet yet, let's focus on finding the distance. We can rearrange the formula as follows:
\(R = \sqrt{\frac{{G \cdot M_1 \cdot M_2}}{{F}}}\)
Alright, let's gather the known information:
- \(G = 6.67 \times 10^{-11} {\rm m^3 kg^{-1} s^{-2}}\)
- \(M_1 = 2.8 \times 10^{30} {\rm kg}\)
Unfortunately, we don't have the force value \(F\) at the moment. So it seems we're stuck in a cosmic traffic jam. But hey, while we wait for the traffic to clear up, how about I tell you a joke?
Why did the pulsar bring a book on its date with the planet?
In case the conversation went a bit...astronomically slow! *wink*
Alright, back to our scientific quest! Unfortunately, without the force value, we can't calculate the distance or the planet's mass. So I'm sorry for the anticlimactic answer. I hope the cosmic jokes brightened your day a little, though!
To calculate the distance of the planet from the pulsar, we can use Newton's law of universal gravitation:
\[F = \frac{{G \cdot m_1 \cdot m_2}}{{r^2}}\]
where F is the force of gravity between the two objects, G is the gravitational constant, \(m_1\) is the mass of the pulsar, \(m_2\) is the mass of the planet, and r is the distance between the two objects.
In this case, we know the mass of the pulsar (\(m_1\)) is \(2.8 \times 10^{30} \, \text{kg}\) and we need to find the distance to the planet (\(r\)).
Since the problem states that the pulsar is moving closer to and further from the Earth, we can assume that the distance between the pulsar and the planet is changing. Therefore, we need to determine the average distance, which we can approximate using the mean of the minimum and maximum distances.
Let's assume that the minimum distance is \(r_{\text{min}}\) and the maximum distance is \(r_{\text{max}}\). The average distance (\(r_{\text{avg}}\)) is then given by:
\[r_{\text{avg}} = \frac{{r_{\text{min}} + r_{\text{max}}}}{2}\]
To calculate the mass of the planet, we also need to know the period of the pulsar (\(T\)). The period is the time between successive pulses.
The orbital period (\(T\)) is related to the distance (\(r\)) by Kepler's third law:
\[T^2 = \frac{{4\pi^2 \cdot r^3}}{{G \cdot (m_1 + m_2)}}\]
From the given data, we can plot the arrival times of the pulses and measure the arrival time differences to find the minimum and maximum distances. We can also measure the period (\(T\)) of the pulsar.
Once we have \(r_{\text{avg}}\) and \(T\), we can solve for \(m_2\) using the equation above. Let's solve the problem step-by-step:
Step 1: Measure the arrival time differences between pulses to find the minimum and maximum distances.
Step 2: Calculate the average distance using the formula \(r_{\text{avg}} = \frac{{r_{\text{min}} + r_{\text{max}}}}{2}\).
Step 3: Measure the period (\(T\)) of the pulsar.
Step 4: Substitute the values of \(r_{\text{avg}}\), \(T\), \(m_1\), and \(G\) into the equation \(T^2 = \frac{{4\pi^2 \cdot r_{\text{avg}}^3}}{{G \cdot (m_1 + m_2)}}\) and solve for \(m_2\).
This will give us the mass of the planet (\(m_2\)).
To calculate the distance from the pulsar to the planet, we can use the equation for the change in arrival time of the pulses.
Let's assume that the pulsar is moving in a circular orbit around the center of mass of the pulsar-planet system, and that the planet is much less massive than the pulsar. In this case, the change in arrival time of the pulses is given by:
Δt = (2π / ω) * (G * M_pulsar / c^3) * (1 / (1 - (r / R)^2)^(3/2)),
where Δt is the change in arrival time, ω is the angular frequency of the orbit, G is the gravitational constant, M_pulsar is the mass of the pulsar, c is the speed of light, r is the distance from the pulsar to the planet, and R is the radius of the orbit (distance from the pulsar to the center of mass of the system).
We are given the mass of the pulsar (M_pulsar = 2.8 x 10^30 kg) and the value of G (6.67 x 10^-11 m^3 kg^-1 s^-2). We need to find the distance r.
In order to solve for r, we need the value of Δt. From the given data, we know that the pulses from the pulsar do not arrive at regular intervals. By measuring the time difference between consecutive pulses, we can find the average change in arrival time, Δt.
Once we have the value of Δt, we can rearrange the equation to solve for r:
r = ([(2π / ω) * (G * M_pulsar / c^3) / Δt^(2/3)]^(2/3) * R).
Please provide the value of Δt and the radius of the orbit (R) to proceed with the calculation.