How do I identify objects in a binary star system?

Question

The question is related to two objects (1 and 2) in a binary system ( with circular orbits). The objects are a pulsar and a white dwarf (it's not specified which object is which). We are given the information that a pulsar timing study shows the orbital period to be 80 days and the orbital velocity of the second object (object 2) is 40 km/s, as well as the objects' combined mass being 3 solar masses. We then must identify which of the objects (1 or 2) is the pulsar.

My best guess is that I should use Newton's version of Kepler's third law, but I don't know. How do I solve this question?

Thank you

Answer 1

We can solve this problem by applying basic orbital mechanics principles, specifically using Kepler’s Third Law (as you mentioned) and Newton’s Law of Gravitation.

Step 1: Calculate the Semi-Major Axis (a)

Using Kepler’s Third Law and Newton's Gravity:

T^2 = (4 * π^2 * a^3) / (G * (M1 + M2))

where:

1. T is the orbital period (80days converted to 6.91e6 seconds)
2. M1/M2 is the mass of each object (the total being 3 solar masses)
3. G is the gravitational constant
4. a is the semi-major axis of rotation (or the distance between the two bodies)

We can solve the above for a, giving us a number for our semi-major axis. Now we can calculate the mass of an object using the fact the motion is circular.

Step 2: Calculate the Mass of Object 1 (M1)

For a circular orbit, the velocity of object 2 can be related to the mass of object 1:

v₂ = √(GM₁/a)

Because we know v₂ is 40km/s (which we should convert to 4e4m/s), we can solve the above for M₁

Step 3: Determine the other Mass

We know that the Mtotal = M₁ + M₂ = 3 solar masses

Rearranging the equation M₂ = 3solarmasses - M₁

When I calculate this (I attached some python code below) I get M₁≈.95 solar masses and M₂≈2.05 solar masses.

Now, to answer your original question of which object is the pulsar: Based on the data we have, and without additional specific observations, we can make an educated guess. Pulsars are generally more massive than white dwarfs, and since object 2 has more than twice the mass of object 1, it’s highly likely that object 2 is the pulsar.

However, there is some room for uncertainty. The masses of pulsars and white dwarfs can sometimes overlap depending on the specific systems, and without more detailed information—such as pulse timing data or spectral analysis—this conclusion isn’t absolutely certain. But, given typical mass ranges, object 2 is the most probable candidate for the pulsar.

import math

# Given data

T = 6.91e6 # orbital period in seconds

v2 = 4.0e4 # velocity of object 2 in m/s

combined_mass = 3 * 1.989e30 # combined mass in kg (3 solar masses)

G = 6.674e-11 # gravitational constant in m^3 kg^-1 s^-2

# Step 1: Calculate the semi-major axis (a) using Kepler's Third Law

a = ((T**2 * G * combined_mass) / (4 * math.pi**2))**(1/3)

# Step 2: Use the orbital velocity formula to calculate M1

M1 = (v2**2 * a) / G

M1sol=M1/1.989e30 #getting the value in terms of solar mass

# Step 3: Calculate M2

M2 = combined_mass - M1

M2sol=M2/1.989e30 #getting the value in terms of solar mass

# Print the results

print(f"Mass of object 1 (M1): {M1:.2e} kg or {M1sol:.2} solar masses") # Print M1 in scientific notation

print(f"Mass of object 2 (M2): {M2:.2e} kg or {M2sol:.3} solar masses") # Print M2 in scientific notation

Answer 2

Identifying objects in a binary star system involves observing and analyzing various astronomical properties. Here's a step-by-step guide:

*Observational Methods:*

1. Visual Observation: Use telescopes or binoculars to observe the system. Binary stars may appear as two distinct points of light or a single, elongated star.

2. Spectroscopy: Analyze light spectra to detect Doppler shifts, indicating orbital motion.

3. Astrometry: Measure positional changes and proper motions to determine orbital parameters.

4. Interferometry: Combine light from multiple telescopes to resolve the binary components.

*Characteristics to Identify:*

1. *Separation*: Measure the angular distance between the two stars.

2. *Magnitude Difference*: Compare brightness of the two stars.

3. *Color Difference*: Observe differences in color, indicating distinct spectral types.

4. *Orbital Period*: Calculate time between successive conjunctions or oppositions.

5. *Eccentricity*: Determine orbital shape (circular, elliptical).

6. *Mass Ratio*: Calculate mass ratio of the two stars.

*Types of Binary Systems:*

1. *Visual Binaries*: Resolvable with telescopes.

2. *Spectroscopic Binaries*: Identified through spectroscopy.

3. *Eclipsing Binaries*: Stars periodically eclipse each other.

4. *Astrometric Binaries*: Detected through positional measurements.

5. *Contact Binaries*: Stars in close contact, sharing material.

*Tools and Resources:*

1. *Telescopes*: Professional or amateur telescopes for observation.

2. *Spectrographs*: Instruments for spectroscopic analysis.

3. *Astrometric software*: Programs for data analysis (e.g., PyEphem, Astropy).

4. *Binary star catalogs*: Online databases (e.g., Washington Double Star Catalog).

5. *Astronomical journals*: Research papers and articles.

*Challenges:*

1. *Resolution limits*: Telescopic resolution may not resolve close binaries.

2. *Observational biases*: Selection biases in data collection.

3. *Data analysis complexities*: Interpreting spectroscopic or astrometric data.

To overcome these challenges, collaborate with experts, utilize advanced equipment, and employ sophisticated data analysis techniques.

You're on the right track with Newton's version of Kepler's third law. Here's a step-by-step solution:

*Given:*

- Orbital period (P) = 80 days

- Orbital velocity of object 2 (v2) = 40 km/s

- Combined mass (M1 + M2) = 3 solar masses (M)

*Newton's version of Kepler's third law:*

(P^2) / (a^3) = (4π^2) / (G * (M1 + M2))

where:

- P = orbital period

- a = semi-major axis

- G = gravitational constant (6.67408e-11 N*m^2/kg^2)

*Orbital velocity:*

v2 = √(G * M1 / r)

where:

- v2 = orbital velocity of object 2

- M1 = mass of object 1

- r = distance between objects (approximately equal to semi-major axis, a)

*Solving for M1 and M2:*

1. Convert orbital period to seconds: P = 80 days * 86400 s/day = 6912000 s

2. Calculate semi-major axis (a) using Kepler's third law:

a^3 = (P^2 * G * (M1 + M2)) / (4π^2)

a^3 = ((6912000 s)^2 * 6.67408e-11 N*m^2/kg^2 * 3 * 1.989e30 kg) / (4π^2)

a ≈ 1.93e11 m or 0.129 AU

1. Calculate mass of object 1 (M1) using orbital velocity:

M1 = (v2^2 * r) / G

M1 ≈ (40,000 m/s)^2 * 1.93e11 m) / (6.67408e-11 N*m^2/kg^2)

M1 ≈ 1.52 * 1.989e30 kg ≈ 1.52 solar masses

*Conclusion:*

Since M1 ≈ 1.52 solar masses, which is closer to the mass of a neutron star (pulsar), and M2 ≈ 1.48 solar masses, which is closer to the mass of a white dwarf:

*Object 1 is the pulsar.*

Object 2, with the measured orbital velocity, is likely the white dwarf.

Great job applying Newton's version of Kepler's third law!