si tacuisses, philosophus mansisses.
Astrophotography by GrandpaJunkrat - Speckle Interferometry
Cosmology from my armchair..............................
I do it metric .............................................................. Location:
Sneek
bastion@protonmail.com The Netherlands
Bortle 7 - 9
Position Measurements on binary stars using speckle interferometry. Focus is on doubles in Cepeheus.
Software used:
NINA with Speckle Interferometry plugin
Toupsky
AstroImageJ
Platesolve 3.80
SpeckleToolBox V10.06 with orbital
GDS 1.02 Gaia Double Star Catalog Selection Tool
Aladin v12.0
Measurements taken until october-16-2025: targets solved 94.
( Mass calculation ) 𝑀 = 4𝜋²𝑟³/𝐺𝑇² T-orbital period r-radius G-(6,67259) × 10−11 m3 s−2 kg−1 x T (gravitational)
M1 + M2 = a3/P2
The original formulation of Kepler's Third Law was: P2 = a3 ,where P = orbital period of a planet (in years), and a = average distance from planet to Sun (in AU).
However, Isaac Newton modified Kepler's Third Law, so that it applied to ANY pair of objects orbiting their mutual center of mass. As modified by Newton, and applied to a binary star system: P2 ( M1 + M2 ) = a3 , where P = orbital period of stars (in years), a = average separation of stars (in AU), M1 = mass of 1st star (in solar masses), and M2 = mass of 2nd star.
Thus, the total mass of a binary system can be found from the relation M1 + M2 = a3/P2 , as long as the values of a & P are known.
Resolution telescope
R=λ /D ( θ ≈ 1.22 λ / D ) (D in mm)
The Rayleigh criterion comes from diffraction theory: two objects can be observed separately if the maximum of one object falls within the diffraction minimum of the second object. For a double star system, this looks like an 8. According to diffraction theory, for a wavelength of 550 nanometers (the highest sensitivity of the human eye): angular resolution in arc seconds = 138 / aperture in mm. A telescope with a 120mm aperture shows a double star system with a 1.15" distance from the two components as an 8.
Calculating the mass of binary or multiple star systems (often called "double stars" when referring to pairs) is a fundamental task in astrophysics. The mass of stars in a binary system can be determined primarily using Kepler’s laws of orbital mechanics and the observed properties of the system, such as the orbital period and the semi-major axis of the orbit.
Here’s a step-by-step guide on how to calculate the mass of stars in a double star system:
To calculate the masses, you typically need the following data for the binary system:
Orbital period (P): The time it takes for one star to complete a full orbit around the other.
Semi-major axis (a): The average distance between the two stars.
Inclination (i): The angle between the line of sight and the orbital plane (for systems where the orbit is not perfectly edge-on, this can be a correction factor).
Eccentricity (e): Describes how elliptical the orbit is. For simplicity, a circular orbit (e=0) is often assumed unless you know otherwise.
Radial velocity data (optional but helpful): If you can measure the Doppler shifts of the stars’ spectra, you can determine their individual velocities in the line of sight.
For a binary star system, Kepler’s Third Law relates the orbital period and semi-major axis to the total mass of the system:
P2=4π2G(M1+M2)a3P^2 = \frac{4\pi^2}{G(M_1 + M_2)} a^3P2=G(M1+M2)4π2a3
Where:
PPP is the orbital period (in years),
aaa is the semi-major axis (in astronomical units, AU),
M1M_1M1 and M2M_2M2 are the masses of the two stars (in solar masses, M⊙M_\odotM⊙),
GGG is the gravitational constant (approximately 6.674×10−11 m3 kg−1 s−26.674 \times 10^{-11} \, \text{m}^3 \, \text{kg}^{-1} \, \text{s}^{-2}6.674×10−11m3kg−1s−2).
Rearranging this equation to solve for the total mass:
M1+M2=4π2G×a3P2M_1 + M_2 = \frac{4\pi^2}{G} \times \frac{a^3}{P^2}M1+M2=G4π2×P2a3
In solar units, the equation simplifies as follows (assuming PPP is in years and aaa is in AU):
M1+M2=a3P2M_1 + M_2 = \frac{a^3}{P^2}M1+M2=P2a3
Where the sum of the masses is in solar masses (M⊙M_\odotM⊙).
If you have enough data to determine the orbital velocities (or you have a spectroscopic binary and can measure the Doppler shifts), you can also compute the mass ratio:
q=M1M2q = \frac{M_1}{M_2}q=M2M1
This can be used to derive the individual masses from the total mass.
For an edge-on orbit (i.e., i=90∘i = 90^\circi=90∘), you can use the observed velocities or radial velocities to calculate the individual masses. The mass of each star can be determined as:
M1=a1a1+a2×(M1+M2)M_1 = \frac{a_1}{a_1 + a_2} \times (M_1 + M_2)M1=a1+a2a1×(M1+M2) M2=a2a1+a2×(M1+M2)M_2 = \frac{a_2}{a_1 + a_2} \times (M_1 + M_2)M2=a1+a2a2×(M1+M2)
Where a1a_1a1 and a2a_2a2 are the distances from the center of mass to the stars.
If the orbit is approximately circular (eccentricity e≈0e \approx 0e≈0), the calculation becomes simpler. For circular orbits, you can use:
M1+M2=a3P2M_1 + M_2 = \frac{a^3}{P^2}M1+M2=P2a3
Where PPP is the period in years, aaa is the semi-major axis in AU, and M1+M2M_1 + M_2M1+M2 will be the total mass in solar masses.
Let’s go through an example:
Suppose the binary star system has a period (P) of 5 years, and the semi-major axis (a) is 2 AU.
The total mass M1+M2M_1 + M_2M1+M2 of the system is:
M1+M2=a3P2=2352=825=0.32M⊙M_1 + M_2 = \frac{a^3}{P^2} = \frac{2^3}{5^2} = \frac{8}{25} = 0.32 M_\odotM1+M2=P2a3=5223=258=0.32M⊙
So the total mass of the system is 0.32 times the mass of the Sun.
Eccentric Orbits: If the orbit is elliptical (eccentricity e>0e > 0e>0), the calculation becomes more complicated, and you might need to use additional corrections for the varying distance over the course of the orbit. In this case, you would typically use radial velocity measurements at different points in the orbit to refine your mass estimates.
Inclination: If the orbit is not edge-on (i.e., i≠90∘i \neq 90^\circi=90∘), the observed velocities need to be corrected for the inclination of the orbit.
Spectroscopic Binaries: If you have spectroscopic data (Doppler shifts), you can derive the individual velocities of the stars along the line of sight, which allows for a more detailed determination of individual star masses.
