\(\boldsymbol{p}\text{(}t\text{)}\) is the position (\({\text{m}}\))
\({m_{1}}\) is the mass of the first star (\({\text{kg}}\))
\({{x_{1}}^{0}}\) is the initial x-position of the first star (\({\text{m}}\))
\({m_{2}}\) is the mass of the second star (\({\text{kg}}\))
\({{x_{2}}^{0}}\) is the initial x-position of the second star (\({\text{m}}\))
\({m_{1}}\) is the mass of the first star (\({\text{kg}}\))
\({{y_{1}}^{0}}\) is the initial y-position of the first star (\({\text{m}}\))
\({m_{2}}\) is the mass of the second star (\({\text{kg}}\))
\({{y_{2}}^{0}}\) is the initial y-position of the second star (\({\text{m}}\))
Notes
The first equation gives the general center-of-mass constraint for a system of \(n\) point masses, where \(i\) indexes each body. In the center-of-mass reference frame (A:inertialFrame), the origin is chosen so that the mass-weighted sum of positions vanishes. The remaining equations specialize to the binary case (n = 2), decomposing the vector constraint into \(x\)-direction and \(y\)-direction components of the initial positions.
\(\boldsymbol{a}\text{(}t\text{)}\) is the acceleration (\(\frac{\text{m}}{\text{s}^{2}}\))
Notes
The net force \(\boldsymbol{F}\) on a body is proportional to the acceleration \(\boldsymbol{a}\text{(}t\text{)}\) of the body, where \(m\) denotes the mass of the body as the constant of proportionality.
\({\boldsymbol{F}_{\boldsymbol{g}}}\) is the force of gravity (\({\text{N}}\))
\(G\) is the gravitational constant (\(\frac{\text{m}^{3}}{\text{kg}\text{s}^{2}}\))
\({m_{1}}\) is the mass of the first star (\({\text{kg}}\))
\({m_{2}}\) is the mass of the second star (\({\text{kg}}\))
\({r_{12}}\) is the separation distance (\({\text{m}}\))
Notes
\({\boldsymbol{F}_{\boldsymbol{g}}}\) is the magnitude of the gravitational force between two bodies; it is proportional to the product of their masses and inversely proportional to the square of the distance between them. The force acts along the relative-position vector (TM:relPosAndSep), directed from one body toward the other. This assumes Newtonian gravitation (A:newtonianGravity) and requires the separation distance to be positive (A:nonzeroSeparation).
\({r_{12}}\) is the separation distance (\({\text{m}}\))
\({x_{1}}\) is the x-position of the first star (\({\text{m}}\))
\({x_{2}}\) is the x-position of the second star (\({\text{m}}\))
\({y_{1}}\) is the y-position of the first star (\({\text{m}}\))
\({y_{2}}\) is the y-position of the second star (\({\text{m}}\))
Notes
The relative position vector is defined as the difference of the two star positions (i.e., r12 = r1 - r2). \({r_{12}}\) is the corresponding separation distance (the magnitude of the relative position vector), computed from the position coordinates \({x_{1}}\), \({y_{1}}\), \({x_{2}}\), \({y_{2}}\) . The motion is confined to a 2D plane (A:planar).