User:Mark Widmer/sandbox

Sandbox. Mark Widmer (talk) 01:17, 5 August 2021 (UTC)

Heat equation

Heat_equation

Define variables when equation is used for temperature: u=temperature, k = k_therm / (c*rho)

Define variable when equation refers to diffusion: u = density???

Draft for additions to Hill_sphere New sections:

Hill sphere and L1 Lagrange point

-- added note in Formulas section

Hill sphere of the Sun

-- added to article

Hill sphere of objects that orbit Earth

The Moon -- added to article

Artificial satellites in low-Earth orbit -- added to article

L1 Lagrange point for comparable-mass objects

Usually, derivations of the L1 point assume a planetary mass that is much less than the star's mass. This no longer applies if the orbiting objects have comparable masses. This is the case for many binary star systems. For example, in the Alpha Centauri system, the stars Alpha Centauri A and B have masses that are 1.1 and 0.9 times that of the Sun, respectively, or a mass ratio of about 0.8.

For two equal-mass objects, let R be the distance between the objects. Each object is then in a circular orbit of radius R/2 about the center of mass, which is halfway between them.

Outline:

We follow the derivation for small planet/star mass ratio given at http://www.phy6.org/stargaze/Slagrang.htm, without making the small-ratio approximations that are incorporated there.

Planet/star mass ratio $\mu =m/M$, with $0<\mu <=1$

Equate the gravitational force (which acts at a distance R) with the centripetal force (for a circle of radius Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R/(1+\mu)} ):

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{GmM}{R^2} = \frac{m v^2}{\frac{R}{1+\mu}} = \frac{m v^2 (1+\mu)}{R}}

Mult by R/m:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{GM}{R} = (1+\mu) v^2}

Substitute for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle v = \frac{2 \pi \frac{R}{1+\mu}}{T}}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{GM}{R} = \frac{(1+\mu) 4 \pi^2 R^2}{T^2 (1+\mu)^2}}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{GM}{R^3} = \frac{4 \pi^2}{T^2 (1+\mu)}}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{GM}{R^3} = \frac{4 \pi^2}{(1+\mu) T^2} }

An small-mass object at the L1 point, a distance r from object m, will have an orbit with radius Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{R}{1+\mu}-r} and the same period T:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{GM}{(R-r)^2} - \frac{Gm}{r^2} = \frac{v^2}{\frac{R}{1+\mu}-r} }

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle v = \frac{2 \pi (\frac{R}{1+\mu}-r)}{T} } ,

so

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{GM}{(R-r)^2} - \frac{Gm}{r^2} = \frac{4 \pi^2 (\frac{R}{1+\mu}-r)^2}{T^2} \frac{1}{\frac{R}{1+\mu}-r} = \frac{4 \pi^2 (\frac{R}{1+\mu}-r)}{T^2} }

Since T is the same for the planet and an object at the L1 point,

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{4 \pi^2}{T^2} = \frac{GM(1+\mu)}{R^3} = \frac{GM}{(R-r)^2 (\frac{R}{1+\mu}-r)} - \frac{Gm}{r^2 (\frac{R}{1+\mu}-r)} }

Divide through by GM

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{(1+\mu)}{R^3} = \frac{1}{(R-r)^2 (\frac{R}{1+\mu}-r)} - \frac{\mu}{r^2 (\frac{R}{1+\mu}-r)}}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{(1+\mu)}{R^3}(\frac{R}{1+\mu}-r) = \frac{1}{(R-r)^2} - \frac{\mu}{r^2}}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{R-r(1+\mu)}{R^3} = \frac{1}{(R-r)^2} - \frac{\mu}{r^2)}}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{R-r(1+\mu)}{R} \frac{(R-r)^2}{R^2} = 1 - \frac{\mu(R-r)^2}{r^2}}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (1 - \rho (1+\mu)) (1 - \rho)^2 = 1 - \mu (\frac{1}{\rho}-1)^2 }

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \rho^2 (1 - \rho (1+\mu)) (1 - \rho)^2 = \rho^2 - \mu (1-\rho)^2 }

Templates for Math Objects

https://en.citizendium.org/wiki/Help:Displaying_mathematical_formulas

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{1}{R} = \frac{1}{T} }

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{1}{r} }

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \alpha \beta \gamma \Delta \theta \pi }

text in math using mbox: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \omega \mbox{ is } 2\pi f }

spaces ignored if using mathrm: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \omega \mathrm{ is } 2\pi f }

space characters using backslash: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \omega \ \mbox{is} \ 2\pi f }

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 6.02 \times 10^{23} }

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle }

Non-math

6.02 x 10²³

small superscript: 10² 10²³

multiplier dot used in units: kg•m/s²

°C °F

Greek characters and other math formatting:

https://en.citizendium.org/wiki/CZ:How_to_edit_an_article#Character_formatting

x² ≥ 0