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Equation 2 · Cosmological Natural Selection: A Theory That Named Its Own Killer

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ΔS(ϕ)=−2r ln⁡ ⁣[1−ssin⁡ϕ]\Delta_S(\phi) = -2r\,\ln\!\left[1 - s\sin\phi\right]

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Inputs and operations-2rln[1 - ssinphi]
Result or conditionDelta_S(phi)
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This equation states an equality: the expressions on both sides have the same value under the article’s assumptions. Read the equation part by part below; each part has a contextual explanation and a link to its mathematical background.

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ΔS\Delta_S

Symbol Delta_S

DeltaSa_S is part of the quantity the equation computes from the expression on the right.

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ϕ\phi

Symbol phi

the orbital phase.

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rr

Symbol r

r is one of the signed contributions combined to compute the quantity on the left.

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ss

Symbol s

s is one of the signed contributions combined to compute the quantity on the left.

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=

=

The expressions on both sides represent the same quantity under the stated assumptions.

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subtraction

subtraction

Subtract the following term or group from the preceding one. A leading minus marks a negative quantity.

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subscript

subscript

The lower label selects a particular version, component, or indexed member of the quantity. For example, x₀ and xₜ can be values at different positions.

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What the article says around this equation

The way you weigh a neutron star from a radio telescope, with no direct access to the star at all, is to time its pulses with enough precision to catch general relativity bending them. In a binary system seen close to edge-on, radio pulses from the neutron star are delayed by a few extra microseconds each orbit as they pass close to the companion star, because the companion’s gravity curves the spacetime the pulse crosses. The delay as a function of orbital phase ϕ\phi is standardly written ΔS(ϕ)=−2r ln⁡ ⁣[1−ssin⁡ϕ]\Delta_S(\phi) = -2r\,\ln\!\left[1 - s\sin\phi\right]. where the “range” parameter r is set directly by the companion’s mass and the “shape” parameter s by the orbit’s inclination. Because r and s are measured independently of the pulsar’s…
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The way you weigh a neutron star from a radio telescope, with no direct access to the star at all, is to time its pulses with enough precision to catch general relativity bending them. In a binary system seen close to edge-on, radio pulses from the neutron star are delayed by a few extra microseconds each orbit as they pass close to the companion star, because the companion’s gravity curves the spacetime the pulse crosses. The delay as a function of orbital phase ϕ\phi is standardly written ΔS(ϕ)=−2r ln⁡ ⁣[1−ssin⁡ϕ]\Delta_S(\phi) = -2r\,\ln\!\left[1 - s\sin\phi\right]. where the “range” parameter r is set directly by the companion’s mass and the “shape” parameter s by the orbit’s inclination. Because r and s are measured independently of the pulsar’s own mass, and Kepler’s laws already fix the total system mass from the orbital period and velocity, a sufficiently edge-on, sufficiently precisely timed binary hands over both masses at once — no assumptions about the neutron star’s interior required. The catch is precision: resolving a mass difference of a few hundredths of a solar mass means resolving a timing residual measured in tens of nanoseconds across years of observation, which is why the equipment built for this work is built around atomic-clock frequency standards and years of accumulated data rather than any single observation.

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