What this Simulation Does
This simulator models how light from a distant supernova is detected on Earth, explicitly showing:
- The Sun-centered (heliocentric) geometry
- The Earth’s orbital position around the Sun
- The observer’s latitude and longitude on Earth
- A photon traveling from the supernova to Earth (visual animation)
- How redshift, distance, Heff, φ, and CST-adjusted φ are computed
- Light-curve timing markers and cosmological time dilation: Δtobs = (1+z)·Δtrest
- Distance-mode switch: DL vs DC vs DA
- Dark energy model: turn dark energy on/off and set equation-of-state w
Equations Used (Paper-style + CST + Time Dilation + Dark Energy)
Distance modulus: μ = 5·log10(DL/Mpc) + 25 ⇒ DL = 10(μ − 25)/5 Mpc
Distance mode switch: DL = (1+z)·DC | DC = DL/(1+z) | DA = DL/(1+z)2
Heff(z) = (c · z) / Dmode(z)
φ(z) = arctan(Heff / H0,ref)
Toy observer redshift: zobs = z + (v⊕/c) · cos(lat) · cos(lon)
Cosmological time dilation: Δtobs = (1+z)·Δtrest
φCST = φ · CST
wCDM expansion history:
E(z)=H(z)/H0 = √(Ωm(1+z)3 + Ωk(1+z)2 + ΩDE(1+z)3(1+w))
DL=(1+z)·(c/H0)·∫0→z dz′/E(z′)
Lookback: tL=(1/H0)·∫0→z dz′/((1+z′)E(z′))
What the Distance Mode Toggle Proves (the (1+z) test):
Switching from DL to DC=DL/(1+z) increases Heff by (1+z):
H_eff,DC = (1+z)·H_eff,DL
So φ(z) lifts upward and the downward bend past z≈0.1 can flatten a lot if the bend was mainly the (1+z) wrapper. Dark energy is tested via the remaining shape from E(z) (the integral).
Switching from DL to DC=DL/(1+z) increases Heff by (1+z):
H_eff,DC = (1+z)·H_eff,DL
So φ(z) lifts upward and the downward bend past z≈0.1 can flatten a lot if the bend was mainly the (1+z) wrapper. Dark energy is tested via the remaining shape from E(z) (the integral).
Observer Location (Earth)
CST
Supernova Source
SN 1987A is nearby (Large Magellanic Cloud), so cosmology distances are not appropriate; this preset uses a local distance calibration.
Light-Curve Template (for Markers)
s scales rest-frame timing: t → s·t. (For Ia, “stretch” is a common way to represent slower/faster light curves.)
Distance Module
Dark Energy (Add to Simulation)
With dark energy ON and flatness ON: ΩDE = 1 − Ωm.
With dark energy OFF: ΩDE=0 (matter/curvature-only expansion).
If flatness is OFF, Ωk is computed so Ωm+ΩDE+Ωk=1.
With dark energy OFF: ΩDE=0 (matter/curvature-only expansion).
If flatness is OFF, Ωk is computed so Ωm+ΩDE+Ωk=1.
Live Clocks
Important: For cosmological distances, “arrival date” in calendar form can exceed JavaScript date limits.
This simulator therefore reports travel times in years while the animation is a visualization.
Dark Energy in this simulator:
You are not just “turning a label on.” You are changing E(z), which changes the distance integral, which changes DL(z), which then changes Heff(z) and φ(z). This is the clean way to represent “dark energy added.”
You are not just “turning a label on.” You are changing E(z), which changes the distance integral, which changes DL(z), which then changes Heff(z) and φ(z). This is the clean way to represent “dark energy added.”