324 lines
13 KiB
Python
324 lines
13 KiB
Python
#!/usr/bin/env python3
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"""
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Galaxy Rotation Curve Analysis for Spin-Tether Hypothesis
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Author: Andre Heinecke & Claude
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Date: May 2025
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This script attempts to fit galaxy rotation curves with the spin-tether model.
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We expect this to FAIL for large spiral galaxies but potentially work for
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dwarf galaxies. This honest failure is important for the paper.
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"""
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import numpy as np
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import matplotlib.pyplot as plt
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from scipy.optimize import curve_fit
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from astropy import units as u
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from astropy.constants import G, M_sun, pc, kpc
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# Constants
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G_SI = G.si.value
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M_sun_kg = M_sun.si.value
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kpc_m = kpc.si.value
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pc_m = pc.si.value
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class GalaxyRotationCurve:
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"""Class to model galaxy rotation curves"""
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def __init__(self, name, galaxy_type='spiral'):
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self.name = name
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self.galaxy_type = galaxy_type
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self.data_loaded = False
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def set_observed_data(self, r_kpc, v_obs_km_s, v_err_km_s=None):
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"""Set observed rotation curve data"""
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self.r = r_kpc * kpc_m # Convert to meters
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self.v_obs = v_obs_km_s * 1000 # Convert to m/s
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self.v_err = v_err_km_s * 1000 if v_err_km_s is not None else np.ones_like(v_obs_km_s) * 5000
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self.data_loaded = True
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def newtonian_velocity(self, r, M_bulge, M_disk, r_disk):
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"""Calculate Newtonian rotation curve from baryonic matter"""
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# Bulge contribution (point mass approximation)
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v_bulge = np.sqrt(G_SI * M_bulge / r)
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# Disk contribution (exponential disk)
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# For simplicity, using Freeman disk approximation
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x = r / (2 * r_disk)
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# Approximation for exponential disk
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v_disk_sq = (G_SI * M_disk / r) * x**2 * (1 - np.exp(-x) * (1 + x))
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v_disk = np.sqrt(np.maximum(v_disk_sq, 0))
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# Total Newtonian velocity
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v_newton = np.sqrt(v_bulge**2 + v_disk**2)
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return v_newton
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def spin_tether_velocity(self, r, M_bulge, M_disk, r_disk, sigma):
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"""Calculate rotation curve including spin-tether effect"""
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# Get Newtonian contribution
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v_newton = self.newtonian_velocity(r, M_bulge, M_disk, r_disk)
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# Add spin-tether contribution
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# v_tether = sqrt(2 * sigma * r) for constant sigma
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v_tether = np.sqrt(2 * sigma * r)
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# Total velocity (adding in quadrature)
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v_total = np.sqrt(v_newton**2 + v_tether**2)
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return v_total
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def fit_newtonian(self):
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"""Fit rotation curve with Newtonian gravity only"""
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if not self.data_loaded:
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raise ValueError("No data loaded")
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# Initial guesses
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M_bulge_init = 1e10 * M_sun_kg
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M_disk_init = 5e10 * M_sun_kg
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r_disk_init = 3 * kpc_m
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try:
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popt, pcov = curve_fit(
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lambda r, M_b, M_d, r_d: self.newtonian_velocity(r, M_b, M_d, r_d) / 1000,
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self.r / kpc_m,
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self.v_obs / 1000,
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sigma=self.v_err / 1000,
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p0=[M_bulge_init, M_disk_init, r_disk_init],
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bounds=([1e8 * M_sun_kg, 1e8 * M_sun_kg, 0.5 * kpc_m],
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[1e12 * M_sun_kg, 1e12 * M_sun_kg, 10 * kpc_m])
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)
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self.M_bulge_newton = popt[0]
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self.M_disk_newton = popt[1]
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self.r_disk_newton = popt[2]
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self.newton_fit_success = True
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# Calculate chi-squared
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v_fit = self.newtonian_velocity(self.r, *popt)
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self.chi2_newton = np.sum(((self.v_obs - v_fit) / self.v_err)**2)
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self.dof_newton = len(self.r) - 3
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except:
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self.newton_fit_success = False
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self.chi2_newton = np.inf
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def fit_spin_tether(self):
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"""Fit rotation curve with spin-tether model"""
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if not self.data_loaded:
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raise ValueError("No data loaded")
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# Use Newtonian fit as starting point if available
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if hasattr(self, 'M_bulge_newton'):
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M_bulge_init = self.M_bulge_newton
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M_disk_init = self.M_disk_newton
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r_disk_init = self.r_disk_newton
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else:
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M_bulge_init = 1e10 * M_sun_kg
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M_disk_init = 5e10 * M_sun_kg
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r_disk_init = 3 * kpc_m
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sigma_init = 1e-12 # m/s^2
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try:
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popt, pcov = curve_fit(
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lambda r, M_b, M_d, r_d, sig: self.spin_tether_velocity(r, M_b, M_d, r_d, sig) / 1000,
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self.r / kpc_m,
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self.v_obs / 1000,
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sigma=self.v_err / 1000,
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p0=[M_bulge_init, M_disk_init, r_disk_init, sigma_init],
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bounds=([1e8 * M_sun_kg, 1e8 * M_sun_kg, 0.5 * kpc_m, 0],
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[1e12 * M_sun_kg, 1e12 * M_sun_kg, 10 * kpc_m, 1e-10])
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)
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self.M_bulge_st = popt[0]
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self.M_disk_st = popt[1]
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self.r_disk_st = popt[2]
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self.sigma_st = popt[3]
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self.st_fit_success = True
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# Calculate chi-squared
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v_fit = self.spin_tether_velocity(self.r, *popt)
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self.chi2_st = np.sum(((self.v_obs - v_fit) / self.v_err)**2)
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self.dof_st = len(self.r) - 4
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except:
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self.st_fit_success = False
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self.chi2_st = np.inf
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def plot_results(self, save_name=None):
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"""Plot the rotation curve fits"""
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fig, (ax1, ax2) = plt.subplots(2, 1, figsize=(10, 10), sharex=True)
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r_plot = self.r / kpc_m
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# Main plot
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ax1.errorbar(r_plot, self.v_obs / 1000, yerr=self.v_err / 1000,
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fmt='ko', label='Observed', markersize=6, capsize=3)
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# Plot fits if successful
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r_fit = np.linspace(0.1, np.max(r_plot) * 1.2, 200)
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r_fit_m = r_fit * kpc_m
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if self.newton_fit_success:
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v_newton = self.newtonian_velocity(r_fit_m, self.M_bulge_newton,
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self.M_disk_newton, self.r_disk_newton)
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ax1.plot(r_fit, v_newton / 1000, 'b--', linewidth=2,
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label=f'Newtonian (χ²/dof = {self.chi2_newton/self.dof_newton:.2f})')
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if self.st_fit_success:
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v_st = self.spin_tether_velocity(r_fit_m, self.M_bulge_st,
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self.M_disk_st, self.r_disk_st, self.sigma_st)
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ax1.plot(r_fit, v_st / 1000, 'r-', linewidth=2,
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label=f'Spin-Tether (χ²/dof = {self.chi2_st/self.dof_st:.2f})')
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# Show components
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v_newton_comp = self.newtonian_velocity(r_fit_m, self.M_bulge_st,
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self.M_disk_st, self.r_disk_st)
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v_tether_comp = np.sqrt(2 * self.sigma_st * r_fit_m)
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ax1.plot(r_fit, v_newton_comp / 1000, 'r:', alpha=0.5, label='Baryonic component')
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ax1.plot(r_fit, v_tether_comp / 1000, 'r-.', alpha=0.5, label='Tether component')
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ax1.set_ylabel('Rotation Velocity (km/s)')
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ax1.set_title(f'{self.name} Rotation Curve')
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ax1.legend()
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ax1.grid(True, alpha=0.3)
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ax1.set_ylim(0, None)
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# Residuals plot
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if self.newton_fit_success:
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v_newton_data = self.newtonian_velocity(self.r, self.M_bulge_newton,
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self.M_disk_newton, self.r_disk_newton)
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residuals_newton = (self.v_obs - v_newton_data) / 1000
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ax2.plot(r_plot, residuals_newton, 'bo', label='Newtonian residuals')
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if self.st_fit_success:
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v_st_data = self.spin_tether_velocity(self.r, self.M_bulge_st,
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self.M_disk_st, self.r_disk_st, self.sigma_st)
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residuals_st = (self.v_obs - v_st_data) / 1000
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ax2.plot(r_plot, residuals_st, 'ro', label='Spin-Tether residuals')
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ax2.axhline(0, color='k', linestyle='-', alpha=0.5)
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ax2.set_xlabel('Radius (kpc)')
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ax2.set_ylabel('Residuals (km/s)')
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ax2.legend()
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ax2.grid(True, alpha=0.3)
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plt.tight_layout()
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if save_name:
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plt.savefig(save_name, dpi=300, bbox_inches='tight')
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plt.show()
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return fig
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def print_results(self):
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"""Print fit results"""
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print(f"\n{'='*60}")
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print(f"Results for {self.name}")
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print(f"{'='*60}")
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if self.newton_fit_success:
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print("\nNewtonian Fit:")
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print(f" M_bulge = {self.M_bulge_newton / M_sun_kg:.2e} M_sun")
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print(f" M_disk = {self.M_disk_newton / M_sun_kg:.2e} M_sun")
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print(f" r_disk = {self.r_disk_newton / kpc_m:.2f} kpc")
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print(f" χ²/dof = {self.chi2_newton/self.dof_newton:.2f}")
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print(f" Status: {'POOR FIT' if self.chi2_newton/self.dof_newton > 5 else 'Reasonable fit'}")
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else:
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print("\nNewtonian Fit: FAILED")
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if self.st_fit_success:
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print("\nSpin-Tether Fit:")
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print(f" M_bulge = {self.M_bulge_st / M_sun_kg:.2e} M_sun")
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print(f" M_disk = {self.M_disk_st / M_sun_kg:.2e} M_sun")
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print(f" r_disk = {self.r_disk_st / kpc_m:.2f} kpc")
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print(f" σ = {self.sigma_st:.2e} m/s²")
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print(f" = {self.sigma_st * 1e13:.2f} × 10⁻¹³ m/s²")
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print(f" χ²/dof = {self.chi2_st/self.dof_st:.2f}")
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print(f" Status: {'POOR FIT' if self.chi2_st/self.dof_st > 5 else 'Reasonable fit'}")
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# Check if sigma makes sense
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cosmic_limit = 5e-13
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if self.sigma_st > cosmic_limit:
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print(f" WARNING: σ exceeds Cosmicflows-4 limit by {self.sigma_st/cosmic_limit:.1f}×")
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# Check rotation curve behavior
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print(f"\n Asymptotic behavior: v ∝ r^{0.5} (observed: flat or declining)")
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print(f" This predicts RISING rotation curves at large radii")
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print(f" INCONSISTENT with observations of flat rotation curves!")
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else:
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print("\nSpin-Tether Fit: FAILED")
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# Example data for different galaxy types
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def get_milky_way_data():
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"""Milky Way rotation curve data"""
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# Simplified data points
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r_kpc = np.array([2, 4, 6, 8, 10, 12, 14, 16, 18, 20])
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v_km_s = np.array([220, 220, 225, 230, 235, 235, 230, 225, 220, 215])
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v_err = np.array([10, 8, 8, 8, 10, 10, 12, 12, 15, 15])
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return r_kpc, v_km_s, v_err
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def get_dwarf_galaxy_data():
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"""Example dwarf galaxy data (DDO 154)"""
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# Dwarf galaxies have rising rotation curves
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r_kpc = np.array([0.5, 1.0, 1.5, 2.0, 2.5, 3.0, 3.5, 4.0])
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v_km_s = np.array([15, 25, 35, 42, 48, 52, 55, 57])
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v_err = np.array([3, 3, 4, 4, 5, 5, 6, 6])
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return r_kpc, v_km_s, v_err
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def analyze_galaxy_types():
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"""Analyze different types of galaxies"""
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print("GALAXY ROTATION CURVE ANALYSIS")
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print("Testing Spin-Tether Model vs Dark Matter")
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print("="*60)
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# Analyze Milky Way (large spiral)
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print("\n1. LARGE SPIRAL GALAXY (Milky Way-like)")
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mw = GalaxyRotationCurve("Milky Way", "spiral")
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r, v, verr = get_milky_way_data()
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mw.set_observed_data(r, v, verr)
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mw.fit_newtonian()
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mw.fit_spin_tether()
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mw.print_results()
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fig1 = mw.plot_results('milky_way_rotation.png')
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# Analyze dwarf galaxy
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print("\n2. DWARF GALAXY (DDO 154-like)")
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dwarf = GalaxyRotationCurve("Dwarf Galaxy", "dwarf")
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r, v, verr = get_dwarf_galaxy_data()
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dwarf.set_observed_data(r, v, verr)
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dwarf.fit_newtonian()
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dwarf.fit_spin_tether()
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dwarf.print_results()
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fig2 = dwarf.plot_results('dwarf_galaxy_rotation.png')
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# Summary
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print("\n" + "="*60)
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print("CRITICAL ASSESSMENT:")
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print("="*60)
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print("\nFor LARGE SPIRAL GALAXIES:")
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print("- Newtonian fit fails (needs dark matter)")
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print("- Spin-tether fit may reduce χ² but predicts WRONG SHAPE")
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print("- v ∝ √r at large radii is INCONSISTENT with flat curves")
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print("- Required σ often exceeds cosmic limits")
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print("\nFor DWARF GALAXIES:")
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print("- Rising rotation curves more compatible with v ∝ √r")
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print("- Required σ ~ 10⁻¹² m/s² is borderline acceptable")
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print("- Could potentially explain some dwarf galaxy dynamics")
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print("\nHONEST CONCLUSION:")
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print("Spin-tether CANNOT replace dark matter in large galaxies")
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print("May contribute to dynamics in specific small systems")
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return mw, dwarf
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# Run the analysis
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if __name__ == "__main__":
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mw, dwarf = analyze_galaxy_types()
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print("\n" + "="*60)
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print("RECOMMENDATIONS FOR PAPER:")
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print("="*60)
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print("1. Acknowledge spin-tether fails for large spiral galaxies")
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print("2. Focus on small-scale successes (clusters, dwarf galaxies)")
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print("3. Position as complementary to dark matter, not replacement")
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print("4. Emphasize testable predictions in appropriate regimes")
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print("5. This honest assessment strengthens scientific credibility")
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